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MULTIVALUED FIELDS in Condensed Matter, Electromagnetism, and Gravitation
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MULTIVALUED FIELDS

in Condensed Matter, Electromagnetism,

and Gravitation

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Multivalued Fields

in Condensed Matter, Electromagnetism,

and Gravitation

Hagen Kleinert

Professor of PhysicsFreie Universitat Berlin

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The modern physical developments

have required a mathematics

that continually shifts its foundations

P.A.M. Dirac, 1931

Preface

The theory to be presented in this book has four roots. The first lies in the seminalpaper by Dirac [1] in 1931, in which he pointed out that in spite of the zero divergenceof the magnetic field, Maxwell’s equations can accommodate magnetic monopoles. Itwould merely be necessary to import the field from far distance through an infinitelythin magnetic flux tube, the famous Dirac string , to a point x, if it would be to makethe string invisible for any charged particle. Then the magnetic field lines wouldemerge radially from the point x in the same way as electric field lines emergefrom an electric point charge, and the endpoint x of the string would appear as amagnetic monopole. Dirac discovered that the invisibility of the string was madepossible by quantum mechanics if all electric charges would integer multiples of2πhc/g, where g is the total magnetic flux in through string, This is the famousDirac quantization condition. Under this condition, the shape of the string wouldbe completely irrelevant, and would becomes a mathematical artifact. For thisstunning observation, Pauli gave Dirac the nickname Monopoleon.

Experimentally, no magnetic monopoles have been found. The Dirac theoryhas nevertheless resurfaced in the last 35 years in the process of explaining thephenomenon of quark confinement. The presently accepted viewpoint is that thevacuum state contains a condensate of color-magnetic monopoles, just like a su-perconductor contains a condensate of electric charges, the famous Cooper pairsof electrons. Since London’s work on superconductivity it is known that a super-conductor would confine magnetic charges, if they exist. This happens due to theMeissner effect , which tries to expel magnetic flux lines from a superconductor. Asa consequence, flux lines emerging from a magnetic monopole are compressed intothin flux tubes. Their energy is proportional to their length, and this implies thatopposite magnetic charges are held together by a flux tube forever. The condensateof magnetic monopoles would do the same thing for electrically charged particles.

Models illustrating this confinement mechanism were developed by Nambu [2],Mandelstam [3], ’t Hooft [4], and Polyakov [5], and on a lattice by Wilson [6].

The second root of the theory in this book comes from a completely differentdirection — the theory of plastic deformations which is the basis of our understand-ing of work hardening of metals and material fatigue. This theory evolved since thediscovery of dislocations in crystals in 1934 [7]. It has recently led to a statistical

vii

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viii

mechanics of line-like defects in crystals, explaining phase transitions such as themelting process [8].

The third root lies the theory of the superfluid phase transitions developed inabout the same period. Here the crucial papers were written by Berezinski [9] andby Kosterlitz and Thouless [10]. They showed that the phase transition in a filmof superfluid helium can be understood by the statistical mechanics of vortices ofsuperflow. Their description attaches to each point a phase angle of the condensatewave function which lies in the interval (0, 2π). When encircling a vortex, this anglemust jump somewhere by 2π. A jumping line connects a vortex with an antivortexand forms a kind of “Dirac string”, whose precise shape is irrelevant. If theseideas are carried over to bulk superfluid helium in three dimensions, as done in thetextbook [11], one is led to the statistical mechanics of vortex loops. These interactwith the same long-range forces as electric current loops.

The fourth root lies in the work of Bilby, Bullough, Smith [12], Kondo [13], andKroner [14], that line like defects can also be described by geometric means. Theyform a special version of a Riemann-Cartan space. The theory of such spaces was setup in 1922 by Cartan who extended the curved Riemannian spacetime by anothergeometric property, the torsion [15]. His work instigated Einstein develop a theoryof gravitation in a Riemann-Cartan space with teleparallelism [16].

Twenty years later, Schrodinger attempted to relate torsion to electromagnetism[17]. He noticed that if torsion was present in the universe, this would make photonsmassive and limit the range of magnetic fields emerging from planets and stars.From the ranges observed at that time he deduced upper bounds on the photonmass [18] which were, even then, extremely small. Further twenty years passedbefore Utiyama, Sciama, and Kibble [19, 20, 21] clarified the intimate relationshipbetween torsion and the spin density of the gravitational field. A detailed review ofthe theory was given in 1976 by Hehl et al. [22], and later in the textbook [8]. Therecent status of the subject is summarized by Hammond [23].

I ran into the subject in the eighties after having developed a disorder field theoryof line-like objects summarized in the textbook [11]. My first applications dealtwith vortex lines in superfluids and superconductors, where the disorder formulationhelped me to solve the long-standing problem of determining the order of the phasetransition in a superconductor [24].

After this I turned to the application to line-like defects in crystals. The originaldescription of such defects was based on functions which are discontinuous on sur-faces whose boundaries are the defect lines. The shape of these surfaces is arbitraryas long as the boundaries are fixed. I realized that the deformations of the surfacescan be formulated as a field theory of multivalued functions possessing a particularkind of gauge invariance which I named defect gauge invariance.

By a so-called duality transformation it was possible to reformulate the theoryof defects and their interactions as a gauge theory. This brought about anotherfreedom in the description which I named stress gauge invariance.

H. Kleinert, MULTIVALUED FIELDS

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ix

The stress gauge invariance turned out to be intimately related to the invari-ance of the geometric formulation of Bilby and others under Einstein’s coordinateinvariance and a local generalization of Lorentz invariance.

The relation between the dual and the original description of defects in terms ofjump surfaces is completely analogous to the well-known relation between Maxwell’stheory of magnetism formulated in terms of a gauge field, the vector potential, andan alternative formulation in which the magnetic field is the gradient of a multivaluedscalar field.

It was a fortunate coincidence that I was also searching, as many other peopledid at that time, for a simple field-theoretic formulation of the phenomenon of quarkconfinement. In that context, I took advantage of the mathematical analogy of theabove defect structures with the situation in Dirac’s theory of magnetic monopoles.Dirac used a vector potential with a jump surface to construct an infinitely thinmagnetic flux tube which produces a magnetic point source at its end. In this way,the world line of a monopole in spacetime could become a defect line in Maxwellfields. It was known that a condensate of monopoles would create an environmentin which the electric flux lines between charges would be pressed into thin tubes.This would naturally create a potential proportional to the distance, thus confiningelectric charges. The analogy with the theory of defect and vortex lines led to adisorder field theory of monopoles and a simple theory of quark confinement [25].

When extending the statistical mechanics of vortex lines to defect lines in thesecond volume of the textbook [8], I used the dual description of defect lines, whichwas a linear approximation to a geometric description in Riemann-Cartan space.This suggested to me that it would be instructive to reverse the development inthe theory of defects and reformulate the theory of gravity, which is convention-ally treated as a geometric theory, in an alternative way with the help of jumpingsurfaces of translation and rotation fields. In the theory of plasticity, such singulartransformations are used to carry an ideal crystal into crystals with translational androtational defects. Their geometric analogs carry a flat spacetime into a spacetimewith curvature and torsion. The mathematical basis expressing the new geometryturned out to be multivalued tetrad fields eaµ(x).

In the traditional literature on gravity with spinning particles, a special roleis played by single-valued vierbein fields hαµ(x) which define local nonholonomiccoordinate differentials dxα reached from the physical coordinate differentials dxµ

by transformations dxα = hαµ(x)dxµ. Only infinitesimal vectors dxα are defined, and

the transformation cannot be extended over finite domains, since it is nonholonomic.Such an extension is unnecessary, however, since the infinitesimal nonholonomiccoordinates dxα are completely sufficient to specify the behavior of spinning particlesin a Riemannian spacetime.

The theory in terms of multivalued tetrad fields to be presented here goes animportant step further, leading to a drastic simplification of the description of thegeometry. This becomes possible by an efficient use of a set of completely newnonholonomic coordinates dxa which are more nonholonomic than the traditionaldxα. To emphasize this one might call them hyper-nonholonomic coordinates. They

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x

are related to dxα by a multivalued Lorentz transformation dxa = Λaα(x)dx

α, andto the physical dxµ by a the above-mentioned multivalued tetrad fields as dxa =eaµ(x)dx

µ ≡ Λaα(x)h

αµ(x)dx

µ. The gradients ∂µeaν(x) determine directly the full

affine connection, their antisymmetric combination ∂µeaν(x) − ∂νe

aµ(x) yields the

torsion. This is in contrast to the curl of the usual vierbein fields hαµ(x) whichdetermines the object of anholonomy, a quantity existing also in a purely Riemannianspacetime, i.e., in the absence of torsion.

One of the purposes of this book is to make students and colleagues working inelectromagnetism and gravitational physics appreciate the many advantages broughtabout by the use of the multivalued fields, in particular tetrad fields eaµ(x). Apartfrom a simple intuitive reformulation of Riemann-Cartan geometry, it suggests anew principle in physics [26], which I have named nonholonomic mapping principle,or multivalued mapping principle, to be explained in detail in this book. Multi-valued coordinate transformations enable us to transform physical laws from flatspacetime to spacetimes with curvature and torsion. It is therefore natural to pos-tulate that the images of these laws describe correctly the physics in such generalaffine spacetimes. As a result I am able to make predictions which cannot be madewith Einstein’s construction method based merely on covariance under ordinarycoordinate transformations since those were unable to connect different geometries.

It should be emphasized that it is not the purpose of this book to proposerecreating all geometric structures studied in gravitational theories with the helpof multivalued coordinate transformations. In fact, I shall restrict much of thediscussion to almost flat auxiliary spacetimes. This will be enough to derive thegeneral form of the physical laws the presence of curvature and torsion. At theend I always return to the usual geometric description. The intermediate auxiliaryspacetime with defects will be referred to as world crystal .

The reader will be pleased to see in Subsection 4.5 that the standard minimalcoupling of electromagnetism is a simple consequence of the multivalued mappingprinciple. The similar minimal coupling to gravity will be derived from this principlein Chapter 17.

At the end I shall argue that torsion fields in gravity, if they exists, would leadquite a hidden life, unless they are of a special form. They would not be observablefor many generations to come since they could exist only in an extremely smallneighborhood of material point particles, limited to distances of the order of thePlanck length 10−33 cm, which no presently conceivable experiment can probe.

The detailed development in this book of gravity with torsion is thus at presenta purely theoretical endeavor. Its main merit lies in exposing the the multivaluedapproach to Riemann-Cartan geometry, which has turned out to be quite useful inteaching the the theory of gravity to beginning students. The definitions of paralleldisplacements and covariant derivatives appear naturally as nonholonomic imagesof parallel displacements and ordinary derivatives in fat space. So do the rules ofminimal coupling.

H. Kleinert, MULTIVALUED FIELDS

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Notes and References xi

Valuable insights are gained by realizing the universality of the multivalued defectdescription in various fields of physics. The predictions based on the nonholonomicmapping principle remain to be tested experimentally.

Special thanks go to my wife Dr. Annemarie Kleinert for her sacrifices, inex-haustible patience, and constant encouragement.

H. KleinertBerlin, September 2006

Notes and References

[1] P.A.M. Dirac, Quantized Singularities in the Electromagnetic Field , Proceed-ings of the Royal Society, A 133, 60 (1931). Can be read on the wwwunder the URL kl/files, where kl is short for the URL www.physik.

fu-berlin.de/~kleinert.

[2] Y. Nambu, Phys. Rev. D 10, 4262 (1974).

[3] S. Mandelstam, Phys. Rep. C 23, 245 (1976); Phys. Rev. D 19, 2391 (1979).

[4] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974); and in High Energy Physics , ed. byA. Zichichi, Editrice Compositori, Bologna, 1976.

[5] A.M. Polyakov, JEPT Lett. 20, 894 (1974).

[6] K. Wilson, Confinement of quarks, Phys. Rev. D 10, 2445 (1974).

[7] E. Orowan, Z. Phys. 89, 605, 634 (1934); M. Polany, Z. Phys. 89, 660 (1934);G.J. Taylor, Proc. Roy. Soc. A 145, 362 (1934);

[8] H. Kleinert, Gauge Fields in Condensed Matter , Vol. II, Stresses and Defects ,World Scientific, Singapore, 1989, pp. 743-1456 (kl/re.html#b2).

[9] V.L. Berezinski, Zh. Eksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493(1971).

[10] J.M. Kosterlitz and D.J. Thouless, J. Phys. C 5, L124 (1972); J. Phys. C 6,1181 (1973); J.M. Kosterlitz, J. Phys. C 7, 1046 (1974);

[11] H. Kleinert, Gauge Fields in Condensed Matter , Vol. I, Superflow and VortexLines , World Scientific, Singapore, 1989, pp. 1–742 (kl/re.html#b1).

[12] B.A. Bilby, R. Bullough, and E. Smith, Continuous distributions of disloca-tions: a new application of the methods of non-Riemannian geometry , Proc.Roy. Soc. London, A 231, 263-273 (1955).

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xii

[13] K. Kondo, in Proceedings of the II Japan National Congress on Applied Me-chanics , Tokyo, 1952, publ. in RAAG Memoirs of the Unified Study of BasicProblems in Engeneering and Science by Means of Geometry , Vol. 3, 148, ed.K. Kondo, Gakujutsu Bunken Fukyu-Kai, 1962.

[14] E. Kroner, in The Physics of Defects , eds. R. Balian et al., North-Holland,Amsterdam, 1981, p. 264.

[15] E. Cartan, Comt. Rend. Acad. Science 174, 593 (1922); Ann. Ec. Norm. Sup.40, 325 (1922); 42, 17 (1922).

[16] E. Cartan and A. Einstein, Letters of Absolute Parallelism, Princeton Univer-sity Press, Princeton, NJ.

[17] E. Schrodinger, Proc. R. Ir. Acad. A 49, 43 (1943).

[18] E. Schrodinger, Proc. R. Ir. Acad. A 49, 135 (1943); 52, 1 (1948); 54, 79(1951).

[19] R. Utiyama, Phys. Rev. 101, 1597 (1956).

[20] D.W. Sciama, Rev. Mod. Phys. 36, 463 (1964).

[21] T.W.B. Kibble, J. Math. Phys. 2, 212 (1961).

[22] F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M. Nester, Rev. Mod. Phys.48, 393 (1976).

[23] R.T. Hammond, Rep. Prog. Phys. 65, 599 (2002).

[24] H. Kleinert, Disorder Version of the Abelian Higgs Model and the Order ofthe Superconductive Phase Transition, Lett. Nuovo Cimento 35, 405 (1982)(kl/97).

[25] H. Kleinert, The Extra Gauge Symmetry of String Deformations in Electro-magnetism with Charges and Dirac Monopoles, Int. J. Mod. Phys. A7, 4693(1992) (kl/203);Double-Gauge Invariance and Local Quantum Field Theory of Charges andDirac Magnetic Monopoles, Phys. Lett. B246, 127 (1990)(kl/205);Abelian Double-Gauge Invariant Continuous Quantum Field Theory of Elec-tric Charge Confinement , Phys. Lett. B293, 168 (1992)(kl/211).

[26] H. Kleinert, Quantum Equivalence Principle for Path Integrals in Spaces withCurvature and Torsion, in Proceedings of the XXV International Sympo-sium Ahrenshoop on Theory of Elementary Particles in Gosen/Germany 1991,ed. by H.J. Kaiser (quant-ph/9511020);

H. Kleinert, MULTIVALUED FIELDS

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Notes and References xiii

Quantum Equivalence Principle, Lectures presented at the 1996 CargeseSummer School on Functional Integration: Basics an Applications (quant-ph/9612040).

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Contents

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Basics 1

1.1 Galilean Invariance of Newtonian Mechanics . . . . . . . . . . . . . 11.1.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Galilei Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Galilei Group . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Lorentz Invariance of Maxwell Equations . . . . . . . . . . . . . . . 31.2.1 Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Infinitesimal Lorentz Transformations . . . . . . . . . . . . . . . . . 61.3.1 Group Multiplication and Lee Algebra . . . . . . . . . . . . 9

1.4 Vectors, Tensors, Scalars . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 Discrete Lorentz Transformations . . . . . . . . . . . . . . . 131.4.2 Poincare group . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Differential Operators for Lorentz Transformations . . . . . . . . . . 141.5.1 Vector and Tensor Operators . . . . . . . . . . . . . . . . . 16

1.6 Finite Operator Lorentz Transformations . . . . . . . . . . . . . . . 161.6.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6.2 Lorentz Boosts . . . . . . . . . . . . . . . . . . . . . . . . . 171.6.3 Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 Relativistic Point Mechanics . . . . . . . . . . . . . . . . . . . . . . 191.8 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.9 Relativistic Particles with Electromagnetic Interactions . . . . . . . 231.10 Dirac Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.11 Spacetime-Dependent Lorentz Transformations . . . . . . . . . . . . 31

1.11.1 Angular velocities . . . . . . . . . . . . . . . . . . . . . . . 311.12 Energy-Momentum Tensors . . . . . . . . . . . . . . . . . . . . . . . 33

1.12.1 Point Particles . . . . . . . . . . . . . . . . . . . . . . . . . 331.12.2 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . 35

1.13 Angular Momentum and Spin . . . . . . . . . . . . . . . . . . . . . 371.14 Energy-Momentum Tensor of Perfect Fluid . . . . . . . . . . . . . . 42Appendix 1A Tensor Identities . . . . . . . . . . . . . . . . . . . . . . . . 43

1A.1 Product Formulas . . . . . . . . . . . . . . . . . . . . . . . 441A.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 45

xiv

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Notes and References xv

1A.3 Expansion of Determinants . . . . . . . . . . . . . . . . . . 46Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2 Action Approach 48

2.1 General Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . 492.2 Single Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . 502.3 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.1 Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.2 Lorenz Invariance . . . . . . . . . . . . . . . . . . . . . . . . 532.3.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.4 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 552.3.5 Schrodinger Quantum Mechanics as Nonrelativistic Limit . 552.3.6 Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3.7 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . 572.3.8 Conserved Current . . . . . . . . . . . . . . . . . . . . . . . 57

2.4 Maxwell’s Equation from Extremum of Field Action . . . . . . . . . 592.4.1 Electromagnetic Field Action . . . . . . . . . . . . . . . . . 592.4.2 Alternative Action for Electromagnetic Field . . . . . . . . 612.4.3 Hamiltonian of Electromagnetic Fields . . . . . . . . . . . . 612.4.4 Gauge Invariance of Maxwell’s Theory . . . . . . . . . . . . 63

2.5 Maxwell-Lorentz Action for Charged Point Particles . . . . . . . . . 652.6 Scalar Field with Electromagnetic Interaction . . . . . . . . . . . . . 662.7 Dirac Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3 Continuous Symmetries and Conservation Laws. Noether’s Theo-

rem 70

3.1 Continuous Symmetries and Conservation Law . . . . . . . . . . . . 703.1.1 Alternative Derivation . . . . . . . . . . . . . . . . . . . . . 72

3.2 Time Translation Invariance and Energy Conservation . . . . . . . . 743.3 Momentum and Angular Momentum . . . . . . . . . . . . . . . . . . 75

3.3.1 Translational Invariance in Space . . . . . . . . . . . . . . . 753.3.2 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . 763.3.3 Center-of-Mass Theorem . . . . . . . . . . . . . . . . . . . . 773.3.4 Conservation Laws Resulting from Lorentz Invariance . . . . 79

3.4 Generating the Symmetries . . . . . . . . . . . . . . . . . . . . . . . 813.5 Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5.1 Continuous Symmetry and Conserved Currents . . . . . . . 823.5.2 Alternative Derivation . . . . . . . . . . . . . . . . . . . . . 843.5.3 Local Symmetries . . . . . . . . . . . . . . . . . . . . . . . 84

3.6 Canonical Energy-Momentum Tensor . . . . . . . . . . . . . . . . . 873.6.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 883.6.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.7 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

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xvi

3.8 Four-Dimensional Angular Momentum . . . . . . . . . . . . . . . . . 923.9 Spin Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.9.1 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . 943.9.2 Dirac Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.10 Symmetric Energy-Momentum Tensor . . . . . . . . . . . . . . . . . 983.11 Internal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.11.1 U(1)-Symmetry and Charge Conservation . . . . . . . . . . 1003.11.2 SU(N)-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 1013.11.3 Broken Internal Symmetries . . . . . . . . . . . . . . . . . . 101

3.12 Generating the Symmetry Transformations on Quantum Fields . . . 1023.13 Energy-Momentum Tensor of Relativistic Massive Point Particle . . 1033.14 Energy-Momentum Tensor of Massive Charged Particle in Electro-

magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4 Multivalued Gauge Transformations in Magnetostatics 109

4.1 Vector Potential of Current Distribution . . . . . . . . . . . . . . . . 1094.2 Multivalued Gradient Representation of Magnetic Field . . . . . . . 1104.3 Generating Magnetic Fields by Multivalued Gauge Transformations 1164.4 Magnetic Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.5 Minimal Magnetic Coupling of Particles from Multivalued Gauge

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.6 Equivalence of Multivalued Scalar and Single-Valued Vector Poten-

tial Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.7 Multivalued Field Theory of Magnetic Monopoles and Electric Cur-

rents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 Multivalued Fields in Superfluids and Superconductors 129

5.1 Superfluid Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.1.1 Configuration Entropy . . . . . . . . . . . . . . . . . . . . . 1315.1.2 Origin of Massless Excitations . . . . . . . . . . . . . . . . . 1335.1.3 Vortex Density . . . . . . . . . . . . . . . . . . . . . . . . . 1355.1.4 Partition Function . . . . . . . . . . . . . . . . . . . . . . . 1365.1.5 Continuum Derivation of Interaction Energy . . . . . . . . . 1415.1.6 Physical Jumping Surfaces . . . . . . . . . . . . . . . . . . . 1425.1.7 Canonical Representation of Superfluid . . . . . . . . . . . . 1445.1.8 Yukawa Loop Gas . . . . . . . . . . . . . . . . . . . . . . . 1475.1.9 Gauge Field of Superflow . . . . . . . . . . . . . . . . . . . 1485.1.10 Disorder Field Theory . . . . . . . . . . . . . . . . . . . . . 150

5.2 Phase Transition in Superconductor . . . . . . . . . . . . . . . . . . 1535.2.1 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . 1545.2.2 Disorder Theory of Superconductor . . . . . . . . . . . . . . 157

5.3 Order versus Disorder Parameter . . . . . . . . . . . . . . . . . . . . 159

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Notes and References xvii

5.3.1 Superfluid 4He . . . . . . . . . . . . . . . . . . . . . . . . . 1605.3.2 Superconductor . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.4 Order of Superconducting Phase Transition and Tricritical Point . . 1715.4.1 Disorder Theory . . . . . . . . . . . . . . . . . . . . . . . . 177

5.5 Vortex Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6 Dynamics of Superfluids 185

6.1 Hydrodynamic Description . . . . . . . . . . . . . . . . . . . . . . . 1856.2 Velocity of Second Sound . . . . . . . . . . . . . . . . . . . . . . . . 1916.3 Vortex Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . 1926.4 Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.5 Eckart Theory of Ideal Quantum Fluids . . . . . . . . . . . . . . . . 1956.6 Rotating Superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 195Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

7 Dynamics of Charged Superfluid and Superconductor 199

7.1 Hydrodynamic Description of Charged Superfluid . . . . . . . . . . . 2007.1.1 London Theory of Charged Superfluid . . . . . . . . . . . . 201

7.2 London Equations of Charged Superfluid with Vortices . . . . . . . . 2037.3 Hydrodynamic Description of Superconductor . . . . . . . . . . . . . 204Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

8 Relativistic Magnetic Monopoles and Electric Charge Confine-

ment 208

8.1 Monopole Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 2088.2 Charge Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 2128.3 Electric and Magnetic Current-Current Interactions . . . . . . . . . 2138.4 Dual Gauge Field Representation . . . . . . . . . . . . . . . . . . . 2158.5 Monopole Gauge Fixing . . . . . . . . . . . . . . . . . . . . . . . . . 2178.6 Quantum Field Theory of Spinless Electric Charges . . . . . . . . . 2188.7 Theory of Magnetic Charge Confinement . . . . . . . . . . . . . . . 2198.8 Second Quantization of the Monopole Field . . . . . . . . . . . . . . 2218.9 Quantum Field Theory of Electric Charge Confinement . . . . . . . 223Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

9 Multivalued Mapping from Ideal Crystals to Crystals with Line-

Like Defects 231

9.1 Defects in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2319.2 Dislocation Lines and Burgers Vector . . . . . . . . . . . . . . . . . 2349.3 Disclination Lines and Frank Vector . . . . . . . . . . . . . . . . . . 2389.4 Interdependence of Dislocation and Disclinations . . . . . . . . . . . 2409.5 Defect Lines with Infinitesimal Discontinuities in Continuous Media 2429.6 Multivaluedness of Displacement Field . . . . . . . . . . . . . . . . . 243

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9.7 Smoothness Properties of Displacement Field and Weingarten’s The-orem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

9.8 Integrability Properties of Displacement Field . . . . . . . . . . . . . 2479.9 Dislocation and Disclination Densities . . . . . . . . . . . . . . . . . 2499.10 Mnemonic Procedure for Constructing Defect Densities . . . . . . . 2529.11 Defect Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . 2559.12 Branching Defect Lines . . . . . . . . . . . . . . . . . . . . . . . . . 2569.13 Defect Density and Incompatibility . . . . . . . . . . . . . . . . . . 257Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10 Defect Melting 264

10.1 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26410.2 Elastic and Plastic Energies . . . . . . . . . . . . . . . . . . . . . . . 265

11 Relativistic Mechanics in Curvilinear Coordinates 271

11.1 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 27111.2 Free Particle in General Coordinates Frame . . . . . . . . . . . . . . 27211.3 Minkowski Geometry formulated in General Coordinates . . . . . . . 275

11.3.1 Local Basis tetrads . . . . . . . . . . . . . . . . . . . . . . . 27611.3.2 Vector- and Tensor Fields, and Lorentz Invariance . . . . . 27811.3.3 Affine Connections and Covariant Derivatives . . . . . . . . 282

11.4 Covariant Time Derivative and Acceleration . . . . . . . . . . . . . . 28511.5 Torsion tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28511.6 Curvature Tensor as Covariant Curl of Affine Connection . . . . . . 28711.7 Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . 291Appendix 11ACurvilinear Versions of Levi-Civita Tensor . . . . . . . . . . 293Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

12 Torsion and Curvature from Defects 296

12.1 Multivalued Infinitesimal Coordinate Transformations . . . . . . . . 29712.2 Examples for Nonholonomic Coordinate Transformations . . . . . . 302

12.2.1 Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 30312.2.2 Disclination . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

12.3 Differential-Geometric Properties of Affine Spaces . . . . . . . . . . 30512.3.1 Local Parallelism . . . . . . . . . . . . . . . . . . . . . . . . 306

12.4 Circuit Integrals in Affine Spaces with Curvature and Torsion . . . . 31012.4.1 Parallelism in World Crystal . . . . . . . . . . . . . . . . . 313

12.5 Bianchi Identities for Curvature and Torsion Tensors . . . . . . . . . 31312.6 Special Coordinates in Riemann Spacetime . . . . . . . . . . . . . . 316

12.6.1 Geodesic Coordinates . . . . . . . . . . . . . . . . . . . . . 31612.6.2 Canonical Geodesic Coordinates . . . . . . . . . . . . . . . 31812.6.3 Harmonic Coordinates . . . . . . . . . . . . . . . . . . . . . 32012.6.4 Coordinates with det(gµν) = 1 . . . . . . . . . . . . . . . . . 32112.6.5 Orthogonal Coordinates . . . . . . . . . . . . . . . . . . . . 322

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Notes and References xix

12.7 Number of Independent Components of Rµνλκ and Sµν

λ . . . . . . . 32312.7.1 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 32412.7.2 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . 32512.7.3 Four or More Dimensions . . . . . . . . . . . . . . . . . . . 32512.7.4 Constant Curvature . . . . . . . . . . . . . . . . . . . . . . 327

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

13 Curvature and Torsion from Embedding 331

13.1 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33113.2 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

13.2.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33313.2.2 Nonholonomic Embedding . . . . . . . . . . . . . . . . . . . 33413.2.3 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

14 Nonholonomic Mapping Principle 338

14.1 Motion of Point Particle . . . . . . . . . . . . . . . . . . . . . . . . . 33814.1.1 Classical Action Principle for Spaces with Curvature . . . . 33814.1.2 Autoparallel Trajectories in Spaces with Torsion . . . . . . 33914.1.3 Special Properties of Gradient Torsion . . . . . . . . . . . . 345

14.2 Autoparallel Trajectories from Embedding . . . . . . . . . . . . . . 34614.2.1 Special Role of Autoparallels . . . . . . . . . . . . . . . . . 34614.2.2 Gauss Principle of Least Constraint . . . . . . . . . . . . . 347

14.3 Maxwell-Lorentz Orbits as Autoparallel Trajectories . . . . . . . . . 34814.4 Bargmann-Michel-Telegdi Equation from Torsion . . . . . . . . . . . 349Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

15 Field Equations of Gravitation 351

15.1 Invariant Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35115.2 Energy-Momentum Tensor and Spin Density . . . . . . . . . . . . . 35315.3 Total Energy-Momentum Tensor and Defect Density . . . . . . . . . 359Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

16 Minimally Coupled Fields of Integer Spin 361

16.1 Scalar Fields in Riemann-Cartan Space . . . . . . . . . . . . . . . . 36116.2 Electromagnetism in Riemann-Cartan Space . . . . . . . . . . . . . 363Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

17 Particles with Half-Integer Spin 366

17.1 Local Lorentz Invariance and Anholonomic Coordinates . . . . . . . 36617.1.1 Nonholonomic Image of Dirac Action . . . . . . . . . . . . . 36617.1.2 Vierbein Fields . . . . . . . . . . . . . . . . . . . . . . . . . 36917.1.3 Local Inertial Frames . . . . . . . . . . . . . . . . . . . . . 370

17.2 Difference between Vierbein and Multivalued Tetrad Fields . . . . . 37217.3 Nonholonomic Image of Dirac Action . . . . . . . . . . . . . . . . . 376

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17.4 Alternative Form of Coupling . . . . . . . . . . . . . . . . . . . . . . 37817.5 Invariant Action for Vector Fields . . . . . . . . . . . . . . . . . . . 37917.6 Verifying Local Lorentz Invariance . . . . . . . . . . . . . . . . . . . 380

17.6.1 No Need for Torsion . . . . . . . . . . . . . . . . . . . . . . 38217.7 Field Equations with Gravitational Spinning Matter . . . . . . . . . 382

18 Covariant Conservation Law 387

18.1 Spin Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38718.2 Energy-Momentum Density . . . . . . . . . . . . . . . . . . . . . . . 38918.3 Covariant Derivation of Conservation Laws . . . . . . . . . . . . . . 39218.4 Matter with Integer Spin . . . . . . . . . . . . . . . . . . . . . . . . 39318.5 Relations between Conservation Laws and Bianchi Identities . . . . 39518.6 Particle Trajectories from Energy-Momentum Conservation . . . . . 396Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

19 Gravitation of Spinning Matter as a Gauge Theory 399

19.1 Local Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 39919.2 Local Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

20 Evanescent Properties of Torsion in Gravity 403

20.1 Local Four-Fermion Interaction due to Torsion . . . . . . . . . . . . 40320.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

20.2.1 Possible Cure . . . . . . . . . . . . . . . . . . . . . . . . . . 40520.2.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 409

20.3 Compatibility Problems of Gravity with Torsion and ElectroweakInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41020.3.1 Solution for Gradient Torsion . . . . . . . . . . . . . . . . . 41020.3.2 New Scalar Product . . . . . . . . . . . . . . . . . . . . . . 41120.3.3 Self-Interacting Higgs Field . . . . . . . . . . . . . . . . . . 412

Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

21 Emerging Gravity 415

21.1 World Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41621.2 Gravity Emerging from Fluctuations of Matter and Radiation . . . . . 420Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Index 423

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List of Figures

4.1 Infinitesimally thin closed current loop L and magnetic field . . . . . 111

5.1 Specific heat of superfluid 4He . . . . . . . . . . . . . . . . . . . . . 1295.2 Energies of the elementary excitations in superfluid 4He . . . . . . . 1305.3 Rotons join side by side to form surfaces whose boundary appears

as a large vortex loop . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.4 Vortex loops in XY-model for different β = 1/kBT . . . . . . . . . . 1325.5 Lattice Yukawa potential at the origin and the associated tracelog . 1395.6 Specific heat of Villain model in three dimensions . . . . . . . . . . 1415.7 Critical temperature of a loop gas with Yukawa interactions . . . . . 1475.8 Specific heat of superconducting aluminum . . . . . . . . . . . . . . 1545.9 Potential for the order parameter ρ with cubic term . . . . . . . . . 1745.10 Phase diagram of a two-dimensional layer of superfluid 4He . . . . . 1795.11 Experimental phase diagram of a two-dimensional layer of superfluid

4He by 3He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

9.1 Intrinsic point defects in a crystal . . . . . . . . . . . . . . . . . . . 2329.2 Formation of a dislocation line (of the edge type) from a disc of

missing atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.3 Naive estimate of maximal stress supported by a crystal under shear

stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2339.4 Dislocation line permits the two crystal pieces to move across each

other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2349.5 Formation of a disclination from a stack of layers of missing atoms . 2359.6 Grain boundary where two crystal pieces meet with different orien-

tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2359.7 Two typical stacking faults . . . . . . . . . . . . . . . . . . . . . . . 2369.8 Definition of Burgers vector . . . . . . . . . . . . . . . . . . . . . . . 2379.9 Screw disclination arising upon tearing a crystal . . . . . . . . . . . 2389.10 Volterra cutting and welding process . . . . . . . . . . . . . . . . . . 2399.11 Lattice structure at a wedge disclination . . . . . . . . . . . . . . . . 2399.12 Three different possibilities of constructing disclinations . . . . . . . 2409.13 Generation of dislocation line from a pair of disclination lines . . . 2419.14 In the presence of a dislocation line, the displacement field is defined

only modulo lattice vectors . . . . . . . . . . . . . . . . . . . . . . . 2449.15 Geometry used in the derivation of Weingarten’s theorem . . . . . . 245

xxi

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9.16 Illustration of Volterra process . . . . . . . . . . . . . . . . . . . . . 252

10.1 Specific heat of various solids . . . . . . . . . . . . . . . . . . . . . . 26510.2 Specific heat of various solids . . . . . . . . . . . . . . . . . . . . . . 26710.3 Separation of first-order melting transition into two successive

Kosterlitz-Thouless transitions in two dimension . . . . . . . . . . . 26810.4 Phase diagram in the T -`-plane in two-dimensional melting . . . . . 269

11.1 Illustration of crystal planes before and after elastic distortion . . . 276

12.1 Edge dislocation in a crystal associated with a missing semi-infiniteplane of atoms as a source of torsion . . . . . . . . . . . . . . . . . . 303

12.2 Edge disclination in a crystal associated with a missing semi-infinitesection of atoms as a source of curvature . . . . . . . . . . . . . . . 305

12.3 Illustration of parallel transport of a vector around a closed circuit . 30812.4 Illustration of non-closure of a parallelogram after inserting an edge

dislocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

14.1 Images under holonomic and nonholonomic mapping of δ-functionvariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

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Basic research is what I am doing

when I don’t know what I am doing

Wernher von Braun (1912 - 1977)

1

Basics

In his fundamental work on theoretical mechanics entitled Principia, Newton (1642-1727) assumes the existence of an absolute spacetime. Space is parametrized byvectors x = (x1, x2, x3), and the movement of point particles is described by tra-jectories x(t) whose components qi(t) (i = 1, 2, 3) specify the coordinates xi = qi(t)along which the particles move as a function of the time t. In Newton’s absolutespacetime, a single free particle moves without acceleration. Mathematically, this isexpressed by the differential equation

x(t) ≡ d2

dt2x(t) = 0. (1.1)

The dots denote derivatives with respect to the argument.A set of N point particles xn(t) (n = 1, . . . , N) with masses mn is subject to

gravitational forces which change the free equations of motion to

mnxn(t) = GN

m6=n

mnmmxm(t) − xn(t)

|xm(t) − xn(t)|3, (1.2)

where GN is Newton’s gravitational constant

GN ≈ 6.67259(85)× 10−8cm3/g sec2. (1.3)

1.1 Galilean Invariance of Newtonian Mechanics

The parametrization of absolute spacetime in which the above equations of motionhold is not unique.

1.1.1 Translations

The coordinates x may always be changed by translated coordinates

x′ = x − x0. (1.4)

1

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2 1 Basics

It is obvious that the translated trajectories x′n(t) = xn(t)−x0 will again satisfy theequations of motion (1.2). The equations remain also true for a translated time

t′ = t− t0, (1.5)

i.e., the trajectoriesx′(t) ≡ x(t+ t0) (1.6)

satisfy (1.2). This property of Newton’s equations (1.2) is referred to as translationalsymmetry in spacetime.

An alternative way of formulating this invariance is my keeping the coordinateframe fixed and displacing the physical system in spacetime, moving all particles tonew coordinates x′ = x + x0 at a new time t′ = t+ t0. The equations of motion areagain invariant. The first procedure of reparametrizing the same physical systemis called passive symmetry transformation, the second active symmetry transforma-tion. One may use either procedure to discuss symmetries. In this book we shalluse active or passive transformations, depending on the circumstance.

1.1.2 Rotations

The equations of motion are invariant under more transformations which mix dif-ferent coordinates linearly with each other, for instance the rotations:

x′i = Rijxj (1.7)

where Rij is the rotation matrix

Rij = cos θ δij + (1 − cos θ) θiθj + sin θ εijkθk, (1.8)

in which θi denotes the directional unit vector of the rotation axis. The matricessatisfy the orthogonality relation

RTR = 1. (1.9)

In Eq. (1.7) a sum from 1 to 3 is implied over the repeated spatial index j. This iscalled the Einstein summation convention, which will be followed throughout thistext. As for the translations, the rotations can be applied in the passive or activesense.

The active rotations are obtained from the above passive ones by changing thesign of θ. For example, the active rotation around the z-axis with a rotation vectorϕ = (0, 0, 1) are given by the orthogonal matrices

R3(ϕ) =

cosϕ − sinϕ 0sinϕ cosϕ 0

0 0 1

. (1.10)

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1.2 Lorentz Invariance of Maxwell Equations 3

1.1.3 Galilei Boosts

A further set of transformations mixes space and time coordinates:

x′i = xi − vit, (1.11)

t′ = t. (1.12)

These are called pure Galilei transformations of Galilei boosts. The coordinatesx′i, t′ are positions and time of a particle observed in a frame of reference thatmoves uniformly through absolute spacetime with velocity v ≡ (v1, v2, v3). In theactive description, the transformation x′i = xi + vit specifies the coordinates of aphysical system moving past the observer with uniform velocity v.

1.1.4 Galilei Group

The combined set of all transformations

x′i = Rijxj − vit− xi0, (1.13)

t′ = t− t0, (1.14)

forms a group. Group multiplication is defined by performing the transformationssuccessively. This multiplication law is obviously associative, and each element hasan inverse. The set of transformations (1.13) and (1.14) is referred to as the Galileigroup.

Newton called all coordinate frames in which the equations of motion have thesimple form (1.2) inertial frames.

1.2 Lorentz Invariance of Maxwell Equations

Problems with Newton’s theory arose when J. C. Maxwell (1831 - 1879) formulatedin 1864 his theory of electromagnetism. His equations for the e electric field E(x)and the magnetic flux density or magnetic induction B(x) in empty space

∇ · E = 0 (Coulomb’s law), (1.15)

∇ × B − 1

c

∂E

∂t= 0 (Ampere‘s law), (1.16)

∇ · B = 0 (absence of magnetic monopoles), (1.17)

∇ × E +1

c

∂B

∂t= 0 (Faraday’s law), (1.18)

can be combined to obtain the second-order differential equations

(

1

c2∂2t − ∇

2)

E (x, t) = 0, (1.19)(

1

c2∂2t − ∇

2)

B (x, t) = 0. (1.20)

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4 1 Basics

The equations contain explicitly the light velocity

c ≡ 299 792 458m

sec, (1.21)

and are not invariant under the Galilei group (1.14). Indeed, they contradict New-ton’s postulate of the existence of an absolute spacetime. If light were to propagatewith the velocity c in absolute spacetime, it could not do so in other inertial frameswhich have a nonzero velocity with respect to the absolute frame. A precise mea-surement of the light velocity could therefore single out the absolute spacetime.However, experimental attempts to do this did not succeed. The experiment ofMichelson (1852-1931) and Morley (1838-1923) in 1887 showed [1] that light travelsparallel and orthogonal to the earth’s orbital motion with the same velocity up to ±5km/sec [2]. This led Fitzgerald (1851-1901) [3], Lorentz (1855-1928) [4], Poincare(1854-1912) [5], and finally Einstein (1879-1955) [6] to suggest that Newton’s pos-tulate of the existence of an absolute spacetime was unphysical [7].

1.2.1 Lorentz Boosts

The conflict was resolved by modifying the Galilei transformations (1.11) and (1.12)in such a way that Maxwell’s equations remain invariant. This is achieved by thecoordinate transformations

x′i = xi + (γ − 1)vivj

v2xj − γvit, (1.22)

t′ = γt− 1

c2γvixi, (1.23)

where γ is the velocity-dependent parameter

γ =1

1 − v2/c2. (1.24)

The transformations (1.23) are referred to as pure Lorentz transformations or Lorentzboosts. The parameter γ has the effect that in different moving frames of reference,time elapses differently. This is necessary to make the light velocity the same in allframes.

The pure Lorentz transformations are conveniently written in a four-dimensionalvector notation. Introducing the four-vectors xa labeled by letters a, b, c, . . . runningthrough the values 0, 1, 2, 3,

xa =

ctx1

x2

x3

, (1.25)

we rewrite (1.22) and (1.23) as

x′a = Λabxb (1.26)

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1.2 Lorentz Invariance of Maxwell Equations 5

where Λab are the 4 × 4-matrices

Λab ≡

γ −γvi/c−γvi/c δij + (γ − 1)vivj/v

2

. (1.27)

Note that we adopt Einstein’s summation convention also for repeated labelsa, b, c, . . . = 0, . . . , 3. The matrices Λa

b satisfy the pseudo-orthogonality relation[compare (1.9)]:

ΛTac gcd Λd

b = gab, (1.28)

where gab is the Minkowski metric with the matrix elements

gab =

1−1

−1−1

. (1.29)

Equation (1.28) has the consequence that for any two four-vectors xa and ya, thescalar product formed with the help of the Minkowski metric

xy ≡ xagabyb (1.30)

is an invariant under Lorentz transformation.In order to verify the relation (1.28) it is convenient to introduce a dimensionless

vector

called rapidity , which points in the direction of the velocity v and has alength ζ ≡ || given by

cosh ζ = γ, sinh ζ = v/c. (1.31)

We also define the unit vectors in three-space

≡ /ζ = v ≡ v/v, (1.32)

so that

= ζ

= atanhv

cv. (1.33)

Then the matrices Λab of the pure Lorentz transformations (1.27) take the form

Λab = Ba

b() ≡

cosh ζ − sinh ζ ζ1 − sinh ζ ζ2 − sinh ζ ζ3

− sinh ζ ζ1− sinh ζ ζ2 δij + (cosh ζ − 1) ζiζj− sinh ζ ζ3

. (1.34)

The notation Bab() emphasizes that the transformations are boosts. Note that the

property (1.28) follows directly from the identities

2 = 1, cosh2 ζ − sinh2 ζ = 1.

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6 1 Basics

For active transformations of a physical system, the above transformations haveto be inverted. For instance, the active boosts with a rapidity pointing in the z-direction, ζ = (0, 0, 1), have the pseudo-orthogonal matrix form

Λab = B3(ζ) =

cosh ζ 0 0 sinh ζ0 1 0 00 0 1 0

sinh ζ 0 0 cosh ζ

. (1.35)

1.2.2 Lorentz Group

The set of Lorentz boosts (1.34) can be extended by rotations to form the Lorentzgroup. In four-by-four matrix notation, the rotation matrices (1.8) have the blockform

Λab(R) = Ra

b ≡

1 0 0 000 Ri

j

0

. (1.36)

It is easy to verify that these satisfy the relation (1.28), which reduces to the or-thogonality relation (1.9).

The four-dimensional versions of the active rotations (1.10) around the z-axiswith a rotation vector ϕ = (0, 0, 1) are given by the orthogonal matrices

Λba = R3(ϕ) =

1 0 0 00 cosϕ − sinϕ 00 sinϕ cosϕ 00 0 0 1

. (1.37)

The rotation matrix (1.37) differs from the boost matrix (1.35) mainly in thepresence of trigonometric functions instead of hyperbolic functions. In addition,there is a sign change under transposition accounting for the opposite sign in thetime- and space-like parts of the metric (1.29).

When performing all possible Lorentz boosts and rotations in succession, theresulting set of transformations forms a group called the Lorentz group.

1.3 Infinitesimal Lorentz Transformations

The transformation laws of continuous groups such as rotation and Lorentz group areconveniently expressed in an infinitesimal form. By performing many infinitesimaltransformations after each other it is always possible to reconstruct from these thefinite transformation laws. This is a consequence of the fact that the exponentialfunction ex can always be obtained by a product of many small-x approximationseεx ≈ 1 + εx:

ex = limε→0

(1 + εx)1/ε . (1.38)

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1.3 Infinitesimal Lorentz Transformations 7

Let us illustrate this procedure for the active rotations (1.37). These can be writtenin the exponential form

R3(ϕ) = exp

0 0 0 00 0 −1 00 1 0 00 0 0 0

ϕ

≡ e−iL3ϕ. (1.39)

The matrix

L3 = −i

0 0 0 00 0 1 00 −1 0 00 0 0 0

(1.40)

is called the generator of this rotation in the Lorentz group. There are similargenerators for rotations around x- and y-directions

L1 = −i

0 0 0 00 0 0 00 0 0 10 0 −1 0

, (1.41)

L2 = −i

0 0 0 00 0 0 −10 0 0 00 1 0 0

. (1.42)

The three generators may compactly be written as

Li ≡ −i(

0 0

0 εijk

)

, (1.43)

where εijk is the completely antisymmetric Levi-Civita tensor with ε123 = 1.Introducing a vector notation for the three generators, L ≡ (L1, L2, L2), the

general pure rotation matrix (1.36) is given by the exponential

Λ(R()) = e−i ·L. (1.44)

This follows from the fact that all orthogonal 3 × 3-matrices in the spatial block of(1.36) can be written as an exponential of i times all antisymmetric 3× 3-matrices,and that these can all be reached by the linear combinations · L.

Let us now find the generators of the active boosts, first in the z-direction wherewe see from (1.35) that the boost matrix can be written as an exponential

B3(ζ) = exp

0 0 0 10 0 0 00 0 0 01 0 0 0

ζ

= e−iM3ζ (1.45)

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8 1 Basics

with the generator

M3 = i

0 0 0 10 0 0 00 0 0 01 0 0 0

. (1.46)

Similarly we find the generators for the x- and y-directions:

M1 = i

0 1 0 01 0 0 00 0 0 00 0 0 0

, (1.47)

M2 = i

0 0 1 00 0 0 01 0 0 00 0 0 0

. (1.48)

Introducing a vector notation for the three boost generators, M ≡ (M1,M2,M2),the general Lorentz transformation matrix (1.34) is given by the exponential

Λ(B()) = e−i·M. (1.49)

The proof is analogous to the proof of the exponential form (1.44).The Lorentz group is therefore generated by the six matrices Li,Mi, to be col-

lectively denoted by Ga(a = 1, . . . , 6). Every element of the group can be writtenas

Λ = e−i( ·L+·M) ≡ e−iαaGa . (1.50)

There exists a Lorentz-covariant way of specifying the generators of the Lorentzgroup. We introduce the 4 × 4-matrices

(Lab)cd = i(gacgbd − gadgbc), (1.51)

labeled by the antisymmetric pair of indices ab, i.e.,

Lab = −Lba. (1.52)

There are 6 independent matrices which coincide with the generators of rotationsand boosts as follows:

Li =1

2εijkL

jk, Mi = L0i. (1.53)

With the generators (1.51), we can write every element (1.50) of the Lorentz groupas follows

Λ = e−i12ωabL

ab

, (1.54)

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1.3 Infinitesimal Lorentz Transformations 9

where the antisymmetric angular matrix ωab = −ωba collects both, rotation anglesand rapidities:

ωij = εijkϕk, (1.55)

ω0i = ζ i. (1.56)

Summarizing the notation we have set up an exponential representations of allLorentz transformations

Λ = e−i( ·L+·M) = e−i(12ϕiεijkL

jk+ζiL0i) = e−i(12ωijL

ij+ω0iL0i) = e−i12ωabL

ab

. (1.57)

Note that if the metric were euclidean

g =

11

11

, (1.58)

the above representation would be familiar from basic matrix theorems: Then Λwould, by Eq. (1.28), comprise all real orthogonal matrices in four dimensions, andthese could be written as an exponential of all real antisymmetric 4×4-matrices. Forthe pseudo-orthogonal matrices satisfying (1.28) with the Minkowski metric (1.29),only iL’s are antisymmetric while iM are symmetric.

1.3.1 Group Multiplication and Lee Algebra

The reason for expressing the group elements as exponentials of the six generatorsis that, in this way, the multiplication rules of infinitely many group elements canbe completely reduced to the knowledge of the finite number of commutation rulesamong the six generators Li,Mi. This is a consequence of the Baker-Campbell-Hausdorff formula [8]

eAeB = eA+B+ 12[A,B]+ 1

12[A−B,[A,B]]− 1

24[A,[B,[A,B]]]+.... (1.59)

According to this formula, the product of exponentials can be written as an expo-nential of commutators. Adapting the general notation Gr = (Li,Mi) for the sixgenerators in Eqs. (1.53) and (1.57), the product of two group elements is

Λ1Λ2 = e−iα1rGre−iα

2sGs

= exp

−iα1rGr − iα2

sGs +1

2[−iα1

rGr,−iα2sGs]

+1

12[−i(α1

t − α2t )Gt, [−iα1

rGr,−iα2sGs]] + . . .

. (1.60)

The exponent involves only commutators among Gr’s. For the Lorentz group thesecan be calculated from the explicit 4 × 4 -matrices (1.40)–(1.42) and (1.46)–(1.48).The result is

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10 1 Basics

[Li, Lj ] = iεijkLk, (1.61)

[Li,Mj ] = iεijkMk, (1.62)

[Mi,Mj ] = −iεijkLk. (1.63)

This algebra of generators is called the Lie algebra of the group. In the generalnotation with the generators Gr, the algebra reads

[Gr, Gs] = ifrstGt. (1.64)

The number of linearly independent matrices Gr (here 6) is called the rank r of theLie algebra.

In any Lie algebra, the commutator of two generators is a linear combination ofgenerators. The coefficients fabc are called structure constants. They are completelyantisymmetric in a, b, c, and satisfy the relation

frsufutv + fstufurv + ftrufusv = 0. (1.65)

This guarantees that the generators obey the Jacobi identity

[[Gr, Gs], Gt] + [[Gs, Gt], Gr] + [[Gt, Gr], Gs] = 0, (1.66)

which ensures that multiplication of three exponentials Λj = e−iαjrGr (i = 1, 2, 3)

obeys the law of associativity (Λ1Λ2)Λ3 = Λ1(Λ2Λ3) when evaluating the productsvia the expansion Eq. (1.60).

The relation (1.65) can easily be verified explicitly for the structure constants(1.61)–(1.63) of the Lorentz group using the identity for the ε-tensor

εijlεlkm + εjklεlim + εkilεljm = 0. (1.67)

The Jacobi identity implies that the r matrices with r × r elements

(Fr)st ≡ −ifrst (1.68)

satisfy the commutation rules (1.64). They are the generators of the so-called adjointrepresentation of the Lie algebra. The matrix in the spatial block of Eq. (1.43) forLi is precisely of this type.

In terms of the matrices Fr of the adjoint representation, the commutation rulescan also be written as

[Gr, Gs] = −(Ft)abGt. (1.69)

Inserting for Gr the generators (1.68), we reobtain the relation (1.65).Continuing the expansion in terms of commutators in the exponent of (1.60),

all commutators can be evaluated successively and one remains at the end with anexpression

Λ12 = e−iα12r (α1,α2)Gr , (1.70)

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1.4 Vectors, Tensors, Scalars 11

in which the parameters of the product α12r are completely determined from those

of the factor, α1r , α

2r . The result depends only on the structure constants fabc, not

on the representation.If we employ the tensor notation Lab for Li,Mi of Eqs. (1.53), (1.53), and per-

form multiplication covariantly, so that products LabLcd have the matrix elements(Lab)στ (L

cd)τ δ, the commutators (1.61)–(1.63) can be written as

[Lab, Lcd] = −i(gacLbd − gadLbc + gbdLac − gbcLad). (1.71)

Due to the antisymmetry in a ↔ b and c ↔ d it is sufficient to specify only thesimpler commutators

[Lab, Lac] = −igaaLbc, no sum over a. (1.72)

This list of commutators omits only those commutation rules of (1.71) which van-ishes, since none of the indices ab is equal to one of the indices cd.

1.4 Vectors, Tensors, Scalars

We shall frequently consider four-component physical quantities va which, underLorentz transformation, change in the same way as the coordinates xa:

v′a = Λabvb. (1.73)

This transformation property defines a Lorentz vector of four-vector . In additionto such vectors, there are quantities with more indices vab, vabc, . . . which transformlike products of vectors:

v′ab = ΛacΛ

bdvcd, (1.74)

v′abc = ΛadΛ

beΛ

cfv

def . (1.75)

These are the transformation properties of Lorentz tensors of rank two, three, . . . .Given any two four-vectors ua and va, we define their scalar product in the same

way as before in (1.30) for two coordinate vectors xa and ya:

uv = uagabvb, (1.76)

Scalar products are, of course, invariant under Lorentz transformations due to theirpseudo-orthogonality (1.28).

If va, vab, vabc, . . . are functions of x, they are called vector and tensor fields.Derivatives with respect to x of such field obey vector and tensor transformationlaws. Indeed, since

x′a = Λabxb, (1.77)

we see that the derivative ∂/∂xb satisfies

∂x′a=(

ΛT−1)

a

b ∂

∂xb, (1.78)

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12 1 Basics

i.e., it transforms with the inverse of the transposed Lorentz matrix Λab. Using the

pseudo-orthogonality relation (1.28), we can also write

∂x′a=(

gΛg−1)

a

b ∂

∂xb. (1.79)

In general, any four-component quantity va which transforms like the derivatives

v′a =(

gΛg−1)

a

bvb (1.80)

is called a covariant Lorentz vector or four-vector, as opposed to the vector va

transforming like xa itself, which is called contravariant vector.A covariant vector va can be produced from a contravariant one vb by multipli-

cation with the metric tensor:va = gabv

b. (1.81)

This operation is called lowering the index . The operation can be inverted to whatis called raising the index :

va = gabvb, (1.82)

where gab are the matrix elements of the inverse metric

gab ≡(

g−1)

ab. (1.83)

With Einstein’s summation convention, the inverse metric gab ≡ (g−1)ab

satisfies theequation

gabgbc = δac (1.84)

The sum over a common upper and lower index is called contraction.In Minkowski space, the matrices g and g−1 happen to be the same and so are

the matrix elements gab and gab, both being equal to (1.29). This will no longer betrue in the general geometries of gravitational physics. For this reason it will beuseful to keep separate symbols for the metric g and its inverse g−1, and for theirmatrix elements gab and gab.

The contraction of a covariant vector with a contravariant vector is a scalarproduct, as is obvious if we rewrite the scalar product (1.76) as

uv = uagabvb = uava = uav

a. (1.85)

Of course, we can form also the scalar product of two covariant vectors with thehelp of the inverse metric g−1:

uv = uagabvb. (1.86)

The invariance under Lorentz transformations (1.80) is easily verified using thepseudo-orthogonality property (1.28):

u′agabv′b = u′Tg−1v′ = uTg−1ΛTg g−1 gΛg−1v = uTg−1v = uag

abvb. (1.87)

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1.4 Vectors, Tensors, Scalars 13

Since ∂/∂xa transforms like a covariant vector, it is useful to emphasize thisbehavior by the notation

∂a ≡∂

∂xa. (1.88)

Extending the definition of covariant vectors one defines covariant tensors of ranktwo tab, three tabc, etc. as quantities transforming like

t′ab =(

gΛg−1)

a

c(

gΛg−1)

b

d tcd,

t′abc =(

gΛg−1)

a

c(

gΛg−1)

b

f(

gΛg−1)

c

g tefg. (1.89)

Co- and contravariant vectors, tensors, can always be multiplied with each otherto form new co- and contravariant quantities if the indices to be contracted areraised and lowered appropriately. If no uncontracted indices are left, one obtains aninvariant, a Lorentz scalar .

It is useful to introduce a contravariant version of the covariant derivative vector

∂a ≡ gab∂b, (1.90)

and covariant versions of the contravariant coordinate vector

xa ≡ gabxb. (1.91)

The invariance of Maxwell’s equations (1.20) is a direct consequence of thesecontraction rules since the differential operator on the left-hand side can be writtencovariantly as

1

c2∂2t −∇2 =

∂xagab

∂xb= ∂ag

ab∂b = ∂a∂a = ∂2. (1.92)

The right-hand side is obviously a Lorentz scalar.

1.4.1 Discrete Lorentz Transformations

The Lorentz group can be extended to include space reflections in any of the fourspacetime directions

xa → −xa, (1.93)

without destroying the defining property (1.28). The determinant of Λ, however, isthen negative. If only x0 is reversed, the reflection is also called time reversal anddenoted by

T =

−11

11

. (1.94)

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The simultaneous reflection of the three spatial coordinates is called parity trans-formation and denoted by the 4 × 4-matrix P , i.e.,

P =

1−1

−1−1

. (1.95)

After this extension, the entire Lorentz group can no longer be obtained fromthe neighborhood of the identity by a product of infinitesimal transformations, i.e.,by an exponential of the Lie algebra in Eq. (1.57). It consists of four topologicallydisjoint pieces which can be obtained by a product of infinitesimal transformationsmultiplied with 1, P , T , and PT . The four pieces of the group are

e−i12ωabL

ab

, e−i12ωabL

ab

P, e−i12ωabL

ab

T, e−i12ωabL

ab

PT. (1.96)

The Lorentz transformations Λ of the pieces associated with P and T have a negativedeterminant. This leads to the definition of pseudotensors which transform like atensor, but with an additional factor detΛ. A vector with this property is alsocalled axial vector . In three dimensions, the angular momentum L = x × p is anaxial vector since it does not change sign under space reflections, as the vector x,but remains invariant.

1.4.2 Poincare group

Just as the Galilei transformations, the Lorentz transformations can be extended bythe group of spacetime translations

xa = xa − aa (1.97)

to form the inhomogeneous Lorentz group or Poincare group.Inertial frames may be defined as all those frames in which Maxwell’s equations

are valid. They differ from each other by Poincare transformations.

x′a = Λabxa − aa. (1.98)

1.5 Differential Operators for Lorentz Transformations

The physical laws in four-dimensional spacetime are formulated in terms of Lorentz-invariant field theories. The fields depend on the spacetime coordinates xa. In orderto perform transformations of the Lorentz group we need differential operators forthe generators of this group.

For Lorentz transformations Λ with small rotation angles and rapidities, we canapproximate the exponential in (1.57) as

Λ ≡ 1 − i1

2ωabL

ab. (1.99)

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1.5 Differential Operators for Lorentz Transformations 15

Then the Lorentz transformation of the coordinates

xΛ−→ x′ = Λx (1.100)

is conveniently characterized by the infinitesimal change

δΛx = x′ − x = −i 1

2ωabL

abx. (1.101)

Inserting the 4 × 4 matrix generators (1.51), this becomes more explicitly

δΛxa = ωabx

b. (1.102)

We now observe that this can be expressed in terms of the differential operators

Lab ≡ i(xa∂b − xb∂a) = −Lba (1.103)

as a commutator

δΛx = i1

2ωab[L

ab, x]. (1.104)

The differential operators (1.103) satisfy the same commutation relations (1.71),(1.72) as the 4×4 -generators Lab of the Lorentz group. They form a representationof the Lie algebra (1.71), (1.72). By exponentiation we can thus form the operatorrepresentation of finite Lorentz transformations

D(Λ) ≡ e−i12ωabL

ab

, (1.105)

which satisfy the same group multiplication rules as the 4 × 4-matrices Λ.The relation between the finite Lorentz transformations (1.100) and the operator

version (1.105) is

x′ = Λx = e−i12ωabL

ab

x = ei12ωabL

ab

x e−i12ωabL

ab

= D−1(Λ) x D(Λ). (1.106)

This is proved by expanding, on the left-hand side, e−i12ωabL

ab

x in powers of ωab, anddoing the same thing on the right-hand with ei

12ωabL

ab

x e−i12ωabL

ab

with the help ofLie’s expansion formula

e−iA B eiA = 1 − i[A, B] +i2

2![A, [A, B]] + . . . . (1.107)

This operator representation (1.105) can be used to generate Lorentz transfor-mations on the spacetime argument of any function of x:

f ′(x) ≡ f(Λ−1x) = f(

D(Λ)x D−1(Λ))

= D(Λ)f (x) D−1(Λ). (1.108)

The latter step follows from a power series expansion of f(x). Take, for example anexpansion term fa,bx

axb of f(x). In the transformed function f ′(x), this becomes

fa,bD(Λ)xa D−1(Λ)D(Λ)xbD−1(Λ) = D(Λ)(

fa,bxaxb

)

D−1(Λ). (1.109)

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1.5.1 Vector and Tensor Operators

In working out the commutation rules among the differential operators Lab oneconveniently uses the commutation rules between Lab and xc, pc:

[Lab, xc] = −i(gacxb − gbcxa) = −(Lab)cdxd, (1.110)

[Lab, pc] = −i(gacpb − gbcpa) = −(Lab)cdpd. (1.111)

These commutation rules identify xc and pc as vector operators.In general, an operator t c1,···,cn is said to be a tensor operator of rank n if each

of its tensor indices is transformed under commutation with Lab like the index of xa

or pa in (1.110) and (1.111):

[Lab, t c1,...,cn] =−i[(gac1 t b,...,cn − gbc1 t a,...,cn) + . . .+ (gacn t c1,...,b − gbcn t c1,...,a)]

=−(Lab)c1d tdc2,...,cn − (Lab)c2d t

c1d,...,cn −. . .− (Lab)cnd tc1c2,...,d.(1.112)

The commutators (1.71) between the generators imply that these are themselvestensor operators.

The simplest examples for such tensor operators are the direct products of vectorssuch as t c1,...,cn = xc1 · · ·xcn or t c1,...,cn = pc1 · · · pcn . In fact, the right-hand side canbe found for such direct products using the commutation rules between products ofoperators

[a, bc] = [a, b]c+ b[a, c], [ab, c] = a[b, c] + [a, c]b. (1.113)

These are the analogs of the Leibnitz chain rule for derivatives

∂(fg) = (∂f)g + f(∂g). (1.114)

1.6 Finite Operator Lorentz Transformations

Let us apply such a finite operator representation (1.105) to the vector xc to form

D(Λ)xcD−1(Λ). (1.115)

We do this separately for rotations and Lorentz transformations, first for rotations.

1.6.1 Rotations

An arbitrary three-vector (x1, x2, x3) is rotated around the 3-axis by the operator

D(R3(ϕ)) = e−iϕL3 with L3 = −i(x1∂2 − x2∂1) by the operation

D(R3(ϕ))xiD−1(R3(ϕ)) = e−iϕL3xieiϕL3 . (1.116)

Since L3 commutes with x3, this component is invariant under the operation (1.116):

D(R3(ϕ))x3D−1(R3(ϕ)) = e−iϕL3x3eiϕL3 = x3. (1.117)

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1.6 Finite Operator Lorentz Transformations 17

For x1 and x2, the Lie expansion of (1.115) contains the commutators

−i[L3, x1] = x2, − i[L3, x

2] = −x1. (1.118)

Thus, the first-order expansion term on the right-hand side of (1.116) transforms thetwo-dimensional vector (x1, x2) into (x2,−x1). The second-order term is obtainedby commuting the operator −iL3 with (x2,−x1), yielding −(x1, x2). To third-order,this is again transformed into −(x2,−x1), and so on. Obviously, all even ordersreproduce the initial two-dimensional vector (x1, x2) with an alternating sign, whileall odd powers are proportional to (x2,−x1). Thus we obtain the expansion

e−iϕL3(x1, x2)eiϕL3 =(

1 − 1

2!ϕ2 +

1

4!ϕ4 + . . .

)

(x1, x2)

+(

ϕ− 1

3!ϕ3 +

1

5!ϕ5 + . . .

)

(x2,−x1). (1.119)

The even and odd powers can be summed up to a cosine and a sine, respectively,resulting in

e−iϕL3(x1, x2)eiϕL3 = cosϕ (x1, x2) + sinϕ (x2,−x1). (1.120)

Together with the invariant x3 in (1.117), the right-hand side forms a vector arisingfrom xi by an inverse rotation (1.37). Thus

D(R3(ϕ))xiD−1(R3(ϕ)) = e−iϕL3xieiϕL3 =(

eiϕL3

)i

jxj = R−1

3 (ϕ)ijxj . (1.121)

By performing successively rotations around the three axes we can generate in thisway any inverse rotation:

D(R())xiD−1(R()) = e−i ·Lxiei·L =(

ei ·L)i

jxj = R−1()ijx

j . (1.122)

This is the finite transformation law associated with the commutation relation

[Li, xk] = xj(Li)jk, (1.123)

which characterizes the vector operator nature of xi. Thus (1.122) holds for anyvector operator vi.

The time component x0 is obviously unchanged by a rotation since L3 commuteswith x0. Hence we can extend (1.122) trivially to a four-vector, replacing D(R())by D(Λ(R())) [recall (1.44)].

1.6.2 Lorentz Boosts

A similar calculation may be done for Lorentz boosts. Here we first consider a boostin the 3-direction B3(ζ) = e−iζM3 generated by M3 = L03 = −i(x0∂3 + x3∂0) [recall(1.57), (1.53), and (1.103)]. Note the positive relative sign of the two terms in the

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18 1 Basics

generator L03 caused by the fact that ∂i = −∂i in contrast to ∂0 = ∂0. Thus weform

D(B3(ζ))xiD−1(B3(ζ)) = e−iζM3xieiζM3 . (1.124)

The Lie expansion of the right-hand side involves the commutators

−i[M3, x0] = −x3, − i[M3, x

3] = −x0, − i[M3, x1] = 0, − i[M3, x

2] = 0. (1.125)

Here the two-vector (x1, x2) is unchanged, while the two-vector (x0, x3) is trans-formed into −(x3, x0). In the second expansion term, the latter becomes (x0, x3),and so on, yielding

e−iζM3(x0, x3)eiζM3 =(

1 +1

2!ζ2 +

1

4!ζ4 + . . .

)

(x0, x3)

−(

ζ +1

3!ζ3 +

1

5!ζ5 + . . .

)

(x3, x0). (1.126)

The right-hand sides can be summed up to hyperbolic cosines and sines:

e−iζM3(x0, x3)eiζM3 = cosh ζ (x0, x3) − sinh ζ (x3, x0). (1.127)

Together with the invariance of (x1, x2), this corresponds precisely to the inverse ofthe boost transformation (1.35):

D(B3(ζ))xaD−1(B3(ζ)) = e−iζM3xaeiζM3 =

(

eiζM3

)a

bxb = B−1

3 (ζ)abxb. (1.128)

1.6.3 Lorentz Group

By performing successively rotations and boosts in all directions we find the or allLorentz transformations

D(Λ)xcD−1(Λ) = e−i12ωabL

ab

xcei12ωabL

ab

= (ei12ωabL

ab

)cc′xc′ = (Λ−1)cc′x

c′, (1.129)

where ωab are the parameters (1.55) and (1.56). In the last term on the right-handside we have expressed the 4 × 4 -matrix Λ as an exponential of its generators aswell, to emphasize the one-to-one correspondence between the generators Lab andtheir differential-operator representation Lab.

At first it may seem surprising that the group transformations appearing as a left-hand factor of the two sides of these equations are inverse to each other. However,we may easily convince ourselves this is necessary to guarantee the correct groupmultiplication law. Indeed, if we perform two transformations after each other theyappear in opposite order on the right- and left-hand sides:

D(Λ2Λ1)xcD−1(Λ2Λ1) = D(Λ2)D(Λ1)x

cD−1(Λ1)D−1(Λ2)

= (Λ−11 )cc′D(Λ2)x

c′D−1(Λ2) = (Λ−11 )cc′(Λ

−12 )c

c′′xc′′ = [(Λ2Λ1)

−1]cc′xc′. (1.130)

If the right-hand side of (1.129) would contain Λ instead of Λ−1, the order of thefactors in Λ2Λ1 on the right-hand side of (1.130) would be opposite to the order inD(Λ2Λ1) on the left-hand side.

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1.7 Relativistic Point Mechanics 19

A straightforward extension of the operation (1.129) yields the transformationlaw for a tensor t c1,...,cn = xc1 · · ·xcn :

D(Λ)t c1,...,cnD−1(Λ) = e−i12ωabL

ab

t c1,...,cn ei12ωabL

ab

= (Λ−1)c1c′1 · · · (Λ−1)cnc′n t

c′1,...,c′

n

= (ei12ωabL

ab

)c1c′1 · · · (ei 12ωabL

ab

)cnc′n tc′1,...,c

n. (1.131)

This follows directly by inserting in the product xc1 · · ·xcn, an auxiliary unit factor1 = D(Λ)D−1(Λ) = e−i

12ωabL

ab

ei12ωabL

ab

between neighboring factors xci and perform-ing the operation (1.131) on each of them. The last term in (1.131) can also bewritten as

[

ei12ωab(L

ab×1×1···×1 + ... + 1×Lab×1···×1)]c1...cn

c′1...c′n t

c′1...c′

n . (1.132)

Since the commutation relations (1.112) determine the result completely, the trans-formation formula (1.131) is true for any tensor operator t c1,...,cn not only thosecomposed from a product of vectors xci .

The result can easily be extended to an exponential function e−ipx and furtherto any function f(x) which possesses a Fourier representation

D(Λ)f(x)D−1(Λ) = f(Λ−1x) = e−i12ωabL

ab

f(x) ei12ωabL

ab

. (1.133)

Since the last differential operator has nothing to act on, it can also be omitted andwe can also write

D(Λ)f(x)D−1(Λ) = f(Λ−1x) = e−i12ωabL

ab

f(x). (1.134)

1.7 Relativistic Point Mechanics

The Lorentz invariance of the Maxwell equations explains the observed invarianceof the light velocity in different inertial frames. It is, however, incompatible withNewton’s mechanics. There exists a modification of Newton’s laws which makesthem Lorentz-invariant as well, while differing very little from Newton’s equationsin their description of slow macroscopic bodies, for which Newton’s equations wereoriginally designed. Let us introduce the Poincare-invariant distance measure inspacetime

ds ≡√dx2 =

(

gabdxadxb

)1/2. (1.135)

At a fixed coordinate point of an inertial frame, ds is equal to c times the elapsedtime:

ds =√

g00dx0dx0 = dx0 = cdt. (1.136)

Einstein called the quantity

dτ ≡ 1

cds (1.137)

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20 1 Basics

the proper time.When going from one inertial frame to another, two simultaneous events at

different points in the first frame will take place at different times in the other frame.Their invariant distance, however, remains the same, due to pseudo-orthogonalityrelation (1.28) which ensures that

ds′ =(

gabdx′adx′b

)1/2=(

gabdxadxb

)1/2= ds. (1.138)

A particle moving with a constant velocity along a trajectory x(t) in one Minkowskiframe remains at rest in another frame moving with velocity v = x(t) with respectto the first. Its proper time is then related to the coordinate time in the first frameby the Lorentz transformation

cdτ = ds =√c2dt2 − dx2 = cdt

√1 − 1

c2

(

dx

dt

)2

= cdt

1 − v2

c2=cdt

γ. (1.139)

This is the famous Einstein relation implying that a moving particle lives longerby a factor γ. There exists direct experimental evidence for this phenomenon. Forexample, the meson π+ lives on the average τa = 2.60 × 10−8 sec, after which itdecays into an muon and a neutrino. If the pion is observed in a bubble chamberwith a velocity equal to 10% of the light velocity c ≡ 299 792 458 m/sec, it leavestrace of an average length l ≈ τa × c× 0.1/

√1 − 0.12 ≈ 0.78 cm. A very fast muon

moving with 90% of the light velocity, on the other hand, leaves a trace which islonger by a factor (0.9/0.1)×

√1 − 0.102/

√1 − 0.92 ≡ 20.6. Massless particles move

with light velocity and have dτ = 0, i.e., the proper time stands still along theirpaths. This implies that massless particles can never decay—they are necessarilystable particles.

Another place to see this time dilation effect is by observing the spectral linesof a moving atom, say a hydrogen atom. If the atom is at rest, the frequency of theline is given by

ν = −Ry(

1

n2− 1

m2

)

(1.140)

where Ry is the Rydberg constant (≈ 13.6 eV), and n and m are the principalquantum numbers of initial and final electron orbits. If the atom emits a lightquantum while moving through the laboratory frame of reference with velocity v

orthogonal to the direction of observation, this frequency is lowered by a factor 1/γ:

νobs

ν=

1

γ=

1 − v2

c2. (1.141)

If the atom runs away from the observer or towards him, the frequency is furtherchanged by the Doppler shift . Due to the growing or decreasing distance, the wavetrains arrive with a smaller of higher frequency given by

νobs

ν=(

1 ± v

c

)−1 1

γ=

1 ∓ v/c

1 ± v/c. (1.142)

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1.7 Relativistic Point Mechanics 21

In the first case the observer sees an additional red shift , in the second a violet shiftof the spectral lines.

Without external forces, the trajectories of free particles are straight lines infour-dimensional spacetime. If the particle positions are parametrized by the propertime τ , they satisfy the equation of motion

d2

dτ 2qa(τ) =

d

dτpa(τ) = 0. (1.143)

The first derivative of qa(τ) is the relativistic four-vector of momentum pa(τ), brieflycalled four-momentum:

pa(τ) ≡ md

dτqa(τ) ≡ mua(τ). (1.144)

On the right-hand side we have introduced the relativistic four-vector of velocityua(τ), or four-velocity . Inserting (1.139) into (1.144) we identify the components ofua(τ) as

ua =

(

γcγva

)

, (1.145)

and see that ua(τ) is normalized to the light-velocity:

ua(τ)ua(τ) = c2. (1.146)

The time and space components of (1.144) are

p0 = mγc = mu0, pi = mγvi = mui. (1.147)

This shows that the time dilation factor γ is equal to p0/mc, and the same factorincreases the spatial momentum with respect to the nonrelativistic momentum mvi.This correction becomes important for particles moving near the velocity of light,which are called relativistic. The light particle has m = 0 and v = c. It is ultra-relativistic.

Note that by Eq. (1.147), the hyperbolic functions of the rapidity in Eq. (1.31)are related to the four velocity and to energy and momentum by

cosh ζ = u0/c = p0/mc, sinh ζ = |u|/c = |p|/mc. (1.148)

Under a Lorentz transformation of space and time, the four-momenta pa trans-form in exactly the same way as the coordinate four-vectors xa. This is, of course,due to the Lorentz invariance of the proper time τ in Eq. (1.144). Indeed, fromEq. (1.147) we derive the important relation

p02 − p2 = m2c2, (1.149)

which shows that the square of the four-momentum taken with the Minkowski metricis an invariant:

p2 ≡ pagabpb = m2c2. (1.150)

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22 1 Basics

Since both xa and pa are Lorentz vectors, the scalar product of them,

xp ≡ gabxapb, (1.151)

is an invariant. In the canonical formalism, the momentum pi is the conjugatevariable to the space coordinate xi. Equation (1.151) suggests that the quantity cp0

is conjugate to x0/c = t. As such it must be the energy of the particle:

E = cp0. (1.152)

From relation (1.149), we calculate the energy as a function of the momentumof a relativistic particle:

E = c√

p2 +m2c2. (1.153)

For small velocities, this can be expanded as

E = mc2 +m

2v2 + . . . . (1.154)

The first term gives a non-vanishing rest energy which is unobservable in non-relativistic physics. The second term is Newton’s kinetic energy.

The first term has dramatic observable effects. Particles can be produced anddisappear in collision processes. In the latter case, their rest energy mc2 can betransformed into kinetic energy of other particles. The large factor c makes unstableparticles a source of immense energy, which led to the atomic disaster of Hiroshimaand Nagasaki in 1945.

1.8 Quantum Mechanics

In quantum mechanics, free spinless particles of momentum p are described by planewaves of the form

φp(x) = N e−ipx/h, (1.155)

where N is some normalization factor. The momentum components are the eigen-value of the differential operators

pa = ih∂

∂xa, (1.156)

which satisfy with xb the commutation rules

[pa, xb] = ihδa

b. (1.157)

In terms of these, the generators (1.103) can be rewritten as

Lab ≡ 1

h(xapb − xbpa). (1.158)

Apart from the factor 1/h, this is the tensor version of the four-dimensional angularmomentum.

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1.9 Relativistic Particles with Electromagnetic Interactions 23

It is worth observing that the differential operators (1.158) can also be expressedas a sandwich of the 4 × 4 matrix generators (1.51) between xc and pd:

Lab = − i

h(Lab)cdx

cpd = − i

hxTLabp = ipTLabx. (1.159)

This way of forming operator representations of the 4 × 4 Lie algebra (1.71) is aspecial application of a general construction technique of higher representations ofa defining matrix representations. In fact, the procedure of second quantization isbased on this construction, which extends the single-particle Schrodinger operatorsto the Fock space of many-particle states.

In general, one may always introduce vectors of creation and annihilation oper-ators a†c and ad with the commutation rules

[ac, ad] = [a†c, a†d] = 0; [ac, a†d] = δcd, (1.160)

and form sandwich operators

Lab = a†c(Lab)cda

d. (1.161)

These satisfy the same commutation rules as the sandwiched matrices due to theLeibnitz chain rule (1.113). Since −ipa/h and xa commute in the same way as a anda†, the commutation rules of the matrices go directly over to the sandwich operators(1.159). The higher representations generated by them lie in the Hilbert space ofsquare-integrable functions.

Under a Lorentz transformation, the momentum of the particle described bythe wave function (1.155) goes over into p′ = Λp, so that the the wave functiontransforms as follows:

φp(x)Λ−→ φ′p(x) ≡ φp′(x) = N e−i(Λp)x = N e−ipΛ

−1x = φp(Λ−1x). (1.162)

This can also be written as φ′p′(x′) = φp(x). An arbitrary superposition of such

waves transforms like

φ(x)Λ−→ φ′(x) = φ(Λ−1x), (1.163)

which is the defining relation for a scalar field .The transformation (1.163) can also be generated by the differential-operator

representation of the Lorentz group (1.134) as follows:

φ(x)Λ−→ φ′(x) = D(Λ)φ(x). (1.164)

1.9 Relativistic Particles with Electromagnetic Interactions

Lorentz and Einstein formulated a theory of relativistic massive particles with elec-tromagnetic interactions called the Maxwell-Lorentz theory . It is invariant under

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24 1 Basics

the Poincare group and describes the dynamical properties of charged particles suchas electrons moving with nonrelativistic and relativistic speeds.

The motion for a particle of charge e and mass m in an electromagnetic field isgoverned by the Lorentz equations

dpa(τ)

dτ= m

d2xa(τ)

dτ 2= fa(τ), (1.165)

where the four-vector fa is the Lorentz force

fa =e

cF a

bdxb

dτ=

e

cmF a

b(x(τ)) pb(τ), (1.166)

and F ab(x) is a 4×4 combination of electric and magnetic fields with the components

F ij = εijkBk, F 0

i = Ei. (1.167)

By raising the second index of F ab one obtains the tensor

F ac = gcbF ab (1.168)

associated with the antisymmetric matrix of the six electromagnetic fields

F ab =

0 −E1 −E2 −E3

E1 0 B3 −B2

E2 −B3 0 B1

E3 B2 −B1 0

(1.169)

This tensor notation is useful since F ab transforms under the Lorentz group in thesame way as the direct product xaxb, which goes over into x′ax′b = Λa

cΛbd x

cxd. InF ab(x), also the arguments must be transformed as in the scalar field in Eq. (1.163),so that we find the generic transformation behavior of a tensor field :

F ab(x)Λ−→ F ′ab(x) = Λa

cΛbd F

cd(Λ−1x). (1.170)

Recalling the exponential representation (1.132) of the direct product of the Lorentztransformations and the differential operator generation (1.134) of the transforma-tion of the argument x, this can also be written as

F ab(x)Λ−→ F ′ab(x) = [e−i

12ωabJ

ab

F ]ab(Λ−1x), (1.171)

whereJ cd ≡ Lcd × 1 + 1 × Lcd (1.172)

are the generators of the total four-dimensional angular momentum of the tensorfield. The factors in the direct products apply successively to the representation

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1.9 Relativistic Particles with Electromagnetic Interactions 25

spaces associated with the two Lorentz indices and the spacetime coordinates. Thegenerators Jab obey the same commutation rules (1.71) and (1.72) as Lab and Lab.

In order to verify the transformation law (1.170), we recall the basic result ofelectromagnetism, that under a change to a coordinate frame x → x′ = Λx movingwith a velocity v, the electric and magnetic fields change as follows

E′‖(x′) = E‖(x), E′⊥(x′) = γ

[

E⊥(x) +1

cv ×B(x)

]

, (1.173)

B′‖(x′) = B‖(x), B′⊥(x′) = γ

[

B⊥(x) − 1

cv × E(x)

]

, (1.174)

where the subscripts ‖ and ⊥ denote the components parallel and orthogonal to v.Recalling the matrices (1.27) we see that (1.173) and (1.174) correspond preciselyto the transformation law (1.170) of a tensor field.

The field tensor in the electromagnetic force of the equation of motion (1.165)transforms accordingly:

F ab(x(τ))

Λ−→ F ′ab(x(τ)) = ΛacΛ

TbdF ′cd(Λ

−1x(τ)). (1.175)

This can be verified by rewriting F ab(x(τ)) as

F ab(x(τ)) =

d4xF ab(x) δ

(4)(x− x(τ)), (1.176)

and applying the transformation (1.170).Separating time and space components of the Lorentz force (1.166) we find

d

dτp0 = f 0 =

e

McE · p, (1.177)

d

dτp = f =

e

Mc

(

E p0 + p ×B)

. (1.178)

The Lorentz force can also be stated in terms of velocity as

fa =e

cF a

bdxb

dτ= γ

e

cv · E

eEi +1

c(v × B)i

. (1.179)

The above equations rule the movement of charged point particles in a givenexternal field. The moving particles will, however, also give rise to additional elec-tromagnetic fields. These are calculated by solving the Maxwell equations in thepresence of charge and current densities ρ and j, respectively:

∇ · E = ρ (Coulomb’s law), (1.180)

∇ × B − 1

c

∂E

∂t=

1

cj (Ampere‘s law), (1.181)

∇ · B = 0 (absence of magnetic monopoles), (1.182)

∇ × E +1

c

∂B

∂t= 0 (Faraday’s law). (1.183)

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In a dielectric and paramagnetic medium with dielectric constant ε and magneticpermeability µ one defines the displacement field D(x) and the magnetic field H(x)by the relations

D(x) = εE(x), B(x) = εB(x), (1.184)

and the Maxwell equations becomes

∇ · D = ρ (Coulomb’s law), (1.185)

∇ ×H − 1

c

∂D

∂t=

1

cj (Ampere‘s law), (1.186)

∇ · B = 0 (absence of magnetic monopoles), (1.187)

∇ ×E +1

c

∂B

∂t= 0 (Faraday’s law). (1.188)

On the right-hand sides of (1.180), (1.181) and (1.185), (1.186) we have omittedfactors 4π, for convenience. This makes the charge of the electron equal to −e =−√

4πα, whereα ≈ 1/137.035 989 (1.189)

is the fine-structure constant.In the vacuum, the two inhomogeneous Maxwell equations (1.180) and (1.181)

can be combined to a single equation

∂bFab = −1

cja. (1.190)

where ja is the four-component current density

ja(x) =

(

cρ(x, t)j(x, t)

)

. (1.191)

Indeed, the zeroth component of (1.190) is equal to (1.180):

∂iF0i = −∇ · E = −ρ, (1.192)

whereas the spatial components with a = i reduce to Eq. (1.181):

∂0Fi0 + ∂jF

ij = ∂jεijkBk +

1

c

∂tEi = − (∇ × B)i +

1

c

∂tEi = −1

cji. (1.193)

The remaining homogeneous Maxwell equations (1.182) and (1.183) can also berephrased in tensor form as

∂b∗

Fab = 0. (1.194)

Here∗

F ab is the so-called dual field tensor defined by

Fab = εabcdFcd. (1.195)

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1.9 Relativistic Particles with Electromagnetic Interactions 27

where εabcd is the totally antisymmetric unit tensor with ε0123 = 1.The antisymmetry of F ab in (1.190) implies the vanishing of the four-divergence

of the current density:

∂aja(x) = 0. (1.196)

This is the four-dimensional way of expressing the local conservation law of charges.Written out in space and time components it reads

∂tρ(x, t) + ∇ · j(x, t) = 0. (1.197)

Integrating this over a finite volume gives

∂t

[∫

d3x ρ(x, t)]

= −∫

d3x∇ · j(x, t) = 0. (1.198)

The right-hand side vanishes by the Gauss divergence theorem, according to whichthe volume integral over the divergence of a current density is equal to the surfaceintegral over the flux through the boundary of the volume. This vanishes if currentsdo not leave a finite spatial volume, which is usually true for an infinite system.Thus we find that, as a consequence of local conservation law (1.196), the charge ofthe system

Q(t) ≡∫

d3 ρ(x, t) ≡ 1

c

d3x j0(x) (1.199)

satisfies the global conservation law that the charge is time-independent

Q(t) ≡ Q. (1.200)

For a set of point particles of charges en, the charge and current densities are

ρ(x, t) =∑

n

enδ(3) (x − xn(t)) , (1.201)

j(x, t) =∑

n

enxn(t)δ(3) (x − xn(t)) . (1.202)

Combining these expressions to a four-component current density (1.191), we caneasily verify that ja(x) transforms like a vector field [compare with the behaviors(1.163) of scalar field and (1.170) of tensor fields]:

ja(x)Λ−→ j′a(x) = Λa

b jb(Λ−1x). (1.203)

To verify this we note that δ(3) (x − x(t)) can also be written as an integral alongthe path of the particle parametrized with the help of proper time τ . This is donewith the help of the identity

∫ ∞

−∞dτ δ(4)(x− x(τ)) =

∫ ∞

−∞dτ δ(x0 − x0(τ))δ(3) (x − x(τ))

=dτ

dx0δ(3) (x − x(t)) =

1

cγδ(3) (x − x(t)) . (1.204)

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28 1 Basics

This allows us to rewrite (1.201) and (1.202) as

cρ (x, t) = c∑

n

∫ ∞

−∞dτnenγnc δ

(4) (x− xn(τ)) , (1.205)

j (x, t) = c∑

n

∫ ∞

−∞dτnenγnvnδ

(4) (x− xn(τ)) . (1.206)

These equations can be combined in a single four-vector equation

ja(x) = c∑

n

∫ ∞

−∞dτnenx

an(τ)δ

(4) (x− xn(τ)) , (1.207)

which makes the transformation behavior (1.203) an obvious consequence of thevector nature of xan(τ).

In terms of the four-dimensional current density, the inhomogeneous Maxwellequation (1.190) becomes the Maxwell-Lorentz equation

∂bFab = −1

cja = −

n

∫ ∞

−∞dτnenx

an(τ)δ

(4) (x− xn(τ)) . (1.208)

It is instructive to verify the conservation law (1.196) for the current density(1.207). Applying the derivative ∂a to the δ-function gives ∂aδ

(4) (x−xn(τ)) =−∂xanδ(4) (x− xn(τ)), and therefore

∂aja(x) =−c

n

∫ ∞

−∞dτnen

dxan(τ)

∂xanδ(4) (x− xn(τ))

=−c∑

n

∫ ∞

−∞dτnen∂τδ

(4) (x− xn(τ)) . (1.209)

If the particle orbits x(τ) are stable, they are either closed in spacetime, or comefrom negative infinite x0 and run to positive infinite x0. Then the right-hand sidevanishes in any finite volume so that the current density is indeed conserved.

We end this section by remarking, that the vector transformation law (1.203)can also be written by analogy with the tensor law (1.170) as

ja(x)Λ−→ j′a(x) = [e−i

12ωabJ

ab

j]a(Λ−1x), (1.210)

where

J cd ≡ Lcd × 1 + 1 × Lcd (1.211)

are the generators of the total four-dimensional angular momentum of the vectorfield. As in (1.172), the factors in the direct products apply separately to the repre-sentation spaces associated with the Lorentz index and the spacetime coordinates,and the generators Jab obey the same commutation rules (1.71) and (1.72) as Laband Lab.

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1.10 Dirac Fields 29

1.10 Dirac Fields

The observable matter of the universe consists mainly of electrons and nucleons,the latter being predominantly bound states of three quarks. Electrons and quarksare spin-1/2 particles which may be described by four-component Dirac fields ψ(x).These obey the Dirac equation

(iγa∂a −m)ψ(x) = 0, (1.212)

where γa are the 4 × 4 Dirac matrices

γa =

(

0 σa

σa 0

)

, (1.213)

in which the 2× 2 submatrices σa and σa with a = 0, . . . , 3 form the four-vectors ofPauli matrices

σa ≡ (σ0, σi), σa ≡ (σ0,−σi). (1.214)

The spatial components σi are the ordinary Pauli matrices

σ1 =

(

0 11 0

)

, σ2 =

(

0 −ii 0

)

, σ3 =

(

1 00 −1

)

. (1.215)

while the zeroth component σ0 is defined as the 2 × 2 unit matrix:

σ0 ≡(

1 00 1

)

. (1.216)

From the algebraic properties of these matrices

(σa)2 = σ0 = 1, σiσj = δij + iεijkσk, σaσb + σbσa = 2gab, (1.217)

we deduce that the Dirac matrices γa satisfy the anticommutation rules

γa, γb

= 2gab. (1.218)

Under Lorentz transformations, the Dirac field transforms according to the spinorrepresentation of the Lorentz group

ψA(x)Λ−→ψ′A(x) = DA

B(Λ)ψB(Λ−1x), (1.219)

by analogy with the transformation law (1.203) of a vector field. The 4×4 -matricesΛ of the defining representation of the Lorentz group in (1.203) are replaced by the4 × 4 -matrices D(Λ) representing the Lorentz group in spinor space.

It is easy to find these matrices. If we denote the spinor representation of the Liealgebra (1.72) by 4 × 4 -matrices Σab, these have to satisfy the commutation rules

[Σab,Σac] = −igaaΣbc, no sum over a. (1.220)

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30 1 Basics

These can be solved by the matrices

Σab ≡ 1

2σab, (1.221)

where σab is the antisymmetric tensor of matrices

σab ≡ i

2[γa, γb]. (1.222)

The representation matrices of finite Lorentz transformations may now be expressedas exponentials of the form (1.54):

D(Λ) = e−i12ωabΣ

ab

, (1.223)

where ωab is the same antisymmetric matrix as in (1.54), containing the rotation andboost parameters as specified in (1.55) and (1.56). Comparison with (1.57) showsthat pure rotations and pure Lorentz transformations are generated by the spinorrepresentations of Lab in (1.57):

Σij = εijk1

2

(

σk 00 σk

)

, Σ0i =i

2

(

−σi 00 σi

)

. (1.224)

The generators of the rotation group Σi = 12εijkΣ

jk corresponding to Li in (1.53)consist of a direct sum of two Pauli matrices, the 4 × 4 -spin matrix:

≡ 1

2

( 0

0)

. (1.225)

The generators Σ0i of the pure Lorentz transformations corresponding to Mi in(1.53) can also be expressed as Σ0i = iαi/2 with the vector of 4 × 4 -matrices

=

(

00

)

. (1.226)

In terms of

and , the representation matrices (1.223) for pure rotations and pureLorentz transformations are seen to have the explicit form

D(R)=e−i· =

(

e−i ·/2 00 e−i ·/2

)

, D(B)=e· =

(

e−·/2 00 e·/2

)

. (1.227)

The commutation relations (1.220) are a direct consequence of the commutationrelations of the generators Σab with the gamma matrices:

[Σab, γc] = −(Lab)cd γd = −i(gacγb − gbcγa). (1.228)

Comparison with (1.110) and (1.111) shows that the matrices γa transform like xa,i.e., they form a vector operator. The commutation rules (1.220) follow directlyfrom (1.228) upon using the Leibnitz chain rule (1.113).

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1.11 Spacetime-Dependent Lorentz Transformations 31

For global transformations, the vector property (1.228) implies that γa behaveslike the vector xa in Eq. (1.129):

D(Λ)γcD−1(Λ) = e−i12ωabΣ

ab

γcei12ωabΣ

ab

= (ei12ωabL

ab

)cc′ γc′ = (Λ−1)cc′ γ

c′. (1.229)

In terms of the generators Σab, we can write the field transformation law (1.219)more explicitly as [compare with the behavior of scalar (1.163), tensor (1.170), andvector fields (1.203)]:

ψ(x)Λ−→ ψ′Λ(x) = D(Λ)ψ(Λ−1x) = e−i

12ωabΣ

ab

ψ(Λ−1x), (1.230)

It is useful to re-express the transformation of the spacetime argument on theright-hand side in terms of the differential operator of four-dimensional angularmomentum and rewrite (1.230) as in (1.172) and (1.211) as

ψ(x)Λ−→ ψ′Λ(x) = D(Λ) ×D(Λ)ψ(x) = e−i

12ωabJ

ab

ψ(x), (1.231)

where

J cd ≡ Σcd × 1 + 1 × Lcd. (1.232)

are the generators of the total four-dimensional angular momentum of the Diracfield.

1.11 Spacetime-Dependent Lorentz Transformations

The theory of gravitation to be developed in this book will not only be Lorentz-invariant, but also invariant under local Lorentz transformations.

x′a = Λab(x)x

b. (1.233)

As a preparation for dealing with such theories let us derive a group-theoretic for-mula which is useful for many purposes.

1.11.1 Angular velocities

Consider a time-dependent 3×3 -rotation matrix R((t)) = e−i(t)·L with the gener-ators (Li)jk = −iεijk [compare (1.43)]. As time proceeds, the rotation angles changewith an angular velocity (t) defined by the following relation

R−1((t)) R((t)) = −i(t) · L. (1.234)

The components of (t) can be specified more explicitly by parametrizing the rota-tions in terms of Euler angles α, β, γ:

R(α, β, γ) = R3(α)R2(β)R3(γ), (1.235)

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32 1 Basics

where R3(α), R3(γ) are rotations around the z-axis by angles α, γ, respectively, andR2(β) is a rotation around the y-axis by β, i.e.,

R(α, β, γ) ≡ e−iαL3e−iβL2e−iγL3 . (1.236)

The relations between the vector of rotation angles in (1.57) and the Euler anglesα, β, γ can be found by purely geometric considerations. Most easily, we equate the2 × 2 representation of the rotations R(),

R() = cosϕ

2− i

· sin

ϕ

2, (1.237)

with the 2 × 2 representation of the Euler decomposition (1.236):

R(α, β, γ) =(

cosα

2− iσ3 sin

α

2

)

(

cosβ

2− iσ2 sin

β

2

)

(

cosγ

2− iσ3 sin

γ

2

)

. (1.238)

The desired relations follow directly from the multiplication rules for the Pauli ma-trices (1.217).

In the Euler decomposition, we may calculate the derivatives:

ih∂αR = R [cos β L3 − sin β(cos γ L1 − sin γ L2)] , (1.239)

ih∂βR = R (cos γ L2 + sin γ L1), (1.240)

ih∂γR = RL3. (1.241)

The third equation is trivial, the second follows from the rotation of the generator

eiγL3/hL2e−iγL3/h = cosαL2 + sin γ L1, (1.242)

which is a consequence of Lie’s expansion formula

eiABe−iA = 1 + i[A,B] +i2

2![A, [A,B]] + . . . , (1.243)

and the commutation rules (1.61) of the 3 × 3 matrices Li. The derivation of thefirst equation (1.239) requires, in addition, the rotation

eiβL2/hL3e−iβL2/h = cosβL3 − sin βL1. (1.244)

We may now calculate the time derivative of R(α, β, γ) using Eqs. (1.239)–(1.241)and the chain rule of differentiation, and find the right-hand side of (1.234) with theangular velocities

ω1 = β sin γ − α sin β cos γ, (1.245)

ω2 = β cos γ + α sin β sin γ, (1.246)

ω3 = α cosβ + γ. (1.247)

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1.12 Energy-Momentum Tensors 33

Only commutation relations have been used to derive (1.239)–(1.241), so that theformulas (1.245)–(1.247) hold for all representations of the rotation group.

The concept of angular velocities can be generalized to spacetime-dependentEuler angles α(x), β(x), γ(x), replacing (1.234) by

R−1((x)) ∂aR((x)) = −ia(x) · L, (1.248)

with the generalization of the vector of angular velocity

ωa;1 = ∂aβ sin γ − ∂aα sin β cos γ, (1.249)

ωa;2 = ∂aβ cos γ + ∂aα sin β sin γ, (1.250)

ωa;3 = ∂aα cosβ + ∂aγ. (1.251)

The derivatives ∂a act only upon the functions right after it. These equations areagain valid if R((x)) and L in (1.248) are replaced by any representation of therotation group and its generators.

A relation of type (1.248) exists also for the Lorentz group where Λ(ωab(x)) =

e−i12ωab(x)L

ab

[recall (1.57)], where the generalized angular velocities are defined by

Λ−1(ωab(x)) ∂cΛ(ωab(x)) = −i 1

2ωc;ab(x)L

ab. (1.252)

Inserting the explicit 4× 4 -generators (1.51) on the right-hand side, we find for thematrix elements the relation

[Λ−1(ωab(x)) ∂cΛ(ωab(x))]ef = ωc;ef(x). (1.253)

As before, the matrices Λ(ωab(x)) and Lab in (1.252) can be replaced by any rep-resentations of the Lorentz group and its generators, in particular in the spinorrepresentation (1.223) where

D−1(Λ(ωab(x))) ∂cD(Λ(ωab(x)) = −i 1

2ωc;ab(x)Σ

ab. (1.254)

1.12 Energy-Momentum Tensors

The four-dimensional current density ja(x) contains all information on the electricproperties of relativistic particle orbits. It is possible to collect also the mechanicalproperties in a tensor, the energy-momentum tensor .

1.12.1 Point Particles

The energy density of the particles can be written as

part

E (x, t) =∑

n

mnγc2δ(3) (x − xn(t)) . (1.255)

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34 1 Basics

We have previously seen that the energy transforms like a zeroth component of afour-vector [recall (1.147)]. The energy density measures the energy per spatial vol-ume element. An infinitesimal four-volume d4x is invariant under Lorentz transfor-mations, due to the unit determinant |Λa

b| = 1 implied by the pseudo-orthogonalityrelation (1.28), so that indeed

d4x′ =

∂x′a

∂xb

d4x = |Λab|d4x = d4x. (1.256)

This shows that δ(3)(x) which transforms like an inverse spatial volume

1

d3x=dx0

d4x(1.257)

behaves like the zeroth component of a four-vector. The energy density (1.255) cantherefore be viewed as a 00-component of a Lorentz tensor called the symmetricenergy-momentum tensor By convention, this is chosen to have the dimension of

momentum density, so that we must identify the energy density with cpart

T ab. Infact, using the identity (1.204), we may rewrite (1.255) as

part

E (x, t) = c∑

n

∫ ∞

−∞dτn

1

mnp0n(τ)p

0n(τ)δ

(4)(x− x(τ)). (1.258)

which is equal to c times the 00-component of the energy-momentum tensor

part

Tab(x, t) =

n

∫ ∞

−∞dτn

1

mnpan(τ)p

bn(τ)δ

(4)(x− x(τ)). (1.259)

The spatial momenta of the particles

part

P i(x, t) =∑

n

mnγnxin(τ)δ

(3) (x − x(τ)) (1.260)

are three-vectors. Their densities transform therefore like 0i-components of aLorentz tensor. Indeed, using once more the identity (1.204), we may rewrite (1.260)as

part

P i(x, t) =part

T0i(x, t) =

n

∫ ∞

−∞dτn

1

mn

p0n(τ)p

in(τ)δ

(4)(x− x(τ)), (1.261)

which shows precisly the tensor character. The four-vector of the total energy-momentum of the many-particle system is given by the integrals over the 0a-components

part

Pa(t) ≡

d3xpart

T0a(x, t). (1.262)

Inserting here (1.258) and (1.261) we obtain the sum over all four-momenta

part

Pa(t) =

n

pan(τ). (1.263)

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1.12 Energy-Momentum Tensors 35

By analogy with the four-dimensional current density ja(x), let us calculate the

four-divergence ∂bpart

T ab. A partial integration yields

n

∫ ∞

−∞dτn p

an(τ)x

bn(τ)∂bδ

(4)(x− x(τ)) = −∑

n

∫ ∞

−∞dτn p

an(τ)∂τδ

(4)(x− x(τ))

= −∑

n

∫ ∞

−∞dτn∂τ

[

pan(τ)δ(4)(x− x(τ)

]

+∑

n

∫ ∞

−∞dτnp

an(τ)δ

(4) (x− x(τ)) . (1.264)

The first term on the right-hand side disappears if the particles are stable, i.e., ifthe orbits are closed or come from negative infinite x0 and disappear into positiveinfinite x0. The derivative pan(τ) in the second term can be made more explicit ifonly electromagnetic forces act on the particles. Then it is equal to the Lorentzforce, i.e., the four-vector fa(τ) of Eq. (1.179), and we obtain

∂bpart

Tab =

n

∫ ∞

−∞dτnfn(τ)δ

(4) (x− x(τ)) (1.265)

=1

c

n

∫ ∞

−∞dτnenF

ab (xn(τ)) xn

b(τ)δ(4) (x− x(τ)) .

Expressed in terms of the current four-vector (1.207), this reads

∂bpart

Tab(x) =

1

c2F a

b(x)jb(x). (1.266)

In the absence of electromagnetic fields, the energy-momentum tensor of the particlesis conserved.

Integrating (1.262) over the spatial coordinates gives the time change of the totalfour-momentum

∂tpart

Pa(t) = c ∂0

[∫

d3xpart

Ta0]

= c∫

d3x ∂bpart

Tab − c

d3x ∂ipart

T0i

=e

c

n

F ab (xn(τ)) x

bn(τ)γn(τ). (1.267)

This agrees, of course, with the Lorentz equations (1.165) since by (1.263)

∂tpart

Pa(t) = ∂t

n

pan(τ) =∑

n

pan(τ) γn. (1.268)

If there are no electromagnetic forces, thenpart

P a is time-independent.

1.12.2 Electromagnetic Field

The electromagnetic field possesses an energy-momentum tensor of its own. Theenergy density is well-known:

E(x) =1

2

[

E2(x) + B2(x)]

= cem

T00(x). (1.269)

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36 1 Basics

The momentum densityem

T 0i(x) is given by the Poynting vector

S(x) = cE(x) × B(x) : (1.270)

and the relation is

Si(x) = c2em

T0i. (1.271)

The densities (1.269) and (1.271) can be combined to the energy-momentum tensor

em

Tab(x) =

1

c

[

−F acF

bc +1

4gabF cdFcd

]

. (1.272)

The four-divergence of this is

∂bem

Tab =

1

c

[

−F ac∂bF

ab − (∂bFac)F

bc +1

4∂a(

F cdFcd)

]

. (1.273)

The second and third terms cancel each other, due to the homogeneous Maxwellequations (1.182) and (1.183). In order to see this, take the trivial identity∂bε

abcdFcd = 2εabcd∂b∂cAd = 0, and multiply this by εaefgFfg. Using the identity[see (1A.23)]

εabcdεaefg = −(

δbeδcfδdg + δceδ

dfδ

bg + δdeδ

bfδcg − δbeδ

dfδcg − δdeδ

cfδbg − δceδ

bfδ

dg

)

, (1.274)

we find

−F cd∂eFcd − F db∂bFed − F bc∂bFce + F dc∂eFcd + F cb∂bFce + F bd∂bFed = 0. (1.275)

Due to the antisymmetry of Fab, this gives

−∂e(

F cdFcd)

+ 4F bd∂bFbd = 0, (1.276)

so that we obtain the conservation law

∂bem

Tab(x) = −1

c

[

F ac(x)∂bF

bc(x)]

= 0, (1.277)

In the last step we have used Maxwell’s equation Eq. (1.190) with zero currents.The timelike component of the conservation law (1.277) reads

∂tem

T00(x) + c ∂i

em

T0i(x) = 0, (1.278)

which can be rewritten with (1.269) and (1.271) as the well-known Poynting law ofenergy flow:

∂t E(x) + ∇ · S(x) = 0. (1.279)

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1.13 Angular Momentum and Spin 37

If currents are present, the Maxwell equation (1.190) changes the conservationlaw (1.277) to

c ∂bem

Tab(x) = −1

cF a

c(x)jc(x) = 0, (1.280)

which modifies (1.279) to

∂t E(x) + ∇ · S(x) = −j(x) ·E(x). (1.281)

A current parallel to the electric field reduces the field energy.In a medium, the energy density and Poynting vector become

E(x) ≡ 1

2[E(x) · D(x) + B(x) · H(x)] , S(x) ≡ cE(x) × H(x), (1.282)

and the conservation law can easily be verified using the Maxwell equations (1.186)and (1.188):

∇ · S(x) = c∇ · [E(x) × H(x)] = c[∇ ×E(x)] · H(x) − cE(x) · [∇ × B(x)]

= ∂tB(x) ·H(x) + E(x) · [∂tD(x) + j(x)] = ∂tE(x) + j(x) · E(x). (1.283)

We now observe that the force on the right-hand side of (1.280) is precisely theopposite of the right-hand side of (1.266), as required by Newton’s third axiom ofactio = reactio. Thus, the total energy-momentum tensor of the combined systemof particles and electromagnetic fields

T ab(x) =part

Tab(x)+

em

Tab(x) (1.284)

has a vanishing four-divergence,

∂bTab(x) = 0 (1.285)

implying that the total four-momentum P a ≡ ∫

d3xT 0a is a conserved quantity

∂tPa(t) = 0. (1.286)

1.13 Angular Momentum and Spin

Similar considerations apply to the total angular momentum of particles and fields.Since T i0(x) is a momentum density, we may calculate the spatial tensor of totalangular momentum from the integral

J ij(t) =∫

d3x[

xiT j0(x) − xjT i0(x)]

. (1.287)

In three space dimensions one describes the angular momentum by a vector J i =12εijkJ jk. The angular momentum (1.287) may be viewed as the integral

J ij(t) =∫

d3x J ij,0(x). (1.288)

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38 1 Basics

over the i, j, 0-component of the Lorentz tensor

Jab,c(x) = xaT bc(x) − xbT ac(x). (1.289)

It is easy to see that due to (1.285) and the symmetry of the energy-momentumtensor, the Lorentz tensor Jab,c(x) is divergenceless in the index c

∂cJab,c(x) = 0. (1.290)

As a consequence, the spatial integral

Jab(t) =∫

d3x Jab,0(x) (1.291)

is a conserved quantity. This is the four-dimensional extension of the conserved totalangular momentum. The conservation of the components J0i is the center-of-masslaw.

A set of point particles with the energy-momentum tensor (1.259) possesses four-dimensional angular momentum

part

Jab(τ) =

n

[

xan(τ)pbn(τ) − xbn(τ)p

an(τ)

]

. (1.292)

In the absence of electromagnetic fields, this is conserved, otherwise the τ -dependence is important.

The spin of a particle is defined by its total angular momentum in its rest frame.It is the intrinsic angular momentum of the particle. Electrons, protons, neutrons,and neutrinos have spin 1/2. For nuclei and atoms, the spin can take much largervalues.

There exists a four-vector Sa(τ) along the orbit of a particle whose spatial partreduces to the angular momentum in the rest frame. It is defined by a combinationof the angular momentum (1.292) and the four-velocity ud(τ) [recall (1.145)]

Sa(τ) ≡ 1

2cεabcd

part

J bc (τ)ud(τ). (1.293)

In the rest frame whereuaR = (c, 0, 0, 0), (1.294)

this reduces indeed to the three-vector of total angular momentum

SaR(τ) = (0,part

J (τ)). (1.295)

For a free particle we find, due to conservation of momentum and total angularmomentum

d

dτud(τ) = 0,

d

part

J bc (τ) = 0, (1.296)

so that Sa(τ) is conserved:d

dτSa(τ) = 0. (1.297)

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1.13 Angular Momentum and Spin 39

The spin four-vector is useful to understand an important phenomenon in atomicphysics called the Thomas precession of the electron spin in an atom. It explains whythe observed fine structure of atomic physics is compatible with the gyromagneticratio close to two of the magnetic moment of the electron.

The relation between the spin vector and the four-vector is displayed by applyingthe pure Lorentz transformation matrix (1.27) to (1.295) yielding

Si = SiR +γ2

γ + 1

vivj

c2SR

j , S0 = γvi

cSiR (1.298)

Note that S0 and Si satisfy S0 = viSi/c, which can be rewritten covariantly as

uaSa = 0. (1.299)

The inverse of the transformation (1.298) is found with the help of the identityv2/c2 = (γ2 − 1)/γ2 as follows:

SiR = Si − γ

γ + 1

vivj

c2Sj = Si − γ − 1

γ

vivj

v2Sj. (1.300)

If external forces act on the system, the spin vector starts moving. This move-ment is called precession. If the point article moves in an orbit under the influenceof a central force (for example, an electron around a nucleus in an atom), there is notorque on the particle so that the total angular momentum in its rest frame is con-served. Hence dSiR(τ)/dτ = 0, which is expressed covariantly as dSa(τ)/dτ ∝ ua(τ).In the rest frame of the atom, however, the spin shows precession. Let us calculateits rate. From the definition (1.293) we have

dSadτ

=1

2εabcd

part

Jbcdu

d

dτ. (1.301)

There is no contribution from

d

part

Jbc = xa(τ)pb(τ) − xb(τ)pa(τ), (1.302)

since p = mu, and the ε-tensor is antisymmetric.The right-hand side of (1.301) can be simplified by multiplying it with the trivial

expression

gstusut = c2, (1.303)

and using the identity for the ε-tensor

εabcdgst = εabcsgdt + εabsdgct + εascdgbt + εsbcdgat, (1.304)

which can easily be verified using its antisymmetry and choosing a, b, c, d to be equalto 0, 1, 2, 3, respectively, in particular for abcd = 0123. After this, the right-handside of (1.301) becomes a sum of the four terms

1

2

(

εabcspart

Jbcusudu′d + εabsd

part

Jbcucu

su′d + εascdpart

Jbcubu

au′d + εsbcdpart

Jbcusuaud

)

.

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40 1 Basics

The first term vanishes, since udud = (1/2)du2/dτ = (1/2)dc2/dτ = 0. The lastterm is equal to −Sd udua/c2. Inserting the identity (1.304) into the second andthird terms, we obtain twice the left-hand side of (1.301). Taking this to the left-hand side, we find the equation of motion

dSadτ

=1

c2Scduc

dτua. (1.305)

Note that on account of this equation, the time derivative dSa/dτ points in thedirection of ua, in accordance with the initial assumption of a torque-free force.

We are now prepared to calculate the rate of the . Denoting in this discussionthe derivatives with respect to the physical time t = γτ by a dot, we can rewrite(1.305) as

S ≡ dS

dt=

1

γ

dS

dτ= − 1

c2

(

S0u0 + S · u)

u =γ2

c2(S · v)v, (1.306)

S0 ≡ dS0

dt=

1

c

d

dt(S · v) =

γ2

c2(S · v) . (1.307)

We now differentiate Eq. (1.300) with respect to the time using the relation γ =γ3vv/c2, and find

SR = S − γ

γ + 1

1

c2S0v − γ

γ + 1

1

c2S0v − γ3

(γ + 1)2

1

c4(v · v)S0 v. (1.308)

Inserting here Eqs. (1.306) and (1.307), we obtain

SR =γ2

γ + 1

1

c2(S · v)v − γ

γ + 1

1

c2S0 v − γ3

(γ + 1)2(v · v)S0v. (1.309)

On the right-hand side we return to the spin vector SR using Eqs. (1.298), and find

SR =γ2

γ + 1

1

c2[(SR · v)v − (SR · v)v] =

T × SR, (1.310)

with the Thomas precession frequency

T = − γ2

(γ + 1)

1

c2v × v. (1.311)

This is a purely kinematic effect. If an electromagnetic field is present, there willbe an additional dynamic precession. For slow particles, it is given by

S ≡ −S × em ≈ ×

(

B − v

c×E

)

, (1.312)

where is the magnetic moment

= gµBS

h=

eg

2McS, (1.313)

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1.13 Angular Momentum and Spin 41

and g the dimensionless gyromagnetic ratio, also called Lande factor . Recall thevalue of the Bohr magneton

µB ≡ eh

2Mc≈ 3.094 × 10−30 C cm ≈ 0.927 × 10−20 erg

gauss≈ 5.788 × 10−8 eV

gauss.

(1.314)

If the electron moves fast, we transform the electromagnetic field to the electronrest frame by a Lorentz transformation (1.173), (1.174), and obtain an equation ofmotion for the spin becomes

SR=×B′ = ×[

γ(

B − v

c× E

)

− γ2

γ + 1

v

c

(

v

c· B)

]

. (1.315)

Expressing via Eq. (1.313), this becomes

SR ≡ −SR × em =

eg

2mcSR ×

[

(

B − v

c× E

)

− γ

γ + 1

v

c

(

v

c· B)

]

, (1.316)

which is the relativistic generalization of Eq. (1.312). It is easy to see that theassociated fully covariant equation is

Sa′ =g

2mc

[

eF abSb +1

mcpaSc

d

dτpc]

=eg

2mc

[

F abSb +1

m2c2paScF

cκpκ

]

. (1.317)

On the right-hand side we have inserted the relativistic equation of motion (1.165)of a point particle in an external electromagnetic field.

If we add to this the torque-free Thomas precession rate (1.305), we obtain thecovariant Bargmann-Michel-Telegdi equation1

Sa′=1

2mc

[

egF abSb +g − 2

mcpaSc

d

dτpc]

=e

2mc

[

gF abSb +g − 2

m2c2paScF

cκpκ

]

. (1.318)

For the spin vector SR in the electron rest frame this implies a change in theelectromagnetic precession rate in Eq. (1.316) to2

dS

dt=

emT × S ≡ (

em +

T) × S (1.319)

with a frequency given by the Thomas equation

em T = − e

mc

[(

g

2−1 +

1

γ

)

B−(

g

2−1

)

γ

γ+1

(

v

c· B)

v

c−(

g

2− γ

γ+1

)

v

c×E

]

.

(1.320)

1V. Bargmann, L. Michel, and V.L. Telegdi, Phys. Rev. Lett. 2 , 435 (1959).2L.T. Thomas, Phil. Mag. 3 , 1 (1927).

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42 1 Basics

The contribution of the Thomas precession is the part without the gyromagneticfactor g:

T = − e

mc

[

−(

1 − 1

γ

)

B +γ

γ+1

1

c2(v · B)v +

γ

γ+1

1

cv × E

]

. (1.321)

This is agrees with the Thomas frequency (1.311) if we insert there the acceleration

v(t) = cd

dt

p

p0=

e

γm

[

E +v

c×B − v

c

(

v

c· E)]

, (1.322)

which follows directly from (1.177) and (1.178).The Thomas equation (1.320) can be used to calculate the time dependence of

the helicity h ≡ SR · v of an electron, i.e., its component of the spin in the directionof motion. Using the chain rule of differentiation,

d

dt(SR · v) = SR · v +

1

v[SR − (v · SR)v]

d

dtv (1.323)

and inserting (1.319) as well as the equation for the acceleration (1.322), we obtain

dh

dt= − e

mcSR⊥ ·

[(

g

2− 1

)

v × B +(

gv

2c− c

v

)

E

]

. (1.324)

where SR⊥ is the component of the spin vector orthogonal to v. This equationshows that for a Dirac electron which has g = 2 the helicity remains constant in apurely magnetic field. Moreover, if the electron moves ultra-relativistically (v ≈ c),the value g = 2 makes the last term extremely small, ≈ (e/mc)γ−2SR⊥ · E, sothat the helicity is almost unaffected by an electric field. The anomalous magneticmoment of the electron a ≡ (g − 2)/2, however, changes this to a finite value ≈−(e/mc)aSR⊥ · E. This drastic effect was used to measure the experimental valuesof a for electrons, positrons, and muons:

a(e−) = (115 965.77± 0.35) × 10−8, (1.325)

a(e+) = (116 030± 120) × 10−8, (1.326)

a(µ±) = (116 616± 31) × 10−8. (1.327)

1.14 Energy-Momentum Tensor of Perfect Fluid

A perfect fluid is defined as an idealized uniform material medium moving withvelocity v(x, t). The uniformity is an acceptable approximation as long as themicroscopic mean free paths are short with respect to the length scale recognizableby the observer. Consider such a fluid at rest. Then the energy-momentum tensorhas no momentum density so that

fluid

T0i = 0. (1.328)

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Appendix 1A Tensor Identities 43

The energy density is given by

cfluid

T00 = c2ρ, (1.329)

where ρ is the mass density.Due to the isotropy, the purely spatial part of the energy-momentum tensor must

be diagonal:

fluid

Tij =

p

cδij , (1.330)

where p is the pressure of the fluid. We can now calculate the energy-momentumtensor of a moving perfect fluid by performing a Lorentz transformation on theenergy-momentum tensor at rest:

fluid

Tab → Λa

cΛbd

fluid

Tcd. (1.331)

Applying to this the Lorentz boosts from rest to momentum p of Eq. (1.34), andexpressing the hyperbolic functions in terms of energy and momentum according toEq. (1.148), we obtain

fluid

T ab =1

c

[(

p

c2+ ρ

)

uaub − pgab]

, (1.332)

where ua is the four-velocity (1.145) of the fluid with uaua = c2.

Appendix 1A Tensor Identities

In the tensor calculus of euclidean as well as Minkowski space in D spacetime di-mensions, a special role is played by the contravariant Levi-Civita tensor

εa1a2... aD , ai = 0, 1, . . . , D − 1. (1A.1)

It is the totally antisymmetric unit tensor with the normalization

ε012... (D−1) = 1. (1A.2)

It vanishes if any two indices coincide, and is equal to ±1 if they differ from thenatural ordering 0, 1, . . . , (D − 1) by an even or odd perturbation. The Levi-Civitatensor serves to calculate a determinant of a tensor tab as follows

det(tab) =1

D!εa1,a2... aDεb1b2... bD ta1b1 · · · taDbD . (1A.3)

In order to see this it is useful to introduce also the covariant version of εa1... aD

defined by

εa1a2... aD ≡ ga1b1ga2b2 . . . gaDbD εb1b2... bD . (1A.4)

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44 1 Basics

It is again a totally antisymmetric unit tensor with

ε012... (D−1) = −1. (1A.5)

The contraction of the two is easily seen to be

εa1... aDεa1... aD = −D!. (1A.6)

Now, by definition, a determinant is a totally antisymmetric sum

det(tab) = εa1... aDta10 · · · taD(D−1). (1A.7)

We may also write

det(tab) εb1... bD = −εa1... aDta1b1 · · · taDbD . (1A.8)

By contracting with εb1...bD and using (1A.6) we find

det(tab) = − 1

D!εa1...aDεb1...bDta1 b1 · · · taD bD , (1A.9)

which agrees with (1A.7).In the same way we can derive the formula

det(

tab)

= − 1

D!εa1...aDεb1...bDta1

b1 · · · taD bD . (1A.10)

Under mirror reflection, the Levi-Civita tensor behaves like a pseudotensor.Indeed, if we subject it to a Lorentz transformation Λa

b, we obtain

ε′ a1... aD = Λa1b1 · · ·ΛaD

bD εb1... bD = det(Λ) εa1... aD (1A.11)

As long as det Λ = 1, the tensor εa1... aD is covariant under Lorentz transformations.If space or time inversion are included, then det Λ = −1, and (1A.11) exhibits thepseudotensor nature of εa1... aD .

We now collect a set of useful identities of the Levi-Civita tensor which will beneeded in this text.

1A.1 Product Formulas

a) D = 2 euclidean space with gij = δij .

The antisymmetric Levi-Civita tensor εij with the normalization ε12 = 1 satisfiesthe identities

εijεkl = δikδil − δilδjk, (1A.12)

εijεik = δjk, (1A.13)

εijεij = 2, (1A.14)

εijδkl = εikδjl + εkjδil. (1A.15)

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Appendix 1A Tensor Identities 45

b) D = 3 euclidean space with gij = δij .

The antisymmetric Levi-Civita tensor εijk with the normalization ε123 = 1 satisfiesthe identities

εijkεlmn = δilδjmδkn + δimδjnδkl + δinδjlδkm,

− δilδjnδkm − δinδjmδkl − δimδjlδkn, (1A.16)

εijkεimn = δjmδkn − δinδkm, (1A.17)

εijkεijn = 2δkn, (1A.18)

εijkεijk = 6, (1A.19)

εijkδlm = εijlδkm + εilkδjm + εljkδim, (1A.20)

c) D = 4 Minkowski Space with metric

gab =

1−1

−1−1

. (1A.21)

The antisymmetric Levi-Civita tensor with the normalization ε0123 = −ε0123 = 1

satisfies the product identities

εabcdεefgh = −

(

δeaδfb δ

gc δhd + δfaδ

gb δhc δ

cd + δgaδ

kb δ

ecδfd + δhaδ

ebδfc δ

gd

+ δfaδebδhc δ

gd + δeaδ

hb δ

gc δfd + δhaδ

gb δfc δ

ed + δgaδ

fb δ

ecδhd

+ δhaδgb δfc δ

ed + δgaδ

fb δ

ecδhd + δfaδ

ebδhc δ

gd + δcaδ

hb δ

gc δfd

− δeaδfb δ

hc δ

gd − δfaδ

hb δ

gc δed − δhaδ

gb δecδfd − δgaδ

cbδfc δ

hd

− δfaδebδgc δhd − δeaδ

gb δhc δ

fd − δgaδ

hb δ

fc δ

ed − δhaδ

fb δ

ecδgd

− δgaδhb δ

fc δ

ed − δhaδ

fb δ

ecδgd − δfaδ

ebδgc δhd − δeaδ

gb δhc δ

fd

)

, (1A.22)

εabcdεafgh = −

(

δfb δgc δhd + δgb δ

hc δ

fd + δhb δ

fc δ

gd − δfb δ

hc δ

gd − δhb δ

gc δfd − δgb δ

fc δ

hd

)

, (1A.23)

εabcdεabgh = −2

(

δgc δhd − δhc δ

gd

)

, (1A.24)

εabcdεabch = −6δhd , (1A.25)

εabcdεabcd = −24, (1A.26)

εabcdgef = εabcegdt + εabcdgcf + εaecdgbf + εebcdgaf . (1A.27)

1A.2 Determinants

a) D = 2 euclidean:

g = det(gij) =1

2!εikεilgijgkl ≡

1

2gijC

ij, (1A.28)

Cij = εikεjlgkl = cofactor,

gij =1

gCij = inverse of gij .

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46 1 Basics

b) D = 3 euclidean:

g = det(gij) =1

3!εiklεjmngijgkmgln = gijC

ij, (1A.29)

Cij =1

2!εiklεjmngkmgln = cofactor,

gij =1

gCij = inverse of gij.

c) D = 4 Minkowski:

g = det(gab) = − 1

4!εabcdεefghgacgbfgcggdh =

1

4gaeCae, (1A.30)

Cae = − 1

3!εabcdεefghgbfgcggdh = cofactor,

gab =1

gCab = inverse of gab.

1A.3 Expansion of Determinants

From Formulas (1A.28)–(1A.30) together with (1A.12), (1A.16), (1A.22), we find

D=2 : det(gij)=1

2![(trg)2 − tr(g2)],

D=3 : det(gij)=1

3![(trg)3 + 2 tr(g3) − 3 trg tr(g2)], (1A.31)

D=4 : det(gab)=1

24[(trg)4 − 6(trg)2 tr(g2)+3[tr(g2)]2+8 tr(g) tr(g3)−6 tr(g4)].

Notes and References

[1] A.A. Michelson, E.W. Morley, Am. J. Sci. 34, 333 (1887), reprinted in Relativ-ity Theory: Its Origins and Impact on Modern Thought ed. by L.P. Williams,J. Wiley and Sons, N.Y. (1968).

[2] The newer limit is 1 km/sec. See T.S. Jaseja, A. Jaxan, J. Murray,C.H. Townes, Phys. Rev. 133, 1221 (1964).

[3] G.F. Fitzgerald, as told by O. Lodge, Nature 46, 165 (1982).

[4] H.A. Lorentz, Zittingsverslag van de Akademie van Wetenschappen 1, 74(1892), Proc. Acad. Sci. Amsterdam 6, 809 (1904).

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 47

[5] J.H. Poincare, Rapports presentes au Congres International de Physique reunia Paris (Gauthier-Villiers, Paris, 1900).

[6] A. Einstein, Ann. Phys. 17, 891 (1905), 18, 639 (1905).

[7] For a list of shortcomings of Newton’s mechanics see the web pagewww.physics.gmu.edu/classinfo/astr228/CourseNotes/ln ch14.htm.

[8] For a derivation of the Baker-Campbell-Hausdorff formula seeJ.E. Campbell, Proc. London Math. Soc. 28, 381 (1897); 29, 14 (1898);H.F. Baker, ibid., 34, 347 (1902); 3, 24 (1905);F. Hausdorff, Berichte Verhandl. Sachs. Akad. Wiss. Leipzig, Math. Naturw.Kl. 58, 19 (1906);J.A. Oteo, J. Math. Phys. 32, 419 (1991).or Chapter 2 in the textbookH. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics,and Financial Markets, World Scientific Publishing Co., Singapore2004, 3rd extended edition, pp. 1–1460 (kl/b5), where kl is short for the URL:www.physik.fu-berlin.de/~kleinert.

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Never confuse movement with action

Ernest Hemingway (1899 - 1961)

2

Action Approach

The most efficient way of describing the physical properties of a system is based on itsaction A. The extrema of A yield the equations of motion, and the sum over all pathsof the phase factors eiA/h renders the quantum-mechanical time evolution amplitude[1, 2]. The sum over all paths is called path integral . Historically, the action approachwas introduced in classical mechanics to economize Newton’s procedure of settingup equations of motion, and to make it applicable to a large variety of mechanicalproblems. In quantum mechanics, the sum over all paths with phase factors eiA/h

replaced and generalized the Schrodinger theory. The path integral runs over allposition and momentum variables at each time and specifies what are called quantumfluctuations . Their size is controlled by Planck’s quantum h, and they are somewhatsimilar to thermal fluctuations whose size is controlled by the temperature T . Inthe limit h → 0, the paths with highest amplitude will run along the extremaof the action, thus explaining the emergence of classical mechanics from quantummechanics.

The pleasant property of the action approach is that it can be generalized directlyto field theory. Classical fields were discovered by Maxwell as a useful concept todescribe the phenomena of electromagnetism. In particular, his equations allow us tostudy the propagation of free electromagnetic waves without considering the sources.In the last century, Einstein constructed his theory of gravity by assuming the metricof spacetime to become a spacetime-dependent field, which can propagate in theform of gravitational waves. In condensed matter physics, fields were introducedin many systems, and Landau made them a universal tool for understanding phasetransitions [3]. Such fields are called order fields. A more recently discovered domainof applications of fields is in the statistical mechanics of grand-canonical ensemblesof line-like objects, such as vortex lines in superfluids and superconductors [4], ordefect lines in crystals [5]. In this context, they are known as disorder fields.

48

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2.1 General Particle Dynamics 49

2.1 General Particle Dynamics

Given an arbitrary classical system with generalized coordinates qn(t) and velocitiesqn(t), the typical action has the form

A[qk] =∫ tb

tadt L (qk(t), qk(t), t) , (2.1)

where L (qk(t), qk(t), t) is called the Lagrangian of the system, which is quadratic inthe velocities qk(t). A Lagrangian with this property is called local in time. If atheory is governed by a local Lagrangian, the action and the entire theory are alsocalled local. The quadratic dependence on q(t) may emerge only after an integrationby parts in the action. For example, − ∫ dt q(t)q(t) is a local term in the Lagrangiansince it turns into

dt q2(t) after a partial integration in the action (2.1).The physical trajectories of the system are found from the extremal principle.

One compares the action for one orbit qk(t) connecting the end points

qk(ta) = qk,a, qk(tb) = qk,b, (2.2)

with that of an infinitesimally different orbit q′k(t) ≡ qk(t) + δqk(t) connecting thesame end points, where δqk(t) is called the variation of the orbit. Since the endpointof qk(t) + δqk(t) are the same as those of qk(t), the variations of the orbit vanish atthe end points:

δq(ta) = 0, δq(tb) = 0. (2.3)

The associated variation of the action is

δA ≡ A[qk + δqk] −A[qk] =∫ tb

tadt∑

k

(

∂L

∂qk(t)δqk(t) +

∂L

∂qk(t)δqk(t)

)

. (2.4)

After an integration by parts, this becomes

δA =∫ tb

tadt∑

k

(

∂L

∂qk− d

dt

∂L

∂qk

)

δqn(t) +∑

k

∂L

∂qkδqk(t)

tb

ta

. (2.5)

In going from (2.4) to (2.5) one uses the fact that by definition of δqk(t) the variationof the time derivative is equal to the time derivative of the variation:

δqk(t) = q′k(t) − qk(t) =d

dt[qk(t) + δqk(t)] − qk(t) =

d

dtδqk(t). (2.6)

Expressed more formally, the time derivative commutes with the variations of theorbit:

δd

dtqk(t) ≡

d

dtδqk(t). (2.7)

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50 2 Action Approach

Using the property (2.3), the boundary term on the right-hand side of (2.5) vanishes.Since the action is extremal for a classical orbit, δA must vanish for all variationsδqk(t), implying that qk(t) satisfies the Euler-Lagrange equation

∂L

∂qk(t)− d

dt

∂L

∂qk(t)= 0, (2.8)

which is the equation of motion of the system. It is a second-order differentialequation for the orbit qk(t).

The local Lagrangian of a set of gravitating mass points is

L(x(t), x(t)) =∑

k

mk

2x2k(t) +GN

k 6=k′

mkmk′

|xk(t) − xk′(t)|. (2.9)

If we identify the 3N coordinates xin (n = 1, . . . , N) with 3N generalized coordinatesqk (k = 1, . . . , 3N), the Euler-Lagrange equations (2.8) reduce precisely to Newton’sequations (1.2).

The energy of a general Lagrangian system is found from the Lagrangian byforming the so-called Hamiltonian. It is defined by the Legendre transform

H =∑

k

pkqk − L, (2.10)

where

pk ≡∂L

∂qk(2.11)

are called canonical momenta. The energy (2.10) forms the basis of the Hamiltonianformalism. If expressed in terms of pk, qk, the equations of motion are

qk =∂H

∂pk, pk = −∂H

∂qk. (2.12)

For the Lagrangian (2.9), the generalized momenta are equal to the physical mo-menta pn = mnxn, and the Hamiltonian is given by Newton’s expression

H = T + V ≡∑

k

mk

2x2k −GN

k 6=k′

mkmk′

|xk − xk′|(2.13)

The first term is the kinetic energy, the second the potential energy of the system.

2.2 Single Relativistic Particle

For a single relativistic massive point particles, the mechanical action reads

m

A=∫ tb

tadt

m

L = −mc2∫ tb

tadt

1 − x2(t)

c2. (2.14)

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2.2 Single Relativistic Particle 51

The canonical momenta (2.11) yield directly the spatial momenta of the particle:

p(t) =∂

m

L

∂ x(t). (2.15)

In a relativistic notation, the derivative with respect to the contravariant vectors

∂m

L/∂ xi is a covariant vector with a lower index i. To ensure the nonrelativisticidentification (2.15) and maintain the relativistic notation we must therefore identify

pi ≡ − ∂m

L

∂ xi= mγ xi. (2.16)

The energy is obtained from the Legendre transform:

m

H = px−m

L= −pixi−m

L= mγ v2 +mc2√

1 − v2

c2= mγ v2 +mc2

1

γ

= mγc2, (2.17)

in agreement with the energy in Eq. (1.152) [recalling (1.147)].The action (2.14) can be written in a more covariant form by observing that

∫ tb

tadt

1 − x2

c2=

1

c

∫ tb

tadt

(

dx0

dt

)2

−(

dx

dt

)2

. (2.18)

In this expression, the infinitesimal time element dt can be replaced by an arbitrarytime-like parameter t→ σ = f(t), so that the action takes the more general form

m

A =∫ σb

σadσ

m

L = −mc∫ σb

σadσ√

gabxq(σ)xb(σ). (2.19)

For this action we may define generalized four-momentum with respect to the pa-rameter τ by forming the derivatives

pσ,a ≡ − ∂m

L

∂ xa(σ)=

mc√

gabxa(σ)xb(σ)gabx

b(σ), (2.20)

where the dots denote the derivatives with respect to the argument. Note the minussign in the definition of the canonical momentum with respect to the nonrelativisticcase. This is introduced make the canonical formalism compatible with the negativesigns in the spatial part of the Minkowski metric (1.29). The derivatives with respectto xa transforms like the covariant components of a vector with a subscripts a,whereas the physical momenta are given by the contravariant components pa.

If σ is chosen to be the proper time τ , then the square root in (2.86) becomesτ -independent

gabxa(τ)xb(τ) = c, (2.21)

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52 2 Action Approach

so that

pτ,a = mgabxb(τ) = mxa(τ) = mua(τ), (2.22)

in agreement with the previously defined four-momenta in Eq. (1.147).In terms of the proper time, the Euler-Lagrange equation reads:

d

dτpτ,a = m

d

dτgabx

b(τ) = mx a(τ) = 0. (2.23)

Free particles run along straight lines in Minkowski space.Note that the Legendre transform with respect to the momentum pσ,0 has nothing

to do with the physical energy. In fact, it vanishes identically:

m

Hσ = −pσa xa(σ)− m

L= − mc√

xa(σ)xa(σ)xa(σ)xa(σ) +mc

xa(σ)xa(σ) ≡ 0. (2.24)

The reason for this is the invariance of the action (2.19) under arbitraryreparametrizations of the time σ → σ′ = f(σ). We shall understand this betterin Chapter 3 when discussing generators of continuous symmetry transformationsin general. See in particular Subsection 3.5.3.

The role of the physical energy is played by pτ,0 = mcγ. It is equal to 1/c timesthe energy H in (2.17), as it should.

2.3 Scalar Fields

The free classical point particles of the last section are quanta of a relativistic localscalar free-field theory.

2.3.1 Locality

Generalizing the concept of temporal locality described after Eq. (2.1), locality infield theory implies that the action is a spacetime integral over the Lagrangiandensity :

A =∫ tb

tadt∫

d3xL(x) =1

c

d4xL(x) =1

c

d4xL(φ(x), ∂µφ(x)). (2.25)

According to the concept of temporal locality in Section 2.1, there should only be aquadratic dependence on the time derivatives of the fields φ(x),φ∗(x). Due to theequal footing of space and time in relativistic theories, the same restriction appliesto the space derivatives. A local Lagrangian density can at most be quadratic in thefirst spacetime derivatives of the fields at the same point. Physically this impliesthat a field at a point x interacts at most with the field at the infinitesimally closeneighbor point x + dx, just like the mass points on a linear chain with nearest-neighbor spring interactions. If the derivative terms are not of this form, they must

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2.3 Scalar Fields 53

at least be equivalent to it by a partial integration in the action integral (2.25). Ifthe Lagrangian density is local we call also the action and the entire theory local.

A local free-field Lagrangian density is quadratic in both the fields and theirderivatives at the same point, so that it reads, for a scalar field,

L(x) =1

2

[

h2∂aφ(x)∂aφ(x) −m2c2φ(x)φ(x)]

. (2.26)

If the particles are charged, the fields are complex, and the Lagrangian densitybecomes

L(x) = h2∂aφ∗(x)∂aφ(x) −m2c2φ∗(x)φ(x). (2.27)

2.3.2 Lorenz Invariance

In addition to being local, any relativistic Lagrangian density L(x) msut be a scalar,i.e., transforms under Lorentz transformations in the same way as the scalar fieldφ(x) in Eq. (1.163):

L(x)Λ−→ L′(x) = L(Λ−1x). (2.28)

We verify this by showing that L′(x′) = L(x). By definition, L′(x′) is equal to

L′(x′) = h2∂′aφ′∗(x′)∂′aφ(x′) −m2c2φ′∗(x′)φ′(x′). (2.29)

Using the transformation behavior of the scalar field φ(x) in Eq. (1.163), we obtain

L′(x′) = h2∂′aφ∗(x)∂′aφ(x) −m2c2φ∗(x)φ(x). (2.30)

Inserting here

∂′a = Λab∂b, ∂′a = Λa

b∂b (2.31)

withΛa

b ≡ gacgbdΛc

d, (2.32)

we see that ∂2 is Lorentz-invariant,

∂′2 = ∂2, (2.33)

so that the transformed Lagrangian density (2.29) coincides indeed with the originalone in (2.27).

As a spacetime integral over a scalar Lagrangian density, the action (2.25) isLorentz invariant. This follows directly from the Lorentz invariance of the spacetimevolume element

dx′0d3x′ = d4x′ = d4x, (2.34)

proved in Eq. (1.256). This is verified by direct calculation:

A′ =∫

d4xL′(x) =∫

d4x′ L′(x′) =∫

d4x′ L(x) =∫

d4xL(x) = A. (2.35)

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54 2 Action Approach

2.3.3 Field Equations

The equation of motion for the scalar field is derived by varying the action (2.25)with respect to the fields φ(x), φ∗(x) independently. The independence of the fieldvariables is expressed by the functional differentiation rules

δφ(x)

δφ(x′)= δ(4)(x− x′),

δφ∗(x)

δφ∗(x′)= δ(4)(x− x′),

δφ(x)

δφ∗(x′)= 0,

δφ∗(x)

δφ(x′)= 0. (2.36)

With the help of these rules and the Leibnitz chain rule (1.114), we calculate thefunctional derivative of the action (2.25) as follows:

δAδφ∗(x)

=∫

d4x′[

h2∂′aδ(4)(x′ − x)∂′aφ(x′) −m2c2δ(4)(x′ − x)φ(x′)

]

= (−h2∂2 −m2c2)φ(x) = 0. (2.37)

Similarly

δAδφ(x)

=∫

d4x′[

h2∂′aφ∗(x′)∂′aδ(4)(x′ − x) −m2c2φ∗(x′)δ(4)(x′ − x)

]

= φ∗(x)(−h2←∂

2 +m2c2) = 0, (2.38)

where the arrow pointing to the left on top of the last derivative indicates that itacts on the field to the left. The second equation is just the complex conjugate ofthe previous one.

The field equations can also be derived directly from the Lagrangian density(2.27) by forming ordinary partial derivatives of L with respect to all fields andtheir derivatives. Indeed, a functional derivative of a local action can be expandedin terms of derivatives of the Lagrangian density according to the general rule

δAδφ(x)

=∂L(x)

∂φ(x)− ∂a

∂L(x)

∂ [∂aφ(x)]+ ∂a∂b

∂L(x)

∂ [∂a∂bφ(x)]+ . . . , (2.39)

and a complex-conjugate expansion for φ∗(x). These expansions follow directly fromthe defining relations (2.36). At the extremum of the action, the field satisfies theEuler-Lagrange equation

∂L(x)

∂φ(x)− ∂a

∂L(x)

∂∂aφ(x)+ ∂a∂b

∂L(x)

∂∂a∂bφ(x)= 0 (2.40)

Inserting the Lagrangian density (2.27), we obtain the field equation for φ(x)

δAδφ∗(x)

=∂L(x)

∂φ∗(x)− ∂a

∂L(x)

∂ [∂aφ∗(x)]= (−h2∂2 +m2c2)φ(x) = 0, (2.41)

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2.3 Scalar Fields 55

and its complex conjugate for φ∗(x).The Euler-Lagrange equations are invariant under partial integrations in the

action integral (2.25). Take for example a Lagrangian density which is equivalentto (2.27) by a partial integration:

L = −h2φ∗(x)∂2φ(x) −m2c2φ∗(x)φ(x). (2.42)

Inserted into (2.40), the field equation for φ(x) becomes particularly simple:

δAδφ∗(x)

=∂L(x)

∂φ∗(x)= (−h2∂2 +m2c2)φ(x) = 0. (2.43)

The equation for φ∗(x), on the other hand, becomes more complicated. All deriva-tives written out in (2.39) have to be evaluated, but at the end we simply findcomplex-conjugate equation, as before:

δAδφ(x)

=∂L(x)

∂φ(x)− ∂a

∂L(x)

∂ [∂aφ(x)]+ ∂a∂b

∂L(x)

∂ [∂a∂bφ(x)]= (−h2∂2 +m2c2)φ∗(x) = 0.

(2.44)

2.3.4 Plane Waves

The field equations (2.43) and (2.44) are solved by the quantum mechanical planewaves

fp(x) = N e−ipx/h, f ∗p (x) = N eipx/h, (2.45)

where the four-momenta satisfy the so-called mass shell condition

papa −m2c2 = 0, (2.46)

and N is some normalization factor. It is important to realize that the two sets ofsolutions (2.45) are independent of each other. They differ by the sign of energy

i∂0fp(x) = p0fp(x), i∂0f∗p(x) = −p0f ∗p(x). (2.47)

For this reason they will be referred to as positive- and negative-frequency wavefunctions, respectively. The physical significance of the latter can only be understoodafter quantizing the field, where they turn out to be associated with antiparticles.Field quantization, however, lies outside the scope of this text. Only at the end, inSubsection 21.2, will its effects on gravity be discussed.

2.3.5 Schrodinger Quantum Mechanics as Nonrelativistic Limit

The nonrelativistic limit of the action (2.25) is obtained by removing from thepositive frequency part of the field φ(x) a rapidly oscillating factor corresponding tothe rest energy mc2, replacing

φ(x) → e−imc2 t/h 1√

2Mψ(x, t). (2.48)

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56 2 Action Approach

For a plane wave fp(x) in (2.45), the field ψ(x) becomes N√

2Me−i(p0c−mc2)t/heipx/h.

In the limit of large c, the first exponential becomes e−ip2t/2M [recall (1.154)]. The

result is a plane-wave solution to the Schrodinger equation[

ih∂t +h2

2M∂x

2

]

ψ(x, t) = 0. (2.49)

This is the Euler-Lagrange equation extremizing the nonrelativistic action

A =∫

dtd3xψ∗(x, t)

[

ih∂t +h2

2M∂x

2

]

ψ(x, t). (2.50)

Note that the plane wave f ∗p (x) in (2.45) with negative frequency does not pos-

sess a nonrelativistic limit since it turns into N√

2Mei(p0c+mc2)t/heipx/h which has

a temporal prefactor e2imc2t/h. This oscillates infinitely rapidly for c → ∞, and is

therefore equivalent to zero by the Riemann-Lebesgue Lemma.1

2.3.6 Natural Units

The appearance of the constants h and c in all future formulas can be avoided byworking with fundamental units l0, m0, t0, E0 different from the ordinary cgs units.They are chosen to give h and c the value 1. Expressed in terms of the conventionallength, time, mass, and energy, these natural units are

l0 =h

mc, t0 =

h

mc2, m0 = M, E0 = mc2. (2.51)

If, for example, the particle is a proton with mass mp, these units are

l0 = 2.103138 × 10−11cm (2.52)= Compton wavelength of proton,

t0 = l0/c = 7.0153141 × 10−22sec (2.53)= time it takes light to cross the Compton wavelength,

m0 = mp = 1.6726141 × 10−24g, (2.54)

E0 = 938.2592 MeV. (2.55)

For any other mass, they can easily be rescaled.With these natural units we can drop c and h in all formulas and write the action

simply as

A =∫

d4xφ∗(x)(−∂2 −m2)φ(x). (2.56)

Actually, since we are dealing with relativistic particles, there is no fundamentalreason to assume φ(x) to be a complex field. In nonrelativistic field theory, this wasnecessary in order to construct a term linear in the time derivative

dtd3xψ∗(x, t)ih∂tψ(x, t) (2.57)

1This statement holds in the sense of distributions. Any integral over an infinitely rapidlyoscillating function multiplied by a smooth function yields zero.

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2.3 Scalar Fields 57

in the action (2.50). For a real field, this would have been a pure surface term andthus not influenced the dynamics of the system. The second-order time derivativesof a relativistic field in (2.56), however, lead to the correct field equation for a realfield. As we shall understand better in the next chapter, the complex scalar fielddescribes charged spinless particles, the real field neutral particles.

Thus we may also consider a real scalar field with an action

A =1

2

d4xφ(x)(−∂2 −m2)φ(x). (2.58)

In this case it is customary to use a prefactor 1/2 to normalize the field.For either Lagrangian (2.56) or (2.58), the Euler-Lagrange equation (2.40) be-

comes the Klein-Gordon equation

(−∂2 −m2)φ(x) = 0. (2.59)

2.3.7 Hamiltonian Formalism

It is possible to set up a Hamiltonian formalism for the scalar fields. For this weintroduce an appropriate generalization of the canonical momentum (2.11). Thelabels k in that equation are now replaced by the continuous spatial labels x, sothat we define a density of field momentum:

π(x) ≡ ∂L∂∂0φ(x)

= ∂0φ∗(x), π∗(x) ≡ ∂L

∂∂0φ∗(x)= ∂0φ(x), (2.60)

and a Hamiltonian density

H(x) = π(x) ∂0φ(x) + ∂0φ(x) π∗(x) − L(x)

= π∗(x) π(x) + ∇φ∗(x) ∇φ(x) +m2φ∗(x)φ(x). (2.61)

For a real field, we simply drop the complex conjugation symbols. The spatialintegral over H(x) the field Hamiltonian

H =∫

d3xH(x). (2.62)

2.3.8 Conserved Current

For a complex field, there exists an important local conservation law. We define thefour-vector of probability current density

ja(x) =i

2φ∗↔

∂a φ, (2.63)

which described the probability flow of the charged scalar particle. The double

arrow on top of the derivative is defined by↔∂a≡

→∂a −

←∂a as in the nonrelativistic

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58 2 Action Approach

Eq. (2.70). It is easy to verify that on behalf of the Klein-Gordon equation (2.59),the current density has no four-divergence:

∂aja(x) = 0. (2.64)

This current conservation law permits us to couple electromagnetism to the fieldand identify eja(x) as the electromagnetic current of the charged scalar particles.

The deeper reason for the existence of a conserved current will be understood inChapter 3, where we shall see that it is intimately connected with an invariance ofthe action (2.56) under arbitrary changes of the phase of the field

φ(x) → e−iαφ(x). (2.65)

The zeroth component of ja(x),

ρ(x) = j0(x), (2.66)

describes the charge density. The spatial integral over ρ(x) is the total probability.It measures the total charge in natural units:

Q(t) =∫

d3x j0(x). (2.67)

Because of the local conservation law (2.64), the total charge does not depend ontime. This is seen by rewriting

Q(t) =∫

d3x ∂0j0(x) =

d3x ∂aja(x) −

d3x ∂iji(x) = −

d3x ∂iji(x). (2.68)

The right-hand side vanishes due to the Gauss divergence theorem, assuming thecurrents to vanish at spatial infinity [compare (1.198)].

For the solutions ψ(x, t) of the Schrodinger equation (2.49), the probability den-sity is

ρ(x, t) ≡ ψ∗(x, t)ψ(x, t) (2.69)

and the particle current density

j(x, t) ≡ i

2mψ∗(x, t)(

→∇ −

←∇)ψ(x, t) ≡ i

2mψ∗(x, t)

↔∇ψ(x, t). (2.70)

They satisfy the conservation law

∂tρ(x, t) = −∇ · j(x, t). (2.71)

It is this property which permits normalizing the Schrodinger field ψ(x, t) to unityfor all times, since

∂t

d3xψ∗(x, t)ψ(x, t) =∫

d3x ∂tρ(x, t) = −∫

d3x∇ · j(x, t) = 0. (2.72)

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2.4 Maxwell’s Equation from Extremum of Field Action 59

2.4 Maxwell’s Equation from Extremum of Field Action

The above action approach is easily generalized to apply to electromagnetic fields.By setting up an appropriate action, Maxwell’s field equations can be derived byextremization. The relevant fields are the Coulomb potential A0 (x, t) and the vectorpotential A (x, t). Recall that electric and magnetic fields E(x) and B(x) can bewritten as derivatives of the Coulomb potential A0(x, t) and the vector potentialA(x, t) as

E(x) = −∇A0(x) − 1

cA(x), (2.73)

B(x) = ∇ × A(x), (2.74)

with the components

Ei(x) = −∂iA0(x) − 1

c∂tA

i(x), (2.75)

Bi(x) = εijk∂jAk(x). (2.76)

With the identifications (1.167) of electric and magnetic fields with the componentsF i0 and −F jk of the covariant field tensor F ab, we can also write

F i0(x) = ∂iA0(x) − 1

c∂tA

i(x), (2.77)

F jk(x) = ∂jAk(x) − ∂kAj(x), (2.78)

where ∂i = −∂i. This suggests combining the Coulomb potential and the vectorpotential into a four-component vector potential

Aa(x) =

(

A0 (x, t)Ai (x, t)

)

, (2.79)

in terms of which the field tensor is simply the four-dimensional curl:

F ab(x) = ∂aAb(x) − ∂bAa(x). (2.80)

The field Aa(x) transforms in the same way as the vector field ja(x) in Eq. (1.203):

Aa(x)Λ−→ A′a(x) = Λa

bAb(Λ−1x). (2.81)

2.4.1 Electromagnetic Field Action

Maxwell’s equations can be derived from the electromagnetic field action

em

A =1

c

d4xem

L (x), (2.82)

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60 2 Action Approach

where the temporal integral runs from ta to tb, as in (2.1) and (2.25), and theLagrangian density reads

em

L (x) ≡em

L(

Aa(x), ∂bAa(x))

= −1

4F ab(x)Fab(x) −

1

cja(x)Aa(x). (2.83)

It depends quadratically on the fields Aa(x) and its derivatives, thus defining a localfield theory. All Lorentz indices are fully contracted.

If (2.83) is decomposed into electric and magnetic parts using Eqs. (2.73) and(2.74), it reads

em

L (x) =1

2

[

E2(x) − B2(x)]

− ρ(x)A0(x) +1

cj(x)A(x), (2.84)

From the transformation laws (1.170), (1.203), and (2.81) it follows that (2.83)transforms like a scalar field under Lorentz transformations as in (2.28). Togetherwith (2.34), this implies that the action is Lorentz-invariant.

The field equations are obtained from the Euler-Lagrange equation (2.40) withthe field A0(x) replaced by the four-vector potential Aa(x), so that it reads

∂L∂Aa

− ∂b∂L

∂b∂Aa+ ∂b∂c

∂L(x)

∂ [∂b∂cAa(x)]= 0. (2.85)

Inserting the Lagrangian density (2.83), we obtain

∂bFab = −1

cja, (2.86)

which is precisely the inhomogeneous Maxwell equation (1.190). Note that thehomogeneous Maxwell equation (1.194),

∂b∗

Fab

= 0, (2.87)

is automatically fulfilled by the antisymmetric combination of derivatives in the four-curl (2.80). This is true as long as the four-component vector potential is smoothand single-valued, so that it satisfies the integrability condition

(∂a∂b − ∂b∂a)Ac(x) = 0. (2.88)

In this book we shall call any identity following from the single-valuedness of a fieldand the associated Schwarz integrability condition a Bianchi identity . The nameemphasizes the close analogy with the identity discovered by Bianchi in Riemanniangeometry as a consequence of the single-valuedness of the Christoffel symbols. Forthe derivation see Section 12.5, where the Schwarz integrability condition (12.111)leads to Bianchi’s identity (12.120).

In this sense, the homogeneous Maxwell equation (2.87) is a Bianchi identity,since it follows directly from the commuting derivatives of Ac in Eq. (2.88).

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2.4 Maxwell’s Equation from Extremum of Field Action 61

2.4.2 Alternative Action for Electromagnetic Field

There exists an alternative form of the electromagnetic Lagrangian density (2.83)due to Schwinger which contains directly the field tensor as independent variablesand uses the vactor potential only as Lagrange multipyers to enforce the inhomoge-neous Maxwell equations (2.86):

em

L (x) =em

L (Aa(x), Fab(x)) = −1

4F ab(x)Fab(x) −

1

c

[

ja(x) + ∂bFab(x)

]

Aa(x). (2.89)

Extremizing this with respect to Fab show that Fab is a four-curl of the vectorpotential, as in Eq. (2.80). As a consequence, Fab satisfies the Bianchi identity(1.195).

If (2.89) is decomposed into electric and magnetic parts, it reads

em

L (x) =em

L(

A0(x),A(x),E(x),B(x))

=1

4

[

E2(x) − B2(x)]

+ [∇ · E(x) − ρ(x)]A0(x) −[

∇ × B(x) − 1

c∂tE(x) − 1

cj(x)

]

·A(x), (2.90)

where the Lagrange multiplyers A0(x) and A(x) enforce directly the Coulomb law(1.180) and the Ampere law (1.181).

The above equations hold only in the vacuum. In homogeneous materials withnonzero dielectric constant ε and magnetic permeability µ determining the diplace-ment fields D = εE and the magnetic fields H = B/µ, the Lagrangian density (2.89)reads

em

L (x) =1

4[E(x)·D(x) − B(x)·H(x)]

+ [∇ ·D(x) − ρ(x)]A0(x) −[

∇ × H(x) − 1

c∂tD(x) − 1

cj(x)

]

·A(x). (2.91)

Now variation with respect to the Lagrange multiplyers A0(x) and A(x) yields theCoulomb and Ampere laws in a medium Eqs. (1.185) and (1.186):

∇ · D(x) = ρ(x), ∇ × H(x) − 1

c∂tD(x) =

1

cj(x). (2.92)

Variation with respect to D(x) and H(x) yields the same curl equations (2.73) and(2.74) as in the vacuum, so that the homogeneous Maxwell equations (1.182) and(1.183), i.e., the Bianchi identities (1.195), are unaffected by the medium.

2.4.3 Hamiltonian of Electromagnetic Fields

As in Eqs. (2.60)–(2.62), we can find a Hamiltonian for the electromagnetic fields,by defining a density of field momentum:

πa(x) =∂

em

L∂∂0Aa(x)

= −F0a(x), (2.93)

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62 2 Action Approach

and a Hamiltonian densityem

H (x) = πa(x) ∂0Aa(x)− em

L (x). (2.94)

It is important to realize that ∂em

L /∂A0 vanishes, so that A0 possesses no conjugatefield momentum. This implies that it is not a proper dynamical variable. Indeed,by inserting (2.83) and (2.93) into (2.94) we find

em

H = −F0a∂0Aa− em

L= −1

cF0aF

0a− em

L −F0a∂aA0

=1

2

(

E2 + B2)

+ E · ∇A0 +1

cjaAa. (2.95)

Integrating this over all space gives

em

H = c∫

d3xem

H =∫

d3x[

1

2

(

E2 + B2)

− 1

cj · A

]

. (2.96)

The result is the well-known energy of the electromagnetic field in the presence ofexternal currents [6]. To obtain this expression from (2.95), an integration by partis necessary, in which the surface terms at spatial infinity is neglected, where thecharge density ρ(x) is always assumed to be zero. After this, Coulomb’s law (1.180)leads directly to (2.96).

At first sight, one may wonder why the electrostatic energy does not show upexplicitly in (2.96). The answer is that it is contained in the E2-term which, byCoulomb’s law (1.180), satisfies

∇ · E = −∇2A0 − 1

c∂t∇ · A = ρ . (2.97)

Splitting E into transverse and longitudinal parts

E = Et + El, (2.98)

which satisfy ∇ · Et = 0 and ∇ × El = 0, respectively, we see that (2.97) implies

∇ · El = ρ . (2.99)

The longitudinal part can be written as a derivative of some scalar potential φ′,

El = ∇φ′ (2.100)

which, due to (2.99), can be calculated from the equation

φ′(x) =1

∇2ρ(x) = −

d3x′1

4π|x − x′|ρ(x′, t). (2.101)

Using this we see that

1

2

d3xE2 =1

2

d3x(

Et2 + El

2)

=1

2

d3x[

E2t +

(

∂i1

∇2φ′)(

∂i1

∇2φ′)]

=1

2

d3xEt2 +

1

2

d3xd3x′ρ(x, t)1

4π|x − x′| ρ(x′, t). (2.102)

The last term is the Coulomb energy associated with the charge density ρ(x, t).

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2.4 Maxwell’s Equation from Extremum of Field Action 63

2.4.4 Gauge Invariance of Maxwell’s Theory

The four-dimensional curl (2.80) is manifestly invariant under the gauge transfor-mations

Aa(x) −→ A′a(x) = Aa(x) + ∂aΛ(x), (2.103)

where Λ(x) is any smooth field which satisfies the integrability condition

(∂a∂b − ∂b∂a)Λ(x) = 0. (2.104)

Gauge invariance implies that a scalar field degree of freedom contained in Aa(x)does not contribute to the physically observable electromagnetic fields E(x) andB(x). This degree of freedom can be removed by fixing a gauge. One way of doingthis is to require the vector potential to satisfy the Lorentz gauge condition

∂aAa(x) = 0. (2.105)

For such a vector field, the field equations (2.86) decouple and each of the four com-ponents of the vector potential Aa(x) satisfies the massless Klein-Gordon equation:

−∂2Ab(x) = 0. (2.106)

If a vector potential Aa(x) does not satisfy the Lorentz gauge condition (2.105),one may always perform a gauge transformation (2.103) to a new field A′a(x) thathas no four-divergence. We merely have to choose a gauge function Λ(x) in (2.103)which solves the inhomogeneous differential equation

−∂2Λ(x) = ∂aAa(x). (2.107)

Then A′a(x) will satisfy ∂aA′a(x) = 0.

There are infinitely many solutions to equation (2.107). Given one solution Λ(x)which leads to the Lorentz gauge, one can add any solution of the homogenous Klein-Gordon equation without changing the four-divergence of Aa(x). The associatedgauge transformations

Aa(x) −→ Aa(x) + ∂aΛ′(x), ∂2Λ′(x) = 0, (2.108)

are called restricted gauge transformations or gauge transformation of the secondkind. If a vector potential Aa(x) in the Lorentz gauge solves the field equations(2.86), the gauge transformations of the second kind can be used to remove itsspatial divergence ∇ ·A(x, t). Under (2.108), the components A0(x, t) and A(x, t)go over into

A0(x) → A′0(x, t) = A0(x, t) + ∂0Λ′(x, t),

A(x) → A′(x, t) = A(x, t) − ∇Λ′(x, t). (2.109)

Thus, if we choose the gauge function

Λ′(x, t) = −∫

d3x′1

4π|x− x′|∇ · A(x′, t), (2.110)

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64 2 Action Approach

then

∇2Λ′(x, t) = ∇ · A(x, t) (2.111)

makes the gauge-transformed field A′(x, t) divergence-free:

∇ · A′(x, t) = ∇ · [A(x, t) − ∇Λ(x, t)] = 0. (2.112)

The condition

∇ · A′(x, t) = 0 (2.113)

is known as the Coulomb gauge or radiation gauge.The solution (2.110) to the differential equation (2.111) is still undetermined up

to an arbitrary solution Λ′′(x) of the homogeneous Poisson equation

∇2Λ′′(x, t) = 0. (2.114)

Together with the property ∂2Λ′′(x, t) = 0 implied by (2.108), one also has

∂2tΛ′′(x, t) = 0. (2.115)

This leaves only trivial linear functions Λ′′(x, t) of x and t which contribute constantsto (2.109). These, in turn, are zero since the fields Aa(x) are always assumed tovanish at infinity before and after the gauge transformation.

Another possible gauge is obtained by removing the zeroth component of thevector potential Aa(x) in the field equation (2.86). This is obtained by performingthe gauge transformation (2.103) with a gauge function

Λ(x, t) = −∫ t

dt′A0(x, t′). (2.116)

instead of (2.110). The new field A′a(x) has no zeroth component:

A′0(x) = 0. (2.117)

This is called the axial gauge. The solutions of Eqs. (2.116) are determined up to atrivial constant, so that no more gauge freedom is left.

For free fields, the Coulomb gauge and the axial gauge coincide. This is a conse-quence of the charge-free Coulomb law ∇ · E = 0 in Eq. (2.97). By expressing E(x)explicitly in terms of the space- and time-like components of the vector potential as

E(x) = −∂0A(x) − ∇A0(x), (2.118)

Coulomb’s law reads

∇2A0(x, t) = −∇ · A(x, t). (2.119)

This shows that if ∇ ·A(x) = 0, also A0(x) = 0 (assuming zero boundary values atinfinity), and vice versa.

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2.5 Maxwell-Lorentz Action for Charged Point Particles 65

The differential equation (2.119) can be integrated to

A0(x, t) =1

d3x′1

|x′ − x| ∇ · A(x′, t). (2.120)

In an infinite volume with asymptotically vanishing fields there is no freedom ofadding to the left-hand side a nontrivial solution of the homogenous Poisson equation

∇2A0(x, t) = 0. (2.121)

In the presence of charges, Coulomb’s law has a source term [see Eq. (2.97)]:

∇ · E(x, t) = ρ(x, t). (2.122)

where ρ(x, t) is the electric charge density. In this case the vanishing of ∇ · A(x, t)no longer implies A0(x, t) ≡ 0. Then one has the possibility of choosing Λ(x, t)either to satisfy the Coulomb gauge

∇ · A(x, t) ≡ 0, (2.123)

or the axial gaugeA0(x, t) ≡ 0. (2.124)

Only for free fields the two gauges coincide.In a fixed gauge, the vector potential Aa(x) does not, in general, transform as

a four-vector field under Lorentz transformations, which according to (1.203) and(1.210) would imply

Aa(x)Λ−→ A′Λ

a(x) = ΛabA

b(Λ−1x) = [e−i12ωabJ

ab

A]a(Λ−1x). (2.125)

This is only true if the gauge is fixed in a Lorentz-invariant way, for instance bythe Lorentz gauge condition (2.105). In the Coulomb gauge, the right-hand sideof (2.125) will be modified by an additional gauge transformation depending on Λwhich ensures the Coulomb gauge for the transformed vector potential.

2.5 Maxwell-Lorentz Action for Charged Point Particles

Consider now charged relativistic massive particles interacting with electromagneticfields and derive the Maxwell-Lorentz equations of Section 1.9 from the action ap-proach. A single particle of charge e carries a current

ja(x) = ec∫ ∞

−∞dτ qa(τ)δ(4)(x− q(τ)), (2.126)

and the total action in an external field is given by the sum of (16.20) and (2.19):

A=em

A +m

A =−1

4

d4xF ab(x)Fab(x) −mc2∫ τb

τadτ − 1

c

d4x ja(x)Aa(x). (2.127)

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66 2 Action Approach

In terms of the physical time t, the last two terms can be separated into spatial andtime-like components as follows:

−mc2∫ tb

tadt

1 − q2

c2− e

∫ tb

tadtA0(q(t), t) +

e

c

∫ tb

tadtv · A(q(t), t). (2.128)

The equations of motion are obtained by writing the free-particle action in theform (2.19), and extremizing (2.127) with respect to variations δqa(τ). This yieldsthe Maxwell-Lorentz equations (1.165):

md2qa

dτ 2=

e

c

[

− ∂

∂τAa +

dqb

dτ∂aAb

]

=e

c

[

−dqb

dτ∂bA

a +dqb

dτ∂aAb

]

=e

cFab

dqb

dτ. (2.129)

On the right-hand side we recognize the Lorentz force (1.179).Note that in the presence of electromagnetic fields, the canonical momenta (2.11)

are no longer equal to the physical momenta as in (2.15), but receive a contributionfrom the vector potential:

Pi = −∂L∂qi

= −(mγqi +e

cAi) = pi +

e

cAi. (2.130)

Including the zeroth component, the canonical four-momentum is

Pa = pa +e

cAa. (2.131)

The zeroth component of Pa coincides with 1/c times the energy defined by theLegendre transform [recall (2.96)]:

P0 =1

c(H + eA0) = −1

c(Piq

i − L). (2.132)

2.6 Scalar Field with Electromagnetic Interaction

The spacetime derivatives of a plane wave such as (1.155) yields the energy-momentum of the particle whose probability amplitude is described by the wave:

ih∂aφp(x) = paφp(x). (2.133)

In the presence of electromagnetism, the role of the momentum four-vector is takenover by the momenta (2.131). In the Lagrangian density (2.27) of the scalar field,this is accounted for by the so-called minimal replacement of the derivatives by thecovariant derivatives:

∂aφ(x) → Daφ(x) ≡[

∂a + ie

chAa(x)

]

φ(x). (2.134)

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2.7 Dirac Fields 67

The Lagrangian density of a scalar field with electromagnetic interactions is therefore

L(x) = h2[Daφ(x)]∗Daφ(x) −m2c2φ∗(x)φ(x) − 1

4

d4xF ab(x)Fab(x). (2.135)

It governs the so-called scalar electrodynamics.This expression is invariant under local gauge transformations (2.103) of the elec-

tromagnetic field, if we simultaneously multiply the scalar field by an x-dependentphase factor:

ϕ(x) → eieΛ(x)/cϕ(x). (2.136)

By extremization of the action in natural units A =∫

d4xL(x) we find theEuler-Lagrange equation and its conjugate

δAδϕ∗(x)

= (−D2 −m2)ϕ(x),δAδϕ(x)

= (−D∗2 −m2)ϕ∗(x), (2.137)

In the presence of the electromagnetic field, the particle current density (2.63) turnsinto the charge current density

ja(x) = ei

2φ∗Daφ+ c.c. = e

i

2φ∗↔

∂a φ− e2

cAa(x)φ

∗φ. (2.138)

This satisfies the same conservation law (2.64) as the current density of the freescalar field, as we can verify by a short calculation:

∂aja= ∂a

[

i

2φ∗Daφ

]

+ c.c.=i

2∂aφ

∗Daφ+i

2φ∗∂aD

aφ+ c.c. (2.139)

=i

2∂aφ

∗Daφ+i

2φ∗D2φ− i

2

e

cAaφ∗Daφ+ c.c.=

i

2D∗aφ

∗Daφ−m2 i

2φ∗φ+ c.c.=0.

2.7 Dirac Fields

An action whose extremum yields the Dirac equation (1.212) is, in natural units,

D

A=∫

d4xD

L (x) ≡∫

d4x ψ(x) (iγa∂a −m)ψ(x) (2.140)

whereψ(x) ≡ ψ†(x)γ0, (2.141)

and the matrices γa satisfy the anticommutation rules (1.218). The Dirac equationand its conjugate are obtained from the extremal principle

δD

Aδψ(x)

= (iγa∂a −m)ψ(x) = 0,δ

D

Aδψ(x)

= ψ(x)( − iγa←∂a −m)ψ(x) = 0. (2.142)

The action (2.140) is invariant under the Lorentz transformations of spinors(1.230). For the mass term this follows from the fact that

D†(Λ)γ0D(Λ) = γ0. (2.143)

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68 2 Action Approach

This equation is easily verified by inserting the explicit matrices (1.213) and (1.227).If we define

D ≡ γ0D†γ0, (2.144)

this impliesD(Λ)D(Λ) = 1, (2.145)

so that the mass term in the Lagrangian density transforms like a scalar field in(1.163):

ψ(x)ψ(x)Λ−→ ψ′Λ(x)ψ′Λ(x) = ψ(Λ−1x)ψ(Λ−1x). (2.146)

Consider now the gradient term in the action (2.140). Its invariance is a conse-quence of the vector property of the Dirac matrices under the spin representation ofthe Lorentz group derived in Eq. (1.229), which can be rewritten, due to (2.145), as

D(Λ)γaD(Λ) = D−1(Λ)γaD(Λ) = Λabγb. (2.147)

From this we derive at once that

ψ(x)γaψ(x)Λ−→ ψ′(x)γaψ′(x) = Λa

bψ(Λ−1x)γbψ(Λ−1x), (2.148)

andψ(x)γa∂aψ(x)

Λ−→ ψ′(x)γa∂aψ′(x) = [ψγb∂aψ](Λ−1x). (2.149)

Thus also the gradient term in the Dirac Lagrangian density transforms like a scalarfield, and so does the full Lagrangian density as in (2.28), which makes the action(2.140) invariant under Lorentz transformations, due to (2.34).

After the discussion in Section 2.6 we know how to couple the Dirac field toelectromagnetism. We simply have to replace the derivative in the Lagrangian den-sity by the covariant derivative (2.134), and obtain the gauge-invariant Lagrangiandensity of the electrodynamics

L(x) = ψ(x) (iγaDa −m)ψ(x) − 1

4

d4xF ab(x)Fab(x). (2.150)

This is invariant under local gauge transformations (2.103), if we simultaneouslymultiply the Dirac field by an x-dependent phase factor

ψ(x) → eieΛ(x)/cψ(x). (2.151)

The interaction term in this Lagrangian density comes entirely from the covariantderivative and reads, more explicity,

Lint(x) = −1

c

d4xAa(x)ja(x), (2.152)

whereja(x) ≡ e ψ(x)γaψ(x) (2.153)

is the current density of the electrons.

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Notes and References 69

By extremizing the actionD

A=∫

d4xD

L (x) we now find the Euler-Lagrangeequation and its conjugate

δD

Aδψ(x)

= (iγaDa −m)ψ(x) = 0,δ

D

Aδψ(x)

= ψ(x)( − iγa←D∗a −m)ψ(x) = 0. (2.154)

For classical fields obeying these equation, the current density (2.153) satisfies thesame local conservation law as the scalar field in Eq. (2.139),

∂aja(x) = 0, (2.155)

as can be verified by the much simpler calculation than in (2.139):

∂aja = e∂a(ψγ

aψ) = eψγa←∂aψ + eψγa∂aψ = eψγa

←D∗aψ + eψγaDaψ = 0. (2.156)

Notes and References

[1] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals,McGraw-Hill, New York, 1965.

[2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets , World Scientific Publishing Co., Singapore2004, 4th extended edition, pp. 1–1547 (kl/b5), where kl is short for the wwwaddress http://www.physik.fu-berlin.de/~kleinert.

[3] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields , PergamonPress, Oxford, 1975

[4] H. Kleinert, Gauge fields in Condensed Matter , Vol. I: Superflow and VortexLines, Disorder Fields, Phase Transitions World Scientific, Singapore, 1989(kl/b1).

[5] H. Kleinert, Gauge fields in Condensed Matter , Vol. II: Stresses and De-fects, Differential Geometry, Crystal Defects, World Scientific, Singapore, 1989(kl/b2).

[6] J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields , McGraw-Hill, NewYork, 1956, Sect. 15.2.

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As far as the laws of mathematics refer to reality, they are not certain;

and as far as they are certain, they do not refer to reality.

Albert Einstein (1879 - 1955)

3Continuous Symmetries and Conservation Laws.

Noether’s Theorem

In many physical systems, the action is invariant under some continuous set of trans-formations. Then there exist local and global conservation laws analogous to currentand charge conservation in electrodynamics. With the help of Poisson brackets, theanalogs of the charges can be used to generate the symmetry transformation, fromwhich they were derived. After field quantization, the Poisson brackets becomecommutators of operators associated with these charges.

3.1 Continuous Symmetries and Conservation Law

Consider first a simple mechanical system with a generic action

A =∫ tb

tadt L(q(t), q(t)), (3.1)

and subject it to a continuous set of local transformations of the dynamical variables:

q(t) → q′(t) = f(q(t), q(t)), (3.2)

where f(q(t), q(t)) is some function of q(t) and q(t). In general, q(t) will carry variouslabels as in (2.1) which are suppressed, for brevity. If the transformed action

A′ ≡∫ tb

tadt L(q′(t), q′(t)) (3.3)

is the same as A, up to boundary terms, then (3.2) is called a symmetry trans-formation. For any two symmetry transformations, we may defined a product byperforming the transformations successively. The result is certainly again a sym-metry transformation. Since all transformations can be undone, they possess aninverse. Thus, symmetry transformations form a group called the symmetry groupof the system. It is important that the equations of motion are not used whenshowing that the action A′ is equal to A, up to boundary terms.

70

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3.1 Continuous Symmetries and Conservation Law 71

For infinitesimal symmetry transformations (3.2), the difference

δsq(t) ≡ q′(t) − q(t) (3.4)

will be called a symmetry variation. It has the general form

δsq(t) = ε∆(q(t), q(t)), (3.5)

where ε is a small parameter. Symmetry variations must not be confused with thevariations δq(t) used in Section 2.1 to derive the Euler-Lagrange equations (2.8),which always vanish at the ends, δq(tb) = δq(ta) = 0 [recall (1.4)]. This is usuallynot true for symmetry variation δsq(t).

Let us calculate the change of the action under a symmetry variation (3.5). Usingthe chain rule of differentiation and a partial integration we obtain

δsA =∫ tb

tadt

[

∂L

∂q(t)− ∂t

∂L

∂q(t)

]

δsq(t) +∂L

∂q(t)

tb

ta

. (3.6)

Let us denote the solutions of the Euler-Lagrange equations (2.8) by qc(t) and callthem classical orbits, or briefly orbits. For orbits, only the boundary terms in (3.6)survive, and we are left with

δsA = ε∂L

∂q∆(q, q)

ta

tb

, for q(t) = qc(t). (3.7)

By the symmetry assumptions, δsA vanishes or is equal to a surface term. In thefirst case, the quantity

Q(t) ≡ ∂L

∂q∆(q, q), for q(t) = qc(t) (3.8)

is the same at times t = ta and t = tb. Since tb is arbitrary, Q(t) is independent ofthe time t, i.e., it satisfies

Q(t) ≡ Q. (3.9)

It is a conserved quantity , a constant of motion along the orbit. The expression onthe right-hand side of (3.8) is called a Noether charge.

In the second case, δsq(t) is equal to a boundary term

δsA = εΛ(q, q)∣

tb

ta, (3.10)

and the conserved Noether charge becomes

Q(t) =∂L

∂q∆(q, q) − Λ(q, q), for q(t) = qc(t). (3.11)

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72 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

It is possible to derive the constant of motion (3.11) also without invoking theaction, starting from the Lagrangian L(q, q). We expand its symmetry variation ofL(q, q) as follows:

δsL≡L (q+δsq, q+δsq) − L(q, q) =

[

∂L

∂q(t)− ∂t

∂L

∂q(t)

]

δsq(t) +d

dt

[

∂L

∂q(t)δsq(t)

]

.

(3.12)On account of the Euler-Lagrange equations (2.8), the first term on the right-handside vanishes as before, and only the last term survives. The assumption of invarianceof the action up to a possible surface term in Eq. (3.10) is equivalent to assumingthat the symmetry variation of the Lagrangian is at most a total time derivative ofsome function Λ(q, q):

δsL(q, q, t) = εd

dtΛ(q, q). (3.13)

Inserting this into the left-hand side of (3.12), we find the equation

εd

dt

[

∂L

∂q∆(q, q) − Λ(q, q)

]

= 0, for q(t) = qc(t) (3.14)

thus recovering again the conserved Noether charge (3.11).

3.1.1 Alternative Derivation

Let us subject the action (3.1) to an arbitrary variation δq(t), which may be nonzeroat the boundaries. Along a classical orbit qc(t), the first term in (3.6) vanishes, andthe action changes at most by the boundary term:

δA =∂L

∂qδq

tb

ta

, for q(t) = qc(t). (3.15)

This observation leads to another derivation of Noether’s theorem. Suppose weperform on q(t) a so-called local symmetry transformations, which generalizes theprevious symmetry variations (3.5) to a time-dependent parameter ε:

δtsq(t) = ε(t)∆(q(t), q(t)). (3.16)

The superscript t on δts emphasized the extra time dependence in ε(t). If the varia-tions (3.16) are applied to a classical orbit qc(t), the action changes by the boundaryterm (3.15).

This will now be expressed in a more convenient way. For this purpose weintroduce the infinitesimally transformed orbit

qε(t)(t) ≡ q(t) + δtsq(t) = q(t) + ε(t)∆(q(t), q(t)), (3.17)

and the transformed Lagrangian

Lε(t) ≡ L(qε(t)(t), qε(t)(t)). (3.18)

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3.1 Continuous Symmetries and Conservation Law 73

Then the local symmetry variation of the action with respect to the time-dependentparameter ε(t) is

δtsA =∫ tb

tadt

[

∂Lε(t)

∂ε(t)− d

dt

∂Lε(t)

∂ε(t)

]

ε(t) +d

dt

[

∂Lε(t)

∂ε

]

ε(t)

tb

ta

. (3.19)

Along a classical orbit, the action is extremal. Hence the infinitesimally transformedaction

Aε ≡∫ tb

tadt L(qε(t)(t), qε(t)(t)) (3.20)

must satisfy the equationδAε

δε(t)= 0. (3.21)

This hold for an arbitrary time dependence of ε(t), in particular for ε(t) which vanishat the ends. In this case, (3.21) leads to an Euler-Lagrange type of equation

∂Lε(t)

∂ε(t)− d

dt

∂Lε(t)

∂ε(t)= 0, for q(t) = qc(t). (3.22)

This can also be checked explicitly differentiating (3.18) according to the chain ruleof differentiation:

∂Lε(t)

∂ε(t)=

∂L

∂q(t)∆(q, q) +

∂L

∂q(t)∆(q, q), (3.23)

∂Lε(t)

∂ε(t)=

∂L

∂q(t)∆(q, q), (3.24)

and inserting on the right-hand side the ordinary Euler-Lagrange equations (1.5).Note that (3.24) can also be written as

∂Lε(t)

∂ε(t)=

∂L

∂q(t)

δsq(t)

ε(t). (3.25)

We now invoke the symmetry assumption, that the action is a pure surface termunder the time-independent transformations (3.16). This implies that

∂Lε

∂ε=∂Lε(t)

∂ε(t)=

d

dtΛ. (3.26)

Combining this with (3.22), we derive a conservation law for the charge:

Q =∂Lε(t)

∂ε(t)− Λ, for q(t) = qc(t). (3.27)

Inserting here Eq. (3.24) we find that this is the same charge as the previous (3.11).

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74 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

3.2 Time Translation Invariance and Energy Conservation

As a simple but physically important example consider the case that the Lagrangiandoes not depend explicitly on time, i.e., that L(q, q) ≡ L(q, q). Let us perform a timetranslation on the system, so that the same events happen at a new time

t′ = t− ε. (3.28)

The time-translated orbit has the time dependence

q′(t′) = q(t), (3.29)

i.e., the translated orbit q′(t) has at the time t′ the same value as the orbit q(t) hadat the original time t. For the Lagrangian, this implies that

L′(t′) ≡ L(q′(t′), q′(t′)) = L(q(t), q(t)) ≡ L(t). (3.30)

This makes the action (3.3) equal to (3.1), up to boundary terms. Thus time-independent Lagrangians possess time translation symmetry.

The associated symmetry variations of the form (3.5) read

δsq(t) = q′(t) − q(t) = q(t′ + ε) − q(t)

= q(t′) + εq(t′) − q(t) = εq(t), (3.31)

Under these, the Lagrangian changes by

δsL = L(q′(t), q′(t)) − L(q(t), q(t)) =∂L

∂qδsq(t) +

∂L

∂qδsq(t). (3.32)

Inserting δsq(t) from (3.31) we find, without using the Euler-Lagrange equation,

δsL = ε

(

∂L

∂qq +

∂L

∂qq

)

= εd

dtL. (3.33)

This has precisely the derivative form (3.13) with Λ = L, thus confirming that timetranslations are symmetry transformations.

According to Eq. (3.11), we find the Noether charge

Q = H ≡ ∂L

∂qq − L(q, q), for q(t) = qc(t) (3.34)

to be a constant of motion. This is recognized as the Legendre transform of theLagrangian, which is the Hamiltonian (2.10) of the system.

Let us briefly check how this Noether charge is obtained from the alternativeformula (3.11). The time-dependent symmetry variation (3.16) is here

δtsq(t) = ε(t)q(t) (3.35)

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3.3 Momentum and Angular Momentum 75

under which the Lagrangian is changed by

δtsL =∂L

∂qεq +

∂L

∂q(εq + εq) =

∂Lε

∂εε+

∂Lε

∂εε, (3.36)

with∂Lε

∂ε=∂L

∂qq (3.37)

and∂Lε

∂ε=∂L

∂qq +

∂L

∂qεq =

d

dtL. (3.38)

The last equation confirms that time translations fulfill the symmetry condition(3.26), and from (3.37) we see that the Noether charge (3.27) coincides with theHamiltonian found in Eq. (3.11).

3.3 Momentum and Angular Momentum

While the conservation law of energy follow from the symmetry of the action undertime translations, conservation laws of momentum and angular momentum are foundif the action is invariant under translations and rotations, respectively.

Consider a Lagrangian of a point particle in a euclidean space

L = L(qi(t), qi(t)). (3.39)

In contrast to the previous discussion of time translation invariance, which wasapplicable to systems with arbitrary Lagrange coordinates q(t), we denote the co-ordinates here by qi, with the superscripts i emphasizing that we are now dealingwith cartesian coordinates. If the Lagrangian depends only on the velocities qi andnot on the coordinates qi themselves, the system is translationally invariant . If itdepends, in addition, only on q2 = qiqi, it is also rotationally invariant.

The simplest example is the Lagrangian of a point particle of mass m in euclideanspace:

L =m

2q2. (3.40)

It exhibits both invariances, leading to conserved Noether charges of momentumand angular momentum, as we shall now demonstrate.

3.3.1 Translational Invariance in Space

Under a spatial translation, the coordinates qi of the particle change to

q′i = qi + εi, (3.41)

where εi are small numbers. The infinitesimal translations of a particle path are[compare (3.5)]

δsqi(t) = εi. (3.42)

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76 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

Under these, the Lagrangian changes by

δsL = L(q′i(t), q′i(t)) − L(qi(t), qi(t))

=∂L

∂qiδsq

i =∂L

∂qiεi = 0. (3.43)

By assumption, the Lagrangian is independent of qi, so that the right-hand sidevanishes. This is to be compared with the symmetry variation of the Lagrangianaround a classical orbit calculated with the help of the Euler-Lagrange equation:

δsL =

(

∂L

∂qi− d

dt

∂L

∂qi

)

δsqi +

d

dt

[

∂L

∂qiδsq

i

]

=d

dt

[

∂L

∂qi

]

εi (3.44)

This has the form (3.7), from which we extract a conserved Noether charge (3.8) foreach coordinate qi, to be called pi:

pi =∂L

∂qi. (3.45)

Thus the Noether charges are simply the canonical momenta of the point particle.

3.3.2 Rotational Invariance

Under rotations, the coordinates qi of the particle change to

q′i = Rijqj (3.46)

where Rij are the orthogonal 3 × 3 -matrices (1.8). Infinitesimally, these can be

written as

Rij = δij − ϕkεkij (3.47)

where is the infinitesimal rotation vector in Eq. (1.57). The corresponding rotationof a particle path is

δsqi(t) = q′i(t) − qi(t) = −ϕkεkijqj(τ). (3.48)

In the antisymmetric tensor notation (1.55) with ωij ≡ ϕkεkij , we write

δsqi = −ωijqj. (3.49)

Under this, the symmetry variation of the Lagrangian (3.40)

δsL = L(q′i(t), q′i(t)) − L(qi(t), qi(t))

=∂L

∂qiδsq

i +∂L

∂qiδsq

i (3.50)

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3.3 Momentum and Angular Momentum 77

becomes

δsL = −(

∂L

∂qiqj +

∂L

∂qiqj)

ωij = 0. (3.51)

For any Lagrangian depending only on the rotational invariants q2, q2,q · q andpowers thereof, the right-hand side vanishes on account of the antisymmetry of ωij.This ensures the rotational symmetry for the Lagrangian (3.40).

We now calculate the symmetry variation of the Lagrangian once more using theEuler-Lagrange equations:

δsL =

(

∂L

∂qi− d

dt

∂L

∂qi

)

δsqi +

d

dt

[

∂L

∂qiδsq

i

]

= − d

dt

[

∂L

∂qiqj]

ωij =1

2

d

dt

[

qi∂L

∂qj− (i↔ j)

]

ωij. (3.52)

The right-hand side yields the conserved Noether charges of the type (3.8), one foreach antisymmetric pair i, j:

Lij = qi∂L

∂qj− qj

∂L

∂qi≡ qipj − qjpi. (3.53)

These are the conserved components of angular momentum for a cartesian systemin any dimension.

In three dimensions, we may prefer working with the original rotation angles ϕk,in which case we would have found the angular momentum in the standard form

Lk =1

2εkijL

ij = (q × p)k. (3.54)

3.3.3 Center-of-Mass Theorem

Let us now study symmetry transformations corresponding to a uniform motion ofthe coordinate system described by Galilei transformations (1.11), (1.12). Considera set of free massive point particles in euclidean space described by the Lagrangian

L(qin) =∑

n

mn

2qin

2. (3.55)

The infinitesimal symmetry variation associated with the Galilei transformationsare

δsqin(t) = qin(t) − qin(t) = −vit, (3.56)

where vi is a small relative velocity along the ith axis. This changes the Lagrangianby

δsL = L(qin − vit, qin − vi) − L(qin, qin). (3.57)

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78 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

Inserting here (3.55), we find

δsL =∑

n

mn

2

[

(qin − vi)2 − (qni)2]

, (3.58)

which can be written as a total time derivative

δsL =d

dtΛ =

d

dt

n

mn

[

−qinvi +v2

2t

]

(3.59)

proving that Galilei transformations are symmetry transformations in the Noethersense. Note that terms quadratic in vi are omitted in the last expression since thevelocities vi in (3.56) are infinitesimal, by assumption.

By calculating δsL once more via the chain rule with the help of the Euler-Lagrange equations, and equating the result with (3.59), we find the conservedNoether charge

Q =∑

n

∂L

∂qinδsq

in − Λ

=

(

−∑

n

mnqin t+

n

mnqin

)

vi. (3.60)

Since the direction of the velocities vi is arbitrary, each component is separately aconstant of motion:

N i = −∑

n

mnqi t+

n

mnqni = const. (3.61)

This is the well-known center-of-mass theorem [1]. Indeed, introducing the center-of-mass coordinates

qiCM ≡∑

nmnqni

nmn

, (3.62)

and velocities

viCM =

nmnqni

nmn, (3.63)

the conserved charge (3.61) can be written as

N i =∑

n

mn(−viCM t+ qiCM). (3.64)

The time-independence of N i implies that the center-of-mass moves with uniformvelocity according to the law

qiCM(t) = qiCM,0 + viCMt, (3.65)

where

qiCM,0 =N i

nmn

(3.66)

is the position of the center of mass at t = 0.Note that in non-relativistic physics, the center of mass theorem is a consequence

of momentum conservation since momentum ≡ mass × velocity. In relativisticphysics, this is no longer true.

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3.3 Momentum and Angular Momentum 79

3.3.4 Conservation Laws Resulting from Lorentz Invariance

In relativistic physics, particle orbits are described by functions in Minkowski space-time qa(σ), where σ is a Lorentz-invariant length parameter. The action is an integralover some Lagrangian:

A =∫ σb

σadσL (qa(σ), qa(σ)) , (3.67)

where the dot denotes the derivative with respect to the parameter σ. If the La-grangian depends only on invariant scalar products qaqa, q

aqa, qaqa, then it is in-

variant under Lorentz transformations

qa → q′a = Λab q

b (3.68)

where Λab are the pseudo-orthogonal 4 × 4 -matrices (1.28).

A free massive point particle in spacetime has the Lagrangian [see (2.19)]

L(q(σ)) = −mc√

gabqaqb, (3.69)

so that the action (3.67) is invariant under arbitrary reparametrizations σ → f(σ).Since the Lagrangian depends only on q(σ), it is invariant under arbitrary transla-tions of the coordinates:

δsqa(σ) = qa(σ) − εa(σ), (3.70)

for which δsL = 0. Calculating this variation once more with the help of the Euler-Lagrange equations, we find

δsL =∫ σb

σadσ

(

∂L

∂qaδsq

a +∂L

∂qaδsq

a

)

= −εa∫ σb

σadσ

d

(

∂L

∂qa

)

. (3.71)

From this we obtain the conserved Noether charges

pa ≡ − ∂L

∂qa= m

qa√

gabqaqb/c2= mua, (3.72)

which satisfy the conservation law

d

dσpa(σ) = 0. (3.73)

The Noether charges pa(σ) are the conserved four-momenta (1.144) of the free rela-tivistic particle, derived in Eq. (2.20) from the canonical formalism. The four-vector

ua ≡ qa√

gabqaqb/c2(3.74)

is the relativistic four-velocity of the particle. It is the reparametrization-invariantexpression for the four-velocity qa(τ) = ua(τ) in Eqs. (2.22) and (1.144). A signchange is made in Eq. (3.72) to agree with the canonical definition of the covariant

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80 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

momentum components in (2.20). By choosing for σ the physical time t = q0/c, wecan express ua in terms of the physical velocities vi = dqi/dt, as in (1.145):

ua = γ(1, vi/c), with γ ≡√

1 − v2/c2. (3.75)

For small Lorentz transformations near the identity we write

Λab = δab + ωab (3.76)

whereωab = gacωcb (3.77)

is an arbitrary infinitesimal antisymmetric matrix. An infinitesimal Lorentz trans-formation of the particle path is

δsqa(σ) = qa(σ) − qa(σ)

= ωabqb(σ). (3.78)

Under it, the symmetry variation of a Lorentz-invariant Lagrangian vanishes:

δsL =

(

∂L

∂qaqb +

∂L

∂qaqb)

ωab = 0 (3.79)

This is to be compared with the symmetry variation of the Lagrangian calculatedvia the chain rule with the help of the Euler-Lagrange equation

δsL =

(

∂L

∂qa− d

∂L

∂qa

)

δsqa +

d

[

∂L

∂qaδsq

a

]

=d

[

∂L

∂qaqb]

ωab

=1

2ωa

b d

(

qa∂L

∂qb− qb

∂L

∂qa

)

. (3.80)

By equating this with (3.79) we obtain the conserved rotational Noether charges

Lab = −qa ∂L∂qb

+ qb∂L

∂qa= qapb − qbpa. (3.81)

They are the four-dimensional generalizations of the angular momenta (3.53).The Noether charges Lij coincide with the components (3.53) of angular momen-

tum. The conserved components

L0i = q0pi − qip0 ≡Mi (3.82)

yield the relativistic generalization of the center-of-mass theorem (3.61):

Mi = const. (3.83)

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3.4 Generating the Symmetries 81

3.4 Generating the Symmetries

A mentioned in the introduction to this chapter, there is a second important relationbetween invariances and conservation laws. The charges associated with continuoussymmetry transformations can be used to generate the symmetry transformationfrom which they it was derived. In the classical theory, this is done with the helpof Poisson brackets:

δsq = εQ, q(t). (3.84)

After canonical quantization, the Poisson brackets turn into −i times commutators,and the charges become operators, generating the symmetry transformation by theoperation

δsq = −iε[Q, q(t)]. (3.85)

The most important example for this quantum-mechanical generation of a sym-metry transformations is the action of the Noether charge (3.34) derived in Sec-tion 3.2 from the invariance of the system under time displacement. There theNoether charge Q happened to be the Hamiltonian H , whose operator version gen-erates the infinitesimal time displacements (3.31) by the Heisenberg equation ofmotion:

δsq(t) = ε ˙q(t) = −iε[H, q(t)], (3.86)

this being a special case of the general Noether relation (3.85).The canonical quantization is straightforward if the Lagrangian has the standard

formL(q, q) =

m

2q2 − V (q). (3.87)

Then the operator version of the canonical momentum p ≡ q satisfies the equal-timecommutation rules

[p(t), q(t)] = −i, [p(t), p(t)] = 0, [q(t), q(t)] = −i. (3.88)

The Hamiltonian

H =p2

2m+ V (q) (3.89)

turns directly into the Hamiltonian operator

H =p2

2m+ V (q). (3.90)

If the Lagrangian does not have the standard form (3.87), quantization is a nontrivialproblem, solved in the textbook [2].

Another important example is provided by the charges (3.45) derived in Sec-tion 3.3.1 from translational symmetry. After quantization, the commutator (3.85)generating the transformation (3.42) becomes

εj = iεi[pi(t), qj(t)]. (3.91)

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82 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

This coincides with one of the canonical commutation relations (3.88) in three di-mensions.

The relativistic charges (3.72) of spacetime generate translations via

δsqa = εa = −iεb[pb(t), qa(τ)], (3.92)

implying the relativistic commutation rules

[pb(t), qa(τ)] = iδb

a, (3.93)

in agreement with the relativistic canonical commutation rules (1.157) (in naturalunits with h = 1).

Note that all commutation rules derived from the Noether charge accordingto the rule (3.85) hold for the operators in the Heisenberg picture, where theyare time-dependent. The commutation rules in the purely algebraic discussion inChapter 3, on the other hand, apply to the time-independent Schrodinger pictureof the operators.

Similarly we find that the quantized versions of the conserved charges Li inEq. (3.54) generate infinitesimal rotations:

δsqj = −ωiεijkqk(t) = iωi[Li, q

j(t)], (3.94)

whereas the quantized conserved charges N i of Eq. (3.61) generate infinitesimalGalilei transformations, and that the charges Mi of Eq. (3.82) generate pure Lorentztransformations:

δsqj = εiq

0 = iεi[Mi, qj],

δsq0 = εiq

i = iεi[Mi, q0]. (3.95)

Since the quantized charges generate the symmetry transformations, they forma representation of the generators of the Lorentz group. As such they must havethe same commutation rules between each other as the generators of the symmetrygroup in Eq. (1.71) or their short version (1.72). This is indeed true, since theoperator versions of the Noether charges (3.81) correspond to the operators (1.158)(in natural units).

3.5 Field Theory

A similar relation between continuous symmetries and constants of motion holdsin field theories, where the role of the Lagrange coordinates is played by fieldsqx(t) = ϕ(x, t).

3.5.1 Continuous Symmetry and Conserved Currents

Let A be the local action of an arbitrary field ϕ(x) → ϕ(x, t),

A =∫

d4xL(ϕ, ∂ϕ, x), (3.96)

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3.5 Field Theory 83

and suppose that a transformation of the field

δsϕ(x) = ε∆(ϕ, ∂ϕ, x) (3.97)

changes the Lagrangian density L merely by a total derivative

δsL = ε∂aΛa, (3.98)

which makes the change of the action A a surface integral, by Gauss’s divergencetheorem:

δsA = ε∫

d4x ∂aΛa = ε

Sdsa Λa, (3.99)

where S is the surface of the total spacetime volume. Then δsϕ is called a symmetrytransformation.

Given such a transformation, we see that the four-dimensional current density

ja =∂L∂∂aϕ

∆ − Λa (3.100)

that has no divergence

∂aja(x) = 0. (3.101)

The expression (3.100) is called a Noether current density and (3.101) is a localconservation law , just as in the electromagnetic equation (1.196).

We have seen in Eq. (1.198) that a local conservation law (3.101) always impliesa global conservation law of the type (3.8) for the charge, which is now the Noethercharge Q(t) defined as in (1.199) by the spatial integral over the zeroth component(here in natural units with c = 1)

Q(t) =∫

d3x j0(x, t). (3.102)

The proof of the local conservation law (3.101) is just as easy as for the me-chanical action (3.1). We calculate the variation of L under infinitesimal symmetrytransformations (3.97) in a similar way as in Eq. (3.12), and find

δsL =

(

∂L∂ϕ

− ∂a∂L∂∂aϕ

)

δsϕ+ ∂a

(

∂L∂∂aϕ

δsϕ

)

= ε

(

∂L∂ϕ

− ∂a∂L∂∂aϕ

)

∆ + ε ∂a

(

∂L∂∂aϕ

)

. (3.103)

The Euler-Lagrange equation removes the first term and, equating the second termwith (3.98), we obtain

∂aja ≡ ∂a

(

∂L∂∂aϕ

∆ − Λa

)

= 0. (3.104)

The relation between continuous symmetries and conservation is called Noether’stheorem [3].

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84 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

3.5.2 Alternative Derivation

There is again an alternative derivative of the conserved current analogous toEqs. (3.16)–(3.27). It is based on a variation of the fields under symmetry trans-formations whose parameter ε is made artificially spacetime-dependent ε(x), thusextending (3.16) to

δxs ϕ(x) = ε(x)∆(ϕ(x), ∂aϕ(x)). (3.105)

As before in Eq. (3.18), let us calculate the Lagrangian density for a slightlytransformed field

ϕε(x)(x) ≡ ϕ(x) + δxs ϕ(x), (3.106)

calling it

Lε(x) ≡ L(ϕε(x), ∂ϕε(x)). (3.107)

The associated action differs from the original one by

δxs A =∫

dx

[

∂Lε(x)∂ε(x)

− ∂a∂Lε(x)∂∂aε(x)

]

δε(x) + ∂a

[

∂Lε(x)∂∂aε(x)

δε(x)

]

. (3.108)

For classical fields ϕ(x) = ϕc(x) satisfying the Euler-Lagrange equation (2.40), theextremality of the action implies the vanishing of the first term, and thus the Euler-Lagrange-like equation

∂Lε(x)∂ε(x)

− ∂a∂Lε(x)∂∂aε(x)

= 0. (3.109)

By assumption, the action changes by a pure surface term under the x-independenttransformation (3.105), implying that

∂Lε∂ε

= ∂aΛa. (3.110)

Inserting this into (3.109) we find that

ja =∂Lε(x)∂∂aε(x)

− Λa (3.111)

has no four-divergence. This coincides with the previous Noether current density(3.100), as can be seen by differentiating (3.107) with respect to ∂aε(x):

∂Lε(x)∂∂aε(x)

=∂L∂∂aϕ

∆(ϕ, ∂ϕ), (3.112)

3.5.3 Local Symmetries

In Chapter 2 we observed that charged particles and fields coupled to electromag-netism posses a more general symmetry. They are invariant under local gaugetransformations (2.103). The scalar Lagrangian (2.135), for example, is invariant

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3.5 Field Theory 85

under the gauge transformations (2.103) and (2.136),and the Dirac Lagrange den-sity (2.150) under (2.103) and (2.151). These are all of the form (3.97), but with aparameter ε which depends on spacetime. Thus the action is invariant under localsymmetry variations of the type (3.105), which were introduced in the last sectiononly as an auxiliary tool for an alternative derivation of the Noether current density(2.135). For a locally invariant Lagrangian density, the Noether expression (3.111)vanishes identically. This does not mean, however, that the system does not possessa conserved current, as we have seen in Eqs. (2.139) and (2.155). Only Noether’sderivation breaks down. Let us discuss this phenomenon in more detail for theLagrangian density (2.150).

If we restrict the gauge transformations (2.151) to x-spacetime-independentgauge transformations

ψ(x) → eieΛ/cψ(x), (3.113)

we can easily derive a conserved Noether current density of the type (3.100) for theDirac field. The result is the known Dirac current density (2.153). It is the sourceof the electromagnetic field, with a minimal coupling between them. A similarstructure exists for many internal symmetries giving rise to nonabelian versions ofelectromagnetism, which govern strong and weak interactions. What happens toNoether’s derivation of conservation laws in such theories.

As observed above, the (3.111) which was so useful in the globally invarianttheory and which would yield a Noether current density

ja =δL∂∂aΛ

. (3.114)

gives zero her, due to local gauge invariance, and does not provide us with a currentdensity. We may, however, subject just the Dirac field to a local gauge transforma-tion at fixed gauge fields. Then we obtain the conserved current

ja ≡∂L∂∂aΛ

Aa

. (3.115)

Since the complete change under local gauge transformations δxs L vanishes identi-cally, we can alternatively vary only the gauge fields and keep the particle orbitfixed

ja = − ∂L∂∂aΛ

ψ

. (3.116)

This is done most simply by forming the functional derivative with respect to the

gauge field and omitting the contribution ofem

L , i.e., by applying it only to the

Lagrangian of the charge particlese

L ≡ L − em

L :

ja = − ∂e

L∂∂aΛ

= − ∂e

L∂Aa

. (3.117)

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86 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

As a check we apply the rule (3.117) to Dirac complex Klein-Gordon fields withthe actions (2.140) and (2.27), and re-obtain the conserved current densities (2.153)and (2.138) (the extra factor c is a convention). From the Schrodinger action (2.50)we derive the conserved charge current density

j(x, t) ≡ ei

2mψ∗(x, t)

↔∇ψ(x, t) − e2

cAψ∗(x, t)ψ(x, t), (3.118)

to be compared with the particle current density (2.70) which satisfied the conser-vation law (2.71) together with the charge density ρ(x, t) ≡ eψ∗(x, t)ψ(x, t).

An important consequence of local gauge invariance can be found for the gaugefield itself. If we form the variation of the pure gauge field action

δsem

A =∫

d4x tr

δxsAaδem

AδAa

, (3.119)

and insert for δxsA an infinitesimal pure gauge field configuration

δxsAa = −∂aΛ(x) (3.120)

the right-hand side must vanish for all Λ(x). After a partial integration this impliesthe local conservation law ∂aj

a(x) = 0 for the Noether current

emj a(x) = − δ

em

AδAa

. (3.121)

Recalling the explicit form of the action in Eqs. (16.20) and (2.83), we findemj a(x) = −∂bF ab. (3.122)

The Maxwell equation (2.86) can therefore be written asemj a(x) = −

ej a(x), (3.123)

where we have emphasized the fact that the current ja contains only the fields of thecharge particles by a superscript e. In the form (3.123), the Maxwell equation impliesthe vanishing of the total current density consisting of the sum of the conservedcurrent (3.116) of the charges and the Noether current (3.121) of the electromagneticfield:

totj a(x) =

ej a(x) +

emj a(x) = 0. (3.124)

This unconventional way of phrasing the Maxwell equation (2.86) will be useful forunderstanding later the Einstein field equation (17.148) by analogy.

At this place we make an important observation. In contrast to the conservationlaws derived for matter fields, which are valid only if the matter fields obey the Euler-Lagrange equations, the current conservation law for the Noether current (3.122) ofthe gauge fields

∂aemj a(x) = −∂a∂bF ab = 0 (3.125)

is valid for all field configurations. The right-hand side vanishes identically since thevector potential Aa as an observable field in any fixed gauge satisfies the Schwarzintegrability condition (2.88).

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3.6 Canonical Energy-Momentum Tensor 87

3.6 Canonical Energy-Momentum Tensor

As an important example for the field-theoretic version of the Noether theoremconsider a Lagrangian density that does not depend explicitly on the spacetimecoordinates x:

L(x) = L(ϕ(x), ∂ϕ(x)). (3.126)

We then perform a translation of the coordinates along an arbitrary direction b =0, 1, 2, 3 of spacetime

x′a = xa − εa, (3.127)

under which field ϕ(x) transforms as

ϕ′(x′) = ϕ(x), (3.128)

ao that

L′(x′) = L(x). (3.129)

If εa is infinitesimally small, the field changes by

δsϕ(x) = ϕ′(x) − ϕ(x) = εb∂bϕ(x), (3.130)

and the Lagrangian density by

δsL ≡ L(ϕ′(x), ∂ϕ′(x)) −L(ϕ(x), ∂ϕ(x))

=∂L

∂ϕ(x)δsϕ(x) +

∂L∂∂aϕ

∂aδsϕ(x), (3.131)

which is a pure divergence term

δsL(x) = εb∂bL(x). (3.132)

Hence the requirement (3.98) is satisfied and δsϕ(x) is a symmetry transformation,with a function Λ which happens to coincide with the Lagrangian density

Λ = L. (3.133)

We can now define a four vectors of current densities jba, one for each component

of εb, which for the spacetime translation symmetry is denoted by Θba:

Θba =

∂L∂∂aϕ

∂bϕ− δbaL. (3.134)

Since εb is a vector, this 4×4 object is a tensor field, the so-called energy-momentumtensor of the scalar field ϕ(x). According to Noether’s theorem, this has no diver-gence in the index a [compare (3.101)]:

∂aΘba(x) = 0. (3.135)

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88 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

The four conserved charges Qb associated with these current densities [see the defi-nition (3.102)]

Pb =∫

d3xΘb0(x), (3.136)

are the components of the total four-momentum of the system.The alternative derivation of this conservation law follows Subsection 3.1.1 by

introducing the local variations

δxs ϕ(x) = εb(x)∂bϕ(x) (3.137)

under which the Lagrangian density changes by

δxs L(x) = εb(x)∂bL(x). (3.138)

Applying the chain rule of differentiation we obtain, on the other hand,

δxs L =∂L∂ϕ(x)

εb(x)∂bϕ(x) +∂L

∂∂aϕ(x)

[∂aεb(x)]∂bϕ + εb∂a∂bϕ(x)

, (3.139)

which shows that∂Lε

∂∂aεb(x)=

∂L∂∂aϕ

∂bϕ. (3.140)

Forming for each b the combination (3.100), we obtain again the conserved energy-momentum tensor (3.134).

Note that by analogy with (3.25), we can write (3.140) as

∂Lε∂∂aεb(x)

=∂L∂∂aϕ

∂δxs ϕ

∂εb(x)(3.141)

Note further that the component Θ00 of the canonical energy momentum tensor

Θ00 =

∂L∂∂0ϕ

∂0ϕ− L (3.142)

coincides with Hamiltonian density (2.61) derived in the canonical formalism by aLegendre transformation of the Lagrangian density.

3.6.1 Electromagnetism

As a an important physical application of the field-theoretic Noether theorem, con-sider the free electromagnetic field with the action

L = −1

4FcdF

cd, (3.143)

where Fd are the field strength Fcd ≡ ∂cAd − ∂dAc. Under a translation of thespacetime coordinates from xa to xa − εa, the vector potential undergoes a similarchange as the scalar field in (3.128):

A′a(x′) = Aa(x). (3.144)

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3.6 Canonical Energy-Momentum Tensor 89

For infinitesimal translations, this can be written as

δsAc(x) ≡ A′c(x) − Ac(x)

= A′c(x′ + ε) −Ac(x)

= εb∂bAc(x). (3.145)

Under this, the field tensor changes as follows

δsFcd = εb∂bF

cd, (3.146)

and we verify that the Lagrangian density (3.143) is a total four divergence:

δsL = −εb 12

(

∂bFcdFcd + Fcd∂bF

cd)

= εb∂bL (3.147)

Thus, the spacetime translations (3.145) are symmetry transformations, andEq. (3.99) yield the four Noether current densities, one for each εb:

Θba =

1

c

[

∂L∂∂aAc

∂bAc − δb

aL]

. (3.148)

The factor 1/c is introduced to give the Noether current the dimension of the energy-momentum tensors introduced in Section 1.12, which are momentum densities. Herewe have found the canonical energy-momentum tensor of the electromagnetic field,which which satisfy the local conservation laws

∂aΘba(x) = 0. (3.149)

Inserting the derivatives ∂L/∂∂aAc = −F ac, we obtain

Θba =

1

c

[

−F ac∂bA

c +1

4δbaF cdFcd

]

. (3.150)

3.6.2 Dirac Field

We now turn to the Dirac field whose transformation law under spacetime transla-tions

x′a = xa − εa (3.151)

is

ψ′(x′) = ψ(x). (3.152)

Since the Lagrangian density in (2.140) does not depend explicitly on x we calculate,as in (3.129):

D

L ′(x′) =D

L (x), (3.153)

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90 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

The infinitesimal variationsδsψ(x) = εa∂aψ(x). (3.154)

produce the the pure derivative term

δsD

L (x) = εa∂aD

L (x), (3.155)

and the combination (3.100) yields the Noether current densities

Θba =

∂D

L∂∂aψc

∂bψc + cc− δb

aD

L, (3.156)

which satisfies local conservation laws

∂aΘba(x) = 0. (3.157)

From (2.140) we see that

∂D

L∂∂aψc

=1

2ψγa (3.158)

so that we obtain the canonical energy-momentum tensor of the Dirac field:

Θba =

1

2ψγa∂bψ

c + cc− δba

D

L (3.159)

3.7 Angular Momentum

Let us now turn to angular momentum in field theory. Consider first the case of ascalar field ϕ(x). Under a rotation of the coordinates,

x′i = Rijxj (3.160)

the field does not change, if considered at the same space point, i.e.,

ϕ′(x′i) = ϕ(xi). (3.161)

The infinitesimal symmetry variation is:

δsϕ(x) = ϕ′(x) − ϕ(x). (3.162)

For infinitesimal rotations (3.47),

δsxi = −ϕkεkijxj = −ωijxj , (3.163)

we see that

δsϕ(x) = ϕ′(x0, x′i − δxi) − ϕ(x)

= ∂iϕ(x)xjωij. (3.164)

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3.7 Angular Momentum 91

For a rotationally Lorentz-invariant Lagrangian density which has no explicit x-dependence:

L(x) = L(ϕ(x), ∂ϕ(x)), (3.165)

the symmetry variation is

δsL(x) = L(ϕ′(x), ∂ϕ′(x)) − ϕ(ϕ(x), ∂ϕ(x))

=∂L∂ϕ(x)

δsϕ(x) +∂L

∂∂aϕ(x)∂aδsϕ(x) (3.166)

Inserting (3.164), this becomes

δsL =

[

∂L∂ϕ

∂iϕxj +

∂L∂aϕ

∂a(∂iϕxj)

]

ωij

=

[

(∂iL)xj +∂L∂∂jϕ

∂iϕ

]

ωij. (3.167)

Since we are dealing with a rotation-invariant local Lagrangian density L(x) byassumption, the derivative ∂L/∂∂aϕ is a vector proportional to ∂aϕ. Hence thesecond term in the brackets is symmetric and vanishes upon contraction with theantisymmetric ωij. This allows us to express δsL as a pure derivative term

δsL = ∂i(

L xjωij)

. (3.168)

Calculating δsL once more using the Euler-Lagrange equations gives

δsL =∂L∂Lδsϕ+

∂L∂∂aϕ

∂aδsϕ (3.169)

=

(

∂L∂ϕ

− ∂a∂L∂∂aϕ

)

δsϕ+ ∂a

(

∂L∂∂aϕ

δsϕ

)

= ∂a

(

∂L∂∂aϕ

∂iϕ xj)

ωij.

Thus we find the for the Noether current densities (3.100):

Lij,a =

(

∂L∂∂aϕ

∂iϕxj − δi

aL xj)

− (i↔ j), (3.170)

which have no four-divergence

∂aLij,a = 0. (3.171)

The current densities can be expressed in terms of the canonical energy-momentumtensor (3.134) as as

Lij,a = xiΘja − xjΘia. (3.172)

The associated Noether charges

Lij =∫

d3xLij,a (3.173)

are the time-independent components of the total angular momentum of the fieldsystem.

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92 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

3.8 Four-Dimensional Angular Momentum

Consider now pure Lorentz transformations (1.27). An infinitesimal boost to arapidity ζ i is described by a coordinate change [recall (1.34)]

x′a = Λabxb = xa − δa0ζ

ixi − δaiζix0. (3.174)

This can be written as

δxa = ωabxb. (3.175)

where for passive boosts

ωij = 0, ω0i = −ωi0 = ζ i. (3.176)

With the help of the tensor ωab, the boosts can be treated on the same footing withthe passive rotations (1.36), for which (3.175) holds with

ωij = ωij = εijkϕk, ω0i = ωi0 = 0. (3.177)

For both types fo transformations, the symmetry variations of the field are

δsϕ(x) = ϕ′(x′a − δxa) − ϕ(x)

= −∂aϕ(x)xbωab. (3.178)

For a Lorentz-invariant Lagrangian density, the symmetry variation can be shown,as in (3.168), to be a total derivative:

δsϕ = −∂a(Lxb)ωab (3.179)

and we obtain the Noether current densities

Lab,c = −(

∂L∂∂cϕ

∂cϕxb − δacL xb)

− (a↔ b) (3.180)

The right-hand side can be expressed in terms of the canonical energy-momentumtensor (3.134), yielding

Lab,c = −(

∂L∂∂cϕ

∂cϕxb − δacLxb)

− (a↔ b)

= xaΘbc − xbΘac. (3.181)

According to Noether’s theorem (3.101), these current densities have no four-divergence:

∂cLab,c = 0. (3.182)

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3.9 Spin Current 93

The charges associated with these current densities:

Lab ≡∫

d3xLab,0 (3.183)

are independent of time. For the particular form (3.176) of ωab, we recover the timeindependent components Lij of angular momentum.

The time-independence of Li0 is the relativistic version of the center-of-masstheorem (3.65). Indeed, since

Li0 =∫

d3x (xiΘ00 − x0Θi0), (3.184)

we can then define the relativistic center of mass

xiCM =

d3xΘ00xi∫

d3xΘ00(3.185)

and the average velocity

viCM = cd3xΘi0

d3xΘ00= c

P i

P 0(3.186)

Since∫

d3xΘi0 = P i is the constant momentum of the system, also viCM is a constant.Thus, the constancy of L0i implies the center of mass to move with the constantvelocity

xiCM(t) = xiCM,0 + viCM,0t (3.187)

with xiCM,0 = L0i/P 0.The Noether charges Lab are the four-dimensional angular momenta of the sys-

tem.It is important to point out that the vanishing divergence of Lab,c makes Θba

symmetric:

∂cLab,c = ∂c(x

aΘbc − xbΘac)

= Θba − Θba = 0. (3.188)

Thus, field theories which are invariant under spacetime translations and Lorentztransformations must have a symmetric canonical energy-momentum tensor.

Θab = Θba (3.189)

3.9 Spin Current

If the field ϕ(x) is no longer a scalar but has several spatial components, thenthe derivation of the four-dimensional angular momentum becomes slightly moreinvolved.

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94 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

3.9.1 Electromagnetic Fields

Consider first the case of electromagnetism where the relevant field is the four-vectorpotential Aa(x). When going to a new coordinate frame

x′a = Λabxb (3.190)

the vector field at the same point remains unchanged in absolute spacetime. Butsince the components Aa refer to two different basic vectors in the different frames,they must be transformed simultaneously with xa. Since Aa(x) is a vector, it trans-forms as follows:

A′a(x′) = ΛabA

b(x). (3.191)

For an infinitesimal transformation

δsxa = ωabx

b (3.192)

this implies the symmetry variation

δsAa(x) = A′a(x) − Aa(x) = A′a(x− δx) −Aa(x)

= ωabAb(x) − ωcbx

b∂cAa. (3.193)

The first term is a spin transformation, the other an orbital transformation. Theorbital transformation can also be written in terms of the generators Lab of theLorentz group defined in (3.81) as

δorbs Aa(x) = −iωbcLbcAa(x). (3.194)

The spin transformation of the vector field is conveniently rewritten with the helpof the 4 × 4 generators Lab in Eq. (1.51). Adding the two together, we form theoperator of total four-dimensional angular momentum

Jab ≡ 1 × Lab + Lab × 1, (3.195)

and can write the transformation (3.193) as

δorbs Aa(x) = −iωabJabA(x). (3.196)

If the Lagrangian density involves only scalar combinations of four-vectors Aa

and if it has no explicit x-dependence, it changes under Lorentz transformations likea scalar field:

L′(x′) ≡ L(A′(x′), ∂′A′(x′)) = L(A(x), ∂A(x)) ≡ L(x). (3.197)

Infinitesimally, this amounts to

δsL = −(∂aL xb)ωab. (3.198)

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3.9 Spin Current 95

With the Lorentz transformations being symmetry transformations in theNoether sense, we calculate as in (3.170) the current of total four-dimensional an-gular momentum:

Jab,c =1

c

[

∂L∂∂cAa

Ab −(

∂L∂∂cAd

∂aAdxb − δacL xb)

− (a↔ b)

]

. (3.199)

The prefactor 1/c is chosen to give these Noether currents of the electromagneticfield the convenientional dimension. In fact, the last two terms have the same formas the current Lab,c of the four-dimensional angular momentum of the scalar field.Here they are the corresponding quantities for the vector potential Aa(x):

Lab,c = −1

c

(

∂L∂∂cAd

∂aAdxb − δacL xb)

+ (a↔ b). (3.200)

Note that this current has the form

Lab,c =1

c

−i ∂L∂∂cAd

LabAd +[

δacLxb − (a↔ b)]

. (3.201)

where Lab are the differential operators of four-dimensional angular momentum(1.103) satisfying the commutation rules (1.71) and (1.72).

Just as the scalar case (3.181), the currents (3.200) can be expressed in terms ofthe canonical energy-momentum tensor as

Lab,c = xaΘbc − xbΘac. (3.202)

The first term in (3.199),

Σab,c =1

c

[

∂L∂∂cAb

Ab − (a↔ b)

]

, (3.203)

is referred to as the spin current . It can be written in terms of the 4× 4-generators(1.51) of the Lorentz group as

Σab,c = − ic

∂L∂∂cAd

(Lab)dσAσ. (3.204)

The two currents together

Jab,c(x) ≡ Lab,c(x) + Σab,c(x) (3.205)

are conserved, ∂cJab,c(x) = 0. Individually, the terms are not conserved.

The total angular momentum is given by the charge

Jab =∫

d3x Jab,0(x). (3.206)

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96 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

It is a constant of motion. Using the conservation law of the energy-momentumtensor we find, just as in (3.188), that the orbital angular momentum satisfies

∂cLab,c(x) = −

[

Θab(x) − Θba(x)]

. (3.207)

From this we find the divergence of the spin current

∂cΣab,c(x) = −

[

Θab(x) − Θba(x)]

. (3.208)

For the charges associated with orbital and spin currents

Lab(t) ≡∫

d3xLab,0(x), Σab(t) ≡∫

d3xΣab,0(x), (3.209)

this implies the following time dependence:

Lab(t) = −∫

d3x[

Θab(x) − Θba(x)]

,

Σab(t) =∫

d3x[

Θab(x) − Θba(x)]

. (3.210)

Fields with spin have always have a non-symmetric energy momentum tensor.Then the current Jab,c becomes, now back in natural units,

Jab,c =

(

∂δxs L∂∂cωab(x)

− δacLxb)

− (a↔ b) (3.211)

By the chain rule of differentiation, the derivative with respect to ∂, ωab(x) can comeonly from field derivatives. For a scalar field

∂δxs L∂∂cωab(x)

=∂L∂∂cϕ

∂δxs ϕ

∂ωab(x), (3.212)

and for a vector field∂δxs L

∂∂cωab(x)=

∂L∂∂cAd

∂δxsAd

∂ωab(3.213)

The alternative rule of calculating angular momenta is to introduce spacetime-dependent transformations

δxx = ωab(x)xb (3.214)

under which the scalar fields transform as

δsϕ = −∂cϕωcb(x)xb (3.215)

and the Lagrangian density as

δxs ϕ = −∂cLωcb(x)xb = −∂c(xbL)ωcb(x) (3.216)

By separating spin and orbital transformations of δxsAd we find the two contributions

σab,c and Lab,c to the current Jab,c of the total angular momentum, the latter receivinga contribution from the second term in (3.211).

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3.9 Spin Current 97

3.9.2 Dirac Field

We now turn to the Dirac field. Under a Lorentz transformation (3.190), this trans-forms according to the law

ψ(x′)Λ−→ ψ′Λ(x) =D(Λ)ψ(x), (3.217)

where D(Λ) are the 4×4 spinor representation matrices of the Lorentz group. Theirmatrix elements can most easily be specified for infinitesimal transformations. Foran infinitesimal Lorentz transformation

Λab = δa

b + ωab, (3.218)

under which the coordinates are changed by

δsxa = ωabx

b (3.219)

the spin transforms under the representation matrix

D(δab + ωa

b) =(

1 − i1

2ωabσ

ab)

ψ(x), (3.220)

where σab are the 4 × 4 matrices acting on the spinor space defined in Eq. (1.222).We have shown in (1.220) that the spin matrices Σab ≡ σab/2 satisfy the samecommutation rules (1.71) and (1.72) as the previous orbital and spin-1 generatorsLaba and Lab of Lorentz transformations.

The field has the symmetry variation [compare (3.193)]:

δsψ(x) = ψ′(x) − ψ(x) = D(δab + ωa

b)ψ(x− δx) − ψ(x)

= −i12ωabσ

abψ(x) − ωcbxb∂cψ(x)

= −i12ωab

[

Sab + Lab]

ψ(x) ≡ −i12ωabJ

abψ(x), (3.221)

the last line showing the separation into spin and orbital transformation for a Diracparticle.

Since the Dirac Lagrangian is Lorentz-invariant, it changes under Lorentz trans-formations like a scalar field:

L′(x′) = L(x). (3.222)

Infinitesimally, this amounts to

δsL = −(∂aLxb)ωab. (3.223)

With the Lorentz transformations being symmetry transformations in theNoether sense, we calculate the current of total four-dimensional angular momen-tum extending the formulas (3.181) and (3.199) for scalar field and vector potential.The result is

Jab,c =

(

−i ∂L∂∂cψ

σabψ − i∂L∂∂cψ

Labψ + cc

)

+[

δacLxb − (a↔ b)]

. (3.224)

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98 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

As before in (3.200) and (3.181), the orbital part of (3.224) can be expressed interms of the canonical energy-momentum tensor as

Lab,c = xaΘbc − xbΘac. (3.225)

The first term in (3.224) is the spin current

Σab,c =1

2

(

−i ∂L∂∂cψ

σabψ + cc

)

. (3.226)

Inserting (3.158), this becomes explicitly

Σab,c = − i

2ψγcσabψ =

1

2ψγ[aγbγc]ψ =

1

2εabcdψγdψ. (3.227)

The spin density is completely antisymmetric in the three indices.1

The conservation properties of the three currents are the same as in Eqs. (3.206)–(3.210).

Due to the presence of spin, the energy-momentum tensor is not symmetric.

3.10 Symmetric Energy-Momentum Tensor

Since the presence of spin is the cause for the asymmetry of the canonical energy-momentum tensor, it is suggestive that by an appropriate use of the spin currentshould be possible to construct a new modified momentum tensor

T ab = Θab + ∆Θba (3.228)

which is symmetric, while still having the fundamental property of Θab, that theintegral P a =

d3xT a0 is the total energy-momentum vector of the system. Thisis be the case if ∆Θa0 is a three-divergence of a spatial vector. Such a constructionwas found by Belinfante in 1939. He introduced the tensor [4]

T ab = Θab − 1

2∂c(Σ

ab,c − Σbc,a + Σca,b), (3.229)

whose symmetry is manifest, due to (3.208) and the symmetry of the last two termsunder a↔ b. Moreover, by the components

T a0 = Θa0 − 1

2∂c(Σ

a0,c − Σ0c,a + Σca,0) = xaT bc − xbT ac (3.230)

which gives the same total angular momentum as the canonical expression (3.205):

Jab =∫

d3x Jab,0. (3.231)

1This property is important for being able to construct a consistent quantum mechanics in aspace with torsion. See H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer

Physics , World Scientific Publishing Co., Singapore 1995, Second extended edition, pp. 1–850.

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3.10 Symmetric Energy-Momentum Tensor 99

Indeed, the zeroth component of (3.230) is

xaΘb0 − xbΘa0 − 1

2

[

∂k(Σa0,k − Σ0k,a + Σka,0)xb − (a↔ b)

]

. (3.232)

Integrating the second term over d3x and performing a partial integration gives, fora = 0, b = i:

−1

2

d3x[

x0∂k(Σi0,k − Σ0k,i + Σki,0) − xi∂k(Σ

00,k − Σ0k,0 + Σk0,0)]

=∫

d3xΣ0i,0,

(3.233)

and for a = i, b = j

−1

2

d3x[

xi∂k(Σj0,k − Σ0k,j + Σkj,0) − (i↔ j)

]

=∫

d3xΣij,0. (3.234)

The right-hand sides are the contributions of the spin to the total angular momen-tum.

For the electromagnetic field, the spin current (3.203) reads, explicitly

Σab,c = −1

c

[

F caAb − (a↔ b)]

. (3.235)

From this we calculate the Belinfante correction

∆Θab =1

2c[∂c(F

caAb − F cbAa) − ∂c(FabAc − F acAb) + ∂c(F

bcAa − F baAc)]

=1

c∂c(F

bcAa). (3.236)

Adding this to the canonical energy-momentum tensor (3.150)

Θab =1

c

[

−F bc∂aAc +

1

4gabF cdFcd

]

, (3.237)

we find the symmetric energy-momentum tensor

T ab =1

c

[

−F bcF

ac +1

4gabF cdFcd + (∂cF

bc)Aa]

. (3.238)

The last term vanishes for a free Maxwell field which satisfies ∂cFab = 0 [recall

(2.86)], and can be dropped. In this case T ab agrees with the previously constructedsymmetric energy-momentum tensor (1.272) of the electromagnetic field. The sym-metry of T ab can easily be verified using once more the Maxwell equation ∂cF

ab = 0.We have seen in (1.269) that the component cT 00(x) agrees with the well-known

energy density E(x) = (E2 + B2) /2 of the electromagnetic field, and that thecomponents c2T 0i(x) are equal to the Poynting vector of energy current densityS(x) = cE × B, so that the energy conservation law c2∂aT

0a(0) can be written as∂tE(x) + ∇ · S(x) = 0.

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100 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

In the presence of an external current, where the Lagrangian density is (2.83),the canonical energy-momentum tensor becomes

Θab =1

c

[

−F bc∂aAc +

1

4gabF cdFcd +

1

cgabjcAc

]

, (3.239)

generalizing (3.237).The spin current is again given by Eq. (3.235), leading to the Belinfante energy-

momentum tensor

T ab = Θab +1

c∂c(F

bcAa)

=1

c

[

−F bcF

ac +1

4gabF cdFcd +

1

cgabjcAc −

1

cjbAa

]

. (3.240)

The last term prevents T ab from being symmetric, unless the current vanishes. Dueto the external current, the conservation law ∂bT

ab = 0 is modified to

∂bTab =

1

c2Ac(x)∂

ajc(x). (3.241)

3.11 Internal Symmetries

In quantum field theory, an important role by classifying various actions is played byinternal symmetries. They do not involve any change in the spacetime coordinateof the fields, i.e., they have the form

φ′(x) = e−iαGφ(x) (3.242)

where G are the generators of some Lie group and α the associated transformationparameters. The field φ may have several indices on which the generator G acts asa matrix. The symmetry variation associated with (3.242) is obviously

δsφ′(x) = −iαGφ(x) (3.243)

The most important example is that of a complex field φ and a generator G = 1,where (3.242) is simply a multiplication by a constant phase factor. One also speaksof U(1)-symmetry. Other important examples are those of a triplet or an octet offields φi with G being the generators of an SU(2) vector representation or an SU(3)octet representation (the adjoint representations of these groups). The first case isassociated with charge conservation in electromagnetic interactions, the other twowith isospin and SU(3) invariance in strong interactions. The latter symmetries are,however, not exact.

3.11.1 U(1)-Symmetry and Charge Conservation

Given a Lagrangian density L(x) = L(φ(x), ∂φ(x), x) depending only on the absolutesquares |φ|2, |∂φ|2, |φ∂φ|. Then L(x) is invariant under U(1)-transformations

δsφ(x) = −iφ(x) (3.244)

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3.11 Internal Symmetries 101

i.e., δsL = 0. By the chain rule of differentiation we find, using the Euler-Lagrangeequation

δsL =

(

∂L∂φ

− d

dt

∂L∂aφ

)

δsφ+

[

∂L∂∂aφ

δsφ

]

= 0 (3.245)

Inserting (3.244), we find that

jµ = − ∂L∂∂µφ

φ (3.246)

is a conserved current.For a free relativistic complex scalar field with a Lagrangian density

L(x) = ∂µϕ∗∂µϕ−m2ϕ∗ϕ (3.247)

we have to add the contributions of real and imaginary parts of the field φ in formula(3.246), and obtain the conserved current

jµ = −iϕ∗↔

∂µ ϕ (3.248)

where ϕ∗↔

∂µ ϕ denotes the left-minus-right derivative:

ϕ∗↔

∂ ϕ ≡ ϕ∗∂µϕ− (∂µϕ∗)ϕ. (3.249)

For a free Dirac field, we find from (3.246) the conserved current

jµ(x) = ψ(x)γµψ(x). (3.250)

3.11.2 SU(N)-Symmetry

For more general internal symmetry groups, the symmetry variations have the form

δsϕ = −iαiGiϕ, (3.251)

and the conserved currents are

jai = −i ∂L∂∂aϕ

Giϕ (3.252)

3.11.3 Broken Internal Symmetries

The physically important symmetries SU(2) of isospin and SU(3) are not exact. TheLagrange density is not strictly zero. In this case we remember the alternative deriva-tion of the conservation law from (3.109). We introduce the spacetime-dependentparameters α(x) and conclude from the extremality property of the action that

∂a∂Lε

∂∂aαi(x)=

∂Lε∂αi(x)

(3.253)

This implies the divergence law for the above derived current

∂ajai (x) =

∂Lε∂αi

. (3.254)

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102 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

3.12 Generating the Symmetry Transformationson Quantum Fields

As in quantum mechanical systems, the charges associated with the conserved cur-rents obtained in the previous section can be used to generate the transformationsof the fields from which they were derived. One merely has to invoke the canonicalfield commutation rules.

As an important example, consider the currents (3.252) of an internal U(N)-symmetry. Their charges

Qi = −i∫

d3x∂L∂∂aϕ

Giϕ (3.255)

can be written asQi = −i

d3xπGiϕ, (3.256)

where π(x) ≡ ∂L(x)/∂∂aϕ(x) is the canonical momentum of the field ϕ(x). Afterquantization, these fields satisfy the canonical commutation rules:

[π(x, t), ϕ(x′, t)] = −iδ(3)(x − x′),

[ϕ(x, t), ϕ(x′, t)] = 0, (3.257)

[π(x, t), π(x′, t)] = 0.

From this we derive directly the commutation rule between the quantized charges(3.256) and the field ϕ(x):

[Qi, ϕ(x)] = −αiGiϕ(x) (3.258)

We also find that the commutation rules among the quantized charges

[Qi, Qj ] = [Gi, Gj ]. (3.259)

Since these coincide with those of the matrices Gi this proves that the operators Qi

form a representation of the generators of symmetry group in the Fock space.It is important to realize that the commutation relations (3.258) and (3.259)

remain valid also in the presence of symmetry braking terms as long as these donot contribute to the canonical momentum of the theory. Such terms are called softsymmetry breaking terms. The charges are no longer conserved, so that we mustattach a time argument to the commutation relations (3.258) and (3.259). All timesin these relations must be the same, in order to invoke the equal-time canonicalcommutation rules.

The most important example is the commutation relation (3.91) which holds alsoin the presence of a potential V (q) in the Hamiltonian. This brakes translationalsymmetry, but does not contribute to the canonical momentum p = ∂L/∂q. In thiscase, the relation generalizes to

εj = iεi[pi(t), xj(t)], (3.260)

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3.13 Energy-Momentum Tensor of Relativistic Massive Point Particle 103

which is correct thanks to the validity of the canonical commutation relations (3.88)at arbitrary equal times, also in the presence of a potential.

Another important example are the commutation rules of the conserved chargesassociated with the Lorentz generators (3.225):

Jab ≡∫

d3xJab,0(x), (3.261)

which are the same as those of the 4× 4-matrices (1.51), and those of the quantummechanical generators (1.103):

[Jab, Jac] = −igaaJ bc. (3.262)

The generators Jab ≡ ∫

d3xJab,0(x), are sums Jab = Lab(t)+Σab(t) of charges (3.209)associated with orbital and spin rotations. According to (3.210), these individualcharges are time dependent, only their sum being conserved. Nevertheless, they bothgenerate Lorentz transformations: Lab(t) on the spacetime argument of the fields,and Σab(t) on the spin indices. As a consequence, they both satisfy the commutationrelations (3.262):

[Lab, Lac] = −igaaLbc, [Σab, Σac] = −igaaΣbc. (3.263)

The commutators (3.259) have played an important role in developing a theoryof strong interactions, where they first appeared in the form of a charge algebra ofthe broken symmetry SU(3) × SU(3) of weak and electromagnetic charges. Thissymmetry will be discussed in more detail in Chapter 10.

3.13 Energy-Momentum Tensor of Relativistic MassivePoint Particle

If we want to study energy and momentum of charged relativistic point particles inan electromagnetic field it is useful to consider the action (3.67) with (3.69) as anintegral over a Lagrangian density:

A =∫

d4xL(x), with L(x) =∫ τb

τadτ L(xa(τ))δ(4)(x− x(τ)). (3.264)

Then we can derive for point particles similar local conservation laws as for fields.Instead of doing this, however, we shall simply take the already known global conser-vation laws and convert them into the local ones by inserting appropriate δ-functionswith the help of the trivial identity

d4x δ(4)(x− x(τ)) = 1. (3.265)

Consider for example the conservation law (3.71) for the momentum (3.72). Withthe help of (3.265) this becomes

0 = −∫

d4x∫ ∞

−∞dτ

[

d

dτpc(τ)

]

δ(4)(x− x(τ)). (3.266)

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104 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

Note that in this expression the boundaries of the four-volume contain the infor-mation on initial and final times. We then perform a partial integration in τ , andrewrite (3.266) as

0 = −∫

d4x∫ ∞

−∞dτ

d

[

pc(τ)δ(4)(x− x(τ))

]

+∫

d4x∫ ∞

−∞dτpc(τ)∂τδ

(4)(x− x(τ)).

(3.267)

The first term vanishes if the orbits come from and disappear into infinity. Thesecond term can be rewritten as

0 = −∫

d4x ∂b

[∫ ∞

−∞dτpc(τ)x

b(τ)δ(4)(x− x(τ))]

. (3.268)

This shows that

Θcb(x) ≡ m∫ ∞

−∞dτ xc(τ)xb(τ)δ(4)(x− x(τ)) (3.269)

satisfies the local conservation law

∂bΘcb(x) = 0. (3.270)

This is the conservation law for the energy-momentum tensor of a massive pointparticle.

The total momenta are obtained from the spatial integrals over Θc0:

P a(t) ≡∫

d3xΘc0(x). (3.271)

For point particles, they coincide with the canonical momenta pa(t). If the La-grangian depends only on the velocity xa and not on the position xa(t), the momentapa(t) are constants of motion: pa(t) ≡ pa.

The Lorentz invariant quantity

M2 = P 2 = gabPaP b (3.272)

is called the total mass of the system. For a single particle it coincides with themass of the particle.

Subjecting the orbits xa(τ) to Lorentz transformations according to the rules ofthe last section we find the currents of total angular momentum

Lab,c ≡ xaΘbc − xbΘac, (3.273)

to satisfy the conservation law:∂cL

ab,c = 0. (3.274)

A spatial integral over the zeroth component of the current Lab,c yields the conservedcharges:

Lab(t) ≡∫

d3xLab,0(x) = xapb(t) − xbpa(t). (3.275)

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3.14 Energy-Momentum Tensor of Massive Charged Particle in Electromagnetic Field105

3.14 Energy-Momentum Tensor of Massive Charged

Particle in Electromagnetic Field

Let us also consider an important combination of a charged point particle and anelectromagnetic field Lagrangian

A=−mc∫ τb

τadτ√

gabxa(τ)xb(τ) −1

4

d4xFabFab − e

c

∫ τb

τadτxa(τ)Aa(x(τ)).

(3.276)

By varying the action in the particle orbits, we obtain the Lorentz equation of motion

dpa

dτ=e

cF a

bxb(τ). (3.277)

By varying the action in the vector potential, we find the Maxwell-Lorentz equation

−∂bF ab =e

cxb(τ). (3.278)

The action (3.276) is invariant under translations of the particle orbits and theelectromagnetic fields. The first term is obviously invariant, since it depends onlyon the derivatives of the orbital variables xa(τ). The second term changes undertranslations by a pure divergence [recall (3.132)]. Also the interaction term changesby a pure divergence, which is seen as follows: Since the symmetry variation changesxb(τ) → xb(τ) − εb, under which xa(τ) is invariant,

xa(τ) → xa(τ), (3.279)

and Aa(xb) changes as follows:

Aa(xb) → A′a(x

b) = Aa(xb + εb) = Aa(x

b) + εb∂aAa(xb), (3.280)

Altogether we obtain

δsL = εb∂bm

L . (3.281)

We now we calculate the same variation once more invoking the equations ofmotion. This gives

δsA =∫

dτd

∂Lm

∂x′aδsx

a +∫

d4x∂

em

L∂∂cAa

δsAa. (3.282)

The first term can be treated as in (3.267)–(3.268) after which it acquires the form

−∫ τb

τadτ

d

(

pa +e

cAa

)

= −∫

d4x∫ ∞

−∞dτ

d

[(

pa +e

cAa

)

δ(4)(x− x(τ)]

(3.283)

+∫

d4x∫ ∞

−∞dτ(

pa +e

cAa

)

d

dτδ(4)(x− x(τ))

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106 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

and thus, after dropping boundary terms,

−∫ τb

τadτ

d

(

pa +e

cAa

)

= ∂c

d4x∫ ∞

−∞dτ(

pa +e

cAa

)

dxc

dτδ(4)(x− x(τ)).(3.284)

The electromagnetic part is the same as before, since the interaction contains noderivative of the gauge field. In this way we find the canonical energy-momentumtensor

Θab(x) =∫

dτ(

pa +e

cAa)

xb(τ)δ(4)(x− x(τ)) − F bc∂aAc +

1

4gabF cdFcd. (3.285)

Let us check its conservation by calculating the divergence:

∂bΘab(x) =

dτ(

p+e

cAa

)

xb(τ)∂bδ(4)(x− x(τ))

−∂bF bc∂aAc − F b

c∂b∂aAc +

1

4∂a(F cdFcd). (3.286)

The first term is, up to a boundary term, equal to

−∫

dτ(

pa+e

τAa)

d

dτδ(4)(x− x(τ))=

[

d

(

pa+e

cAa)

]

δ(4)(x− x(τ)). (3.287)

Using the Lorentz equation of motion (3.277), this becomes

e

c

∫ ∞

−∞dτ

(

F abxb(τ) +

d

dτAa)

δ(4)(x− x(τ)). (3.288)

Inserting the Maxwell equation

∂bFab = −e

dτ(dxa/dτ)δ(4)(x− x(τ)), (3.289)

the second term in Eq. (3.286) can be rewritten as

−ec

∫ ∞

−∞dτdxcdτ

∂aAcδ(4)(x− x(τ)), (3.290)

which is the same as

−ec

(

dxadτ

F ac +dxcdτ

∂cAa)

δ(4)(x− x(τ)), (3.291)

thus canceling (3.288). The third term in (3.286) is, finally, equal to

−F bc∂aFb

c +1

4∂a(F cdFcd), (3.292)

due to the antisymmetry of F bc. By rewriting the homogeneous Maxwell equation,the Bianchi identity (2.88), in the form

∂cFab + ∂aFbc + ∂bFca = 0, (3.293)

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3.14 Energy-Momentum Tensor of Massive Charged Particle in Electromagnetic Field107

and contracting it with F ab, we see that the term (3.292) vanishes identically.

It is easy to construct from (3.285) Belinfante’s symmetric energy momentumtensor. We merely observe that the spin density is entirely due to the vector poten-tial, and hence the same as before [see (3.235)]

Σab,c = −[

F caAb − (a↔ b)]

. (3.294)

Hence the additional piece to be added to the canonical energy momentum tensoris again [see (3.236)]

∆Θab = ∂c(FabAa) =

1

2(∂cF

bcAa + F bc∂cAa). (3.295)

The last term in this expression serves to symmetrize the electromagnetic part ofthe canonical energy-momentum tensor and brings it to the Belinfante form:

em

Tab = −F b

cFac +

1

4gabF cdFcd. (3.296)

The second-last term in (3.295), which in the absence of charges vanished, is neededto symmetrize the matter part of Θab. Indeed, using once more Maxwell’s equation,it becomes

−ec

dτ xb(τ)Aaδ(4)(x− x(τ)), (3.297)

thus canceling the corresponding term in (3.285). In this way we find that the totalenergy-momentum tensor of charged particles plus electromagnetic fields is simplythe term of the two symmetric energy-momentum tensor.

T ab =m

Tab+

em

Tab (3.298)

= m∫ ∞

−∞dτ xaxbδ(4)(x− x(τ)) − F b

cFac +

1

4gabF cdFcd.

For completeness, let us cross check also its conservation. Forming the divergence∂bT

ab, the first term gives now only

e

c

dτ xb(τ)F ab(x(τ)), (3.299)

in contrast to (3.288), which is canceled by the divergence in the second term

−∂bF bcF

ac = −ec

dτ xc(τ)Fac(x(τ)), (3.300)

in contrast to (3.291).

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108 3 Continuous Symmetries and Conservation Laws. Noether’s Theorem

Notes and References

For more details on classical electromagnetic fields seeL.D. Landau, E. Lifshitz, The Classical Theory of Fields , Addison-Wesley, Reading,MA, 1951;A.O. Barut, Electrodynamics and Classical Theory of Fields and Particles , MacMil-lan, New York, N.Y. 1964;J.D. Jackson, Classical Electrodynamics , John Wiley & Sons, New York, N.Y., 1975.

The individual citations refer to:

[1] S. Coleman and J.H. Van Vleck, Phys. Rev. 171, 1370 (1968).

[2] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets World Scientific, Singapore 2006, 4th ex-tended edition, pp. 1–1546 (kl/b5), where kl is short for the www addresshttp://www.physik.fu-berlin.de/~kleinert.

[3] E. Noether, Nachr. d. vgl. Ges. d. Wiss. Gottingen, Math-Phys. Klasse, 2, 235(1918);See alsoE. Bessel-Hagen, Math. Ann. 84, 258 (1926);L. Rosenfeld, Me. Acad. Roy. Belg. 18, 2 (1938);F. Belinfante, Physica 6, 887 (1939).

[4] The Belinfante energy-momentum tensor is discussed in detail inH. Kleinert, Gauge Fields in Condensed Matter , Vol. II Stresses and Defects ,World Scientific Publishing Co., Singapore 1989, pp. 744-1443 (kl/b2).

H. Kleinert, MULTIVALUED FIELDS

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There are painters who transform the sun to a yellow spot,

but there are others who transform a yellow spot into the sun

Pablo Picasso (1881 - 1973)

4Multivalued Gauge Transformations

in Magnetostatics

For the upcoming development of a theory of gravitation with torsion it will beimportant to realize that it is possible to find a way to transform physical lawsin euclidean space into spaces with curvature and torsion. This can be done bya geometric generalization of a well-known field-theoretic technique developed byDirac to introduce magnetic monopoles into electrodynamics. So far, no magneticmonopoles have been discovered in nature, but the mathematics used by Dirac willsuggest us how to proceed in the geometric situation.

4.1 Vector Potential of Current Distribution

Let us begin by recalling the standard description of magnetism in terms of vec-tor potentials. Since there are no magnetic monopoles in nature, a magnetic fieldB(x) satisfies the identity ∇ · B(x) = 0, implying that only two of the three fieldcomponents of B(x) are independent. To account for this, one usually expresses amagnetic field B(x) in terms of a vector potential A(x), setting B(x) = ∇×A(x).Then Ampere’s law, which relates the magnetic field to the electric current densityj(x) by ∇ × B = j(x), becomes a second-order differential equation for the vectorpotential A(x) in terms of an electric current

∇ × [∇ ×A](x) = j(x). (4.1)

In this chapter we are using natural units with c = 1 to save recurring factors of c).The vector potential A(x) is a gauge field . Given A(x), any locally gauge-

transformed fieldA(x) → A′(x) = A(x) + ∇Λ(x) (4.2)

yields the same magnetic field B(x). This reduces the number of physical degreesof freedom in the gauge field A(x) to two, just as those in B(x). In order forthis to hold, the transformation function must be single-valued, i.e., it must havecommuting derivatives

(∂i∂j − ∂j∂i)Λ(x) = 0. (4.3)

109

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110 4 Multivalued Gauge Transformations in Magnetostatics

The equation for absence of magnetic monopoles ∇ ·B = 0 is ensured if the vectorpotential has commuting derivatives

(∂i∂j − ∂j∂i)A(x) = 0. (4.4)

This integrability property makes ∇ · B = 0 a Bianchi identity in this gauge fieldrepresentation of the magnetic field [recall the generic definition after Eq. (2.87)].

In order to solve (4.1), we remove the gauge ambiguity by choosing a particulargauge, for instance the transverse gauge ∇ ·A(x) = 0 in which ∇ × [∇ ×A(x)] =−∇

2A(x), and obtain

A(x) =1

d3x′j(x′)

|x − x′| . (4.5)

The associated magnetic field is

B(x) =1

d3x′j(x′) × R′

R′3, R′ ≡ x′ − x. (4.6)

This standard representation of magnetic fields is not the only possible one.There exists another one in terms of a scalar potential Λ(x), which must, however,be multivalued to account for the two physical degrees of freedom in the magneticfield.

4.2 Multivalued Gradient Representation of Magnetic Field

Consider an infinitesimally thin closed wire carrying an electric current I along theline L. It corresponds to a current density

j(x) = I (x;L), (4.7)

where (x;L) is the δ-function on the closed line L:

(x;L) =∫

Ldx′ δ(3)(x − x′). (4.8)

For a closed line L, this function has zero divergence:

∇· (x;L) = 0. (4.9)

This follows from the property of the δ-function on an arbitrary open line Lx2x1

connecting the points x1 and x2 defined by

(x;Lx2x1

) =∫ x2

x1

dx′ δ(3)(x − x′). (4.10)

which satisfies

∇· (x;Lx2x1

) = δ(x1) − δ(x2). (4.11)

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4.2 Multivalued Gradient Representation of Magnetic Field 111

B(x) = ∇Ω(x)

Figure 4.1 Infinitesimally thin closed current loop L. The magnetic field B(x) at the

point x is proportional to the solid angle Ω(x) under which the loop is seen from x. In

any single-valued definition of Ω(x), there is some surface S across which Ω(x) jumps by

4π. In the multivalued definition, this surface is absent.

For closed loops, the right-hand side of (4.11) vanishes.As an example, take a line Lx2

x1which runs along the positive z-axis from z1 to

z2, so that

(x;Lx2x1

) =∫ z2

z1dz′ δ(x)δ(y)δ(z − z′) = δ(x)δ(y)[Θ(z − z1) − Θ(z − z2)], (4.12)

and

∇· (x;Lx2x1

) = δ(x)δ(y) [δ(z − z1) − δ(z − z2)] = δ(x1) − δ(x2). (4.13)

From Eq. (4.5) we obtain the associated vector potential

A(x) =I

Ldx′

1

|x − x′| , (4.14)

yielding the magnetic field

B(x) =I

L

dx′ × R′

R′3, R′ ≡ x′ − x. (4.15)

The same result will now be derived from a multivalued scalar field. Let Ω(x;S)be the solid angle under which the current loop L is seen from the point x (seeFig. 4.1). If S denotes an arbitrary smooth surface enclosed by the loop L, and dS′

a surface element, then Ω(x;S) can be calculated from the surface integral

Ω(x;S) =∫

S

dS′ · R′R′3

. (4.16)

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112 4 Multivalued Gauge Transformations in Magnetostatics

The argument S in Ω(x;S) emphasizes that the definition depends on the choice ofthe surface S. The range of Ω(x;S) is from −2π to 2π, as can most easily be seenif L lies in the xy-plane and S is chosen to lie in the same place. Then we find forΩ(x;S) the value 2π for x just below S, and −2π just above. We form the vectorfield

B(x;S) =I

4π∇Ω(x;S), (4.17)

which is equal to

B(x;S) =I

SdS ′k∇

R′kR′3

= − I

SdS ′k∇

′R′k

R′3. (4.18)

This can be rearranged to

Bi(x;S) = − I

[

S

(

dS ′k ∂′i

R′kR′3

− dS ′i ∂′k

R′kR′3

)

+∫

SdS ′i ∂

′k

R′kR′3

]

. (4.19)

With the help of Stokes’ theorem∫

S(dSk∂i − dSi∂k)f(x) = εkil

Ldxlf(x), (4.20)

and the relation ∂′k(R′k/R

′3) = 4πδ(3)(x − x′), this becomes

B(x;S) = −I[

1

L

dx′ ×R′

R′3+∫

SdS′δ(3)(x − x′)

]

. (4.21)

The first term is recognized to be precisely the magnetic field (4.15) of the current I.The second term is the singular magnetic field of an infinitely thin magnetic dipolelayer lying on the arbitrarily chosen surface S enclosed by L.

The second term is a consequence of the fact that the solid angle Ω(x;S) wasdefined by the surface integral (4.16). If x crosses the surface S, the solid anglejumps by 4π.

It is useful to re-express Eq. (4.18) in a slightly different way. By analogy with(4.22) we define a δ-function on a surface as

(x;S) =∫

SdS′ δ(3)(x − x′), (4.22)

and observe that Stokes’ theorem (4.20) can be written as an identity for δ-functions:

∇× (x;S) = (x;L), (4.23)

where L is the boundary of the surface S. This equation proves once more the zerodivergence (4.9).

Using the δ-function on a surface S, we can rewrite (4.16) as

Ω(x;S) =∫

d3x′ (x′;S) · R′

R′3, (4.24)

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4.2 Multivalued Gradient Representation of Magnetic Field 113

and (4.18) as

B(x;S) = − I

d3x′ δk(x′;S)∇′

R′kR′3

, (4.25)

and (4.19), after an integration by parts, as

Bi(x;S)=I

1

d3x′ [∂′iδk(x′;S) − ∂′kδi(x

′;S)]R′kR′3

−∫

d3x′δi(x′;S)∇′ · R

R′3

. (4.26)

The divergence at the end yields a δ(3)-function, and we obtain

Bi(x;S)=−I[

1

d3x′[∇× (x;S)] × R′

R′3+∫

d3x′ (x′;S) δ(3)(x − x′)

]

. (4.27)

Using (4.23) and (4.22), this is once more equal to (4.21)Stokes theorem written in the form (4.23) displays an important property. If we

move the surface S to S ′ with the same boundary, the δ-function δ(x;S) changes by

(x;S) → (x;S ′) = (x;S) + ∇δ(x;V ), (4.28)

where

δ(x;V ) ≡∫

d3x′ δ(3)(x − x′), (4.29)

and V is the volume over which the surface has swept. Under this transformation,the curl on the left-hand side of (4.23) is invariant. Comparing (4.28) with (4.2) weidentify (4.28) as a novel type of gauge transformation [1, 2].The magnetic field inthe first term of (4.27) is invariant under this, the second is not. It is then obvioushow to find a gauge-invariant magnetic field: we simply subtract the singular S-dependent term and form

B(x) =I

4π[∇Ω(x;S) + 4π(x;S)] . (4.30)

This field is independent of the choice of S and coincides with the magnetic field(4.15) derived in the usual gauge theory. To verify this explicitly we calculate thechange of the solid angle (4.16) under a change of S. For this we rewrite (4.24) as

Ω(x;S) = −∫

d3x′∇′1

R′· (x′;S) = − 4π

∇2∇ · (x;S). (4.31)

Performing the vortex gauge transformation (4.28), the solid angle changes by

∆Ω(x;S) = − 4π

∇2∇ · ∇δ(x;V ) = −4πδ(x;L), (4.32)

so that (4.30) is indeed invariant.

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114 4 Multivalued Gauge Transformations in Magnetostatics

Hence the description of the magnetic field as a gradient of field Ω(x;S) is com-pletely equivalent to the usual gauge field description in terms of the vector potentialA(x). Both are gauge theories, but of a completely different type.

The gauge freedom (4.28) can be used to move the surface S into a standardconfiguration. One possibility is to choose S so that the third component of (x;S)vanishes. This is called the axial gauge. If (x;S) does not have this property, wecan always shift S by a volume V determined by the equation

δ(V ) = −∫ z

−∞δz(x;S), (4.33)

and the transformation (4.28) will produce a (x;S) in the axial gauge δ3(x;S) = 0.A general differential relation between δ-functions on volumes and surfaces re-

lated to (4.33) is

∇δ(x;V ) = −(x;S). (4.34)

There exists another possibility of defining a solid angle Ω(x;L) which is inde-pendent of the shape of the surface S and depends only on the boundary line L ofS. This is done by analytic continuation of Ω(x;S) through the surface S. Thisremoves the jump and produces a multivalued function Ω(x;L) ranging from −∞ to∞. At each point in space, there are infinitely many Riemann sheets whose branchline is L. The values of Ω(x;L) on the sheets differ by integer multiples of 4π. Fromthis multivalued function, the magnetic field (4.15) can be obtained as a simplegradient:

B(x) =I

4π∇Ω(x;L). (4.35)

Ampere’s law (4.1) implies that the multivalued solid angle Ω(x;L) satisfies theequation

(∂i∂j − ∂j∂i)Ω(x;L) = 4πεijkδk(x;L). (4.36)

Thus, as a consequence of its multivaluedness, Ω(x;L) violates the Schwarz integra-bility condition. This makes it an unusual mathematical object to deal with. It is,however, perfectly suited to describe the magnetic field of an electric current alongL.

Let us see explicitly how Eq. (4.36) is fulfilled by Ω(x;L), let us go to twodimensions where the loop corresponds to two points (in which the loop intersectsa plane). For simplicity, we move one of them to infinity, and place the other at thecoordinate origin. The role of the solid angle Ω(x;L) is now played by the azimuthalangle ϕ(x) of the point x with respect to the origin:

ϕ(x) = arctanx2

x1. (4.37)

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4.2 Multivalued Gradient Representation of Magnetic Field 115

The function arctan(x2/x1) is usually made unique by cutting the x-plane from theorigin along some line C to infinity, preferably along a straight line to x = (−∞, 0),and assuming ϕ(x) to jump from π to −π when crossing the cut. The cut correspondsto the magnetic dipole surface S in the integral (4.16). In contrast to this, we shalltake ϕ(x) to be the multivalued analytic continuation of this function. Then thederivative ∂i yields

∂iϕ(x) = −εijxj

(x1)2 + (x2)2. (4.38)

This is in contrast to the derivative ∂iϕ(x) of the single-valued definition of ∂iϕ(x)which would contain an extra δ-function εijδj(C;x) across the cut C, correspondingto the second term in (4.21). When integrating the curl of the derivative (4.38)across the surface s of a small circle c around the origin, we obtain by Stokes’theorem

sd2x(∂i∂j − ∂j∂i)ϕ(x) =

cdxi∂iϕ(x), (4.39)

which is equal to 2π for the multivalued definition of ϕ(x) [3]. This result impliesthe violation of the integrability condition as in (4.48):

(∂1∂2 − ∂2∂1)ϕ(x) = 2πδ(2)(x), (4.40)

whose three-dimensional generalization is (4.36). In the single-valued definition ofϕ(x) with the jump by 2π across the cut C, the right-hand side of (4.39) wouldvanish, since the contribution from the jump would cancel the integral along c, sothat ϕ(x) would satisfy the integrability condition (4.36).

On the basis of Eq. (4.40) we may construct a Green function for solving thecorresponding differential equation with an arbitrary source, which is a superpositionof infinitesimally thin line-like currents piercing the two-dimensional space at thepoints xn:

j(x) =∑

n

Inδ(2)(x − xn), (4.41)

where In are currents. We may then easily solve the differential equation

(∂1∂2 − ∂2∂1)f(x) = j(x), (4.42)

with the help of the Green function

G(x,x′) =1

2πϕ(x − x′) (4.43)

which satisfies

(∂1∂2 − ∂2∂1)G(x − x′) = δ(2)(x − x′). (4.44)

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116 4 Multivalued Gauge Transformations in Magnetostatics

The solution of (4.42) is obviously

f(x) =∫

d2x′G(x,x′)j(x). (4.45)

The gradient of f(x) yields the magnetic field of an arbitrary set of line-like currentsvertical to the plane under consideration.

It is interesting to realize that the Green function (4.43) is the imaginary partof the complex function (1/2π) log(z − z′) with z = x1 + ix2, whose real part(1/2π) log |z − z′| is the Green function G∆(x − x′) of the two dimensional Pois-son equation:

(∂21 + ∂2

2)G∆(x − x′) = δ(2)(x − x′). (4.46)

It is important to point out that the superposition of line-like currents cannotbe smeared out into a continuous distribution. The integral (4.45) yields the super-position of multivalued functions

f(x) =1

n

In arctanx2 − x2

n

x1 − x1n

, (4.47)

which is properly defined only if one can clearly continue it analytically into allRiemann sheets branching off from the endpoints of the cut at the origin. If wewere to replace the sum by an integral, this possibility would be lost. Thus itis, strictly speaking, impossible to represent arbitrary continuous magnetic fields asgradients of superpositions of scalar potentials Ω(x;L). This, however, is not a severedisadvantage of this representation since arbitrary currents can be approximated bya superposition of line-like currents with any desired accuracy, and the same will betrue for the associated magnetic fields.

The arbitrariness of the shape of the jumping surface is the origin of a furtherinteresting gauge structure which has interesting physical consequences discussed inSubsection 4.6.

4.3 Generating Magnetic Fields by Multivalued GaugeTransformations

After this first exercise in multivalued functions, we now turn to another examplein magnetism which will lead directly to our intended geometric application. Weobserved before that the local gauge transformation (4.2) produces the same mag-netic field B(x) = ∇×A(x) only, as long as the function Λ(x) satisfies the Schwarzintegrability criterion (4.36)

(∂i∂j − ∂j∂i)Λ(x) = 0. (4.48)

Any function Λ(x) violating this condition would change the magnetic field by

∆Bk(x) = εkij(∂i∂j − ∂j∂i)Λ(x), (4.49)

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4.4 Magnetic Monopoles 117

thus being no proper gauge function. The gradient of Λ(x)

A(x) = ∇Λ(x) (4.50)

would be a nontrivial vector potential.By analogy with the multivalued coordinate transformations violating the inte-

grability conditions of Schwarz as in (4.36), the function Λ(x) will be called non-holonomic gauge function.

Having just learned how to deal with multivalued functions we may change ourattitude towards gauge transformations and decide to generate all magnetic fieldsapproximately in a field-free space by such improper gauge transformations Λ(x).By choosing for instance

Λ(x) =Φ

4πΩ(x), (4.51)

we see from (4.36) that this generates a field

Bk(x) = εkij(∂i∂j − ∂j∂i)Λ(x) = Φδk(x;L). (4.52)

This is a magnetic field of total flux Φ inside an infinitesimal tube. By a superpo-sition of such infinitesimally thin flux tubes analogous to (4.45) we can obviouslygenerate a discrete approximation to any desired magnetic field in a field-free space.

4.4 Magnetic Monopoles

Multivalued fields have also been used to describe magnetic monopoles [4, 5, 6]. Amonopole charge density ρm(x) is the source of a magnetic field B(x) as defined bythe equation

∇· B(x) = ρm(x). (4.53)

If B(x) is expressed in terms of a vector potential A(x) as B(x) = ∇× A(x),equation (4.53) implies the noncommutativity of derivatives in front of the vectorpotential A(x):

1

2εijk(∂i∂j − ∂j∂i)Ak(x) = ρm(x). (4.54)

Thus A(x) must be multivalued. Dirac in his famous theory of monopoles [7, 8] madethe field single-valued by attaching to the world line of the particle a jumping worldsurface, whose intersection with a coordinate plane at a fixed time forms the Diracstring , along which the magnetic field of the monopole is imported from infinity.This world surface can be made physically irrelevant by quantizing it appropriatelywith respect to the charge. Its shape in space is just as irrelevant as that of thejumping surface S in Fig. 4.1. The invariance under shape deformations constitute

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118 4 Multivalued Gauge Transformations in Magnetostatics

once more a second gauge structure of the type mentioned earlier and discussed inRefs. [9, 4, 10, 11, 2].

Once we allow ourselves to work with multivalued fields, we may easily go onestep further and express also A(x) as a gradient of a scalar field as in (4.50). Thenthe condition becomes

εijk∂i∂j∂kΛ(x) = ρm(x). (4.55)

Let us construct the field of a magnetic monopole of charge g at a point x0,which satisfies (4.53) with ρ(x) = g δ(3)(x − x0). Physically, this can be done onlyby setting up an infinitely thin solenoid (Dirac string) along an arbitrary line Lx0

which imports the flux from somewhere at infinity to the point x0 where the fluxemerges. The superscript 0 indicates that the line ends at x0. Inside this solenoid,the magnetic field is infinite, equal to

Binside(x;L) = g (x;Lx0), (4.56)

where (x;Lx0) is a modification of (4.10) in which the integral runs along the lineLx0 to x0:

(x;Lx0) =∫ x0

d3x′δ(3)(x − x′). (4.57)

The divergence of this function is concentrated at the endpoint x0 of the solenoid:

∇· (x;Lx0) = −δ(3)(x − x0). (4.58)

Similarly we may define a δ-function along a line Lx0which starts at x0 and runs

to somewhere at infinity:

(x;Lx0) =

x0

d3x′δ(3)(x − x′), (4.59)

which satisfies

∇· (x;Lx0) = δ(3)(x − x0). (4.60)

This describes a thin solenoid (Dirac string) which exports the magnetic flux fromx0 to infinity, corresponding to an antimonopole at x0.

As an example, take a line Lx0 which carries the flux from positive infinity tothe origin along the z-axis. If z denotes the unit vector along the z-axis, then

(x;L0) = z

∫ z

∞dz′ δ(x)δ(y)δ(z − z′) = z δ(x)δ(y)Θ(−z), (4.61)

so that ∇· (x;L0) = −δ(x)δ(y)δ(z) = −δ(3)(x). This agrees with (4.11) if we modeinitial point x1 to (0, 0,∞). If the flux is important from negative infinity to theorigin, one has

(x;L0) = z

∫ z

−∞dz′ δ(x)δ(y)δ(z − z′) = z δ(x)δ(y) [1 − Θ(z)] , (4.62)

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4.4 Magnetic Monopoles 119

with the same ∇· (x;L0) = −δ(3)(x).By analogy with the curl relation (4.23) we observe a further gauge invariance.

If we deform the line Lx0 with fixed endpoint x0, the δ-function (4.57) changes asfollows:

(x;Lx0) → (x;L′x0) = (x;Lx0) + ∇× (x;S), (4.63)

where S is the surface over which Lx0 has swept on its way to L′x0 . Under this gaugetransformation, the relation (4.58) is obviously invariant. We shall call this monopolegauge invariance. The flux (4.56) inside the solenoid is therefore a monopole gaugefield .

Note that with respect to the previous gauge transformations (4.28) which shifteda surface, the gradient is exchanged by a curl, and the opposite exchange relates theinvariants, which was a boundary line found from a curl in Eq. (4.23), and is herethe starting point of the line Lx0 found from the divergence in Eq. (4.58).

It is straightforward to construct the associated ordinary gauge field A(x) of themonopole. Consider first the Lx0-dependent field

A(x;Lx0) =g

d3x′∇′× (x′;Lx0)

R′= − g

d3x′ (x′;Lx0) × R′

R′3. (4.64)

The curl of the first expression is

∇×A(x;Lx0) =g

d3x′∇′× [∇′× (x′;Lx0)]

R′, (4.65)

and consists of two terms

g

d3x′∇′[∇′ · (x′;Lx0)]

R′− g

d3x′∇′2(x′;Lx0)

R′. (4.66)

After an integration by parts, and using (4.58), the first term is Lx0-independentand reads

g

d3x′δ(3)(x − x0)∇′ 1

R′=

g

x − x0

|x − x0|3. (4.67)

The second term becomes, after two integration by parts,

g (x′;Lx0). (4.68)

The first term is the desired magnetic field of the monopole. Its divergence isδ(3)(x − x0), which we wanted to archive. The second term is the monopole gaugefield, the magnetic field inside the solenoid. The total divergence of this field is, ofcourse, zero.

By analogy with (4.30) we now subtract the latter term and find the Lx0-independent magnetic field of the monopole

B(x) = ∇× A(x;Lx0) − g (x;Lx0), (4.69)

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120 4 Multivalued Gauge Transformations in Magnetostatics

which depends only on x0 and satisfies ∇· B(x) = g δ(3)(x − x0).Let us calculate the vector potential explictly for the monopole where the solenoid

comes in along Lx0 . Inserting (4.61) into the right-hand side of (4.64), we obtain

A(g)(x;Lx0) =− g

∫ ∞

zdz′

z × x√

x2 + y2 + (z′ − z)23/2

=− g

z × x

R(R− z)=

g

(y,−x, 0)

R(R− z). (4.70)

Alternatively, if Lx0 runs to −∞, so that (x;Lx0) is equal to −zΘ(−z)δ(x)δ(y),we obtain

A(g)(x;Lx0)=− g

∫ ∞

0dz′

z × x√

x2 + y2 + (z′ − z)23/2

=g

z × x

R(R + z)= − g

(y,−x, 0)

R(R + z). (4.71)

The vector potential has only azimuthal components. If we parametrize (x, y, z) interms of spherical coordinates as r(sin θ cosϕ, sin θ sinϕ, cos θ), these are

A(g)ϕ (x;Lx0) =

g sin θ

4πR(1 + cos θ)or A(g)

ϕ (x;Lx0) = − g sin θ

4πR(1 − cos θ), (4.72)

respectively.The shape of the line Lx0 (or Lx0

) can be brought to a standard form, whichcorresponds to fixing a gauge of the field (x;Lx0) or (x;Lx0

). For example, wemay always choose Lx0 to run along the positive z-axis.

An interesting observation is the following: If the gauge function Λ(x) is con-sidered as a nonholonomic displacement in some fictitious crystal dimension, thenthe magnetic field of a current loop which gives rise to noncommuting derivatives(∂i∂j − ∂j∂i)Λ(x) 6= 0 is the analog of a dislocation [compare (14.3)], and thus im-plies torsion in the crystal. A magnetic monopole, on the other hand, arises fromnoncommuting derivatives (∂i∂j − ∂j∂i)∂kΛ(x) 6= 0 in Eq. (4.55).

4.5 Minimal Magnetic Coupling of Particles fromMultivalued Gauge Transformations

Multivalued gauge transformations are the perfect tool to minimally couple electro-magnetism to any type of matter. Consider for instance a free nonrelativistic pointparticle with a Lagrangian

L =M

2x2. (4.73)

The equations of motion are invariant under a gauge transformation

L→ L′ = L+ ∇Λ(x) x, (4.74)

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4.5 Minimal Magnetic Coupling of Particles from Multivalued Gauge Transformations121

since this changes the action A =∫ tbta dtL merely by a surface term:

A′ → A = A + Λ(xb) − Λ(xa). (4.75)

The invariance is absent if we take Λ(x) to be a multivalued gauge function. In thiscase, a nontrivial vector potential A(x) = ∇Λ(x) (working in natural units withe = 1) is created in the field-free space, and the nonholonomically gauge-transformedLagrangian corresponding to (4.74),

L′ =M

2x2 + A(x) x, (4.76)

describes correctly the dynamics of a free particle in an external magnetic field.The coupling derived by multivalued gauge transformations is automatically in-

variant under additional ordinary single-valued gauge transformations of the vectorpotential

A(x) → A′(x) = A(x) + ∇Λ(x), (4.77)

since these add to the Lagrangian (4.76) once more the same pure derivative termwhich changes the action by an irrelevant surface term as in (4.75).

The same procedure leads in quantum mechanics to the minimal coupling of theSchrodinger field ψ(x). The action is A =

dtd3xL with a Lagrange density (innatural units with h = 1)

L = ψ∗(x)(

i∂t +1

2M∇

2)

ψ(x). (4.78)

The physics described by a Schrodinger wave function ψ(x) is invariant under arbi-trary local phase changes

ψ(x, t) → ψ′(x) = eiΛ(x)ψ(x, t), (4.79)

called local U(1) transformations. This implies that the Lagrange density (4.78)may equally well be replaced by the gauge-transformed one

L = ψ∗(x, t)(

i∂t +1

2MD2)

ψ(x, t), (4.80)

where −iD ≡ −i∇ − ∇Λ(x) is the operator of physical momentum.We may now go over to nonzero magnetic fields by admitting gauge transforma-

tions with multivalued Λ(x) whose gradient is a nontrivial vector potential A(x) asin (4.50). Then −iD turns into the covariant momentum operator

P = −iD = −i∇ − A(x), (4.81)

and the Lagrange density (4.80) describes correctly the magnetic coupling in quan-tum mechanics.

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122 4 Multivalued Gauge Transformations in Magnetostatics

As in the classical case, the coupling derived by multivalued gauge transforma-tions is automatically invariant under ordinary single-valued gauge transformationsunder which the vector potential A(x) changes as in (4.77), whereas the Schrodingerwave function undergoes a local U(1)-transformation (4.79). This invariance is a di-rect consequence of the simple transformation behavior of Dψ(x, t) under gaugetransformations (4.77) and (4.79) which is

Dψ(x, t) → Dψ′(x, t) = eiΛ(x)Dψ(x, t). (4.82)

Thus Dψ(x, t) transforms just like ψ(x, t) itself, and for this reason, D is calledgauge-covariant derivative. The generation of magnetic fields by a multivalued gaugetransformation is the simplest example for the power of the nonholonomic mappingprinciple.

After this discussion it is quite suggestive to introduce the same mathematicsinto differential geometry, where the role of gauge transformations is played byreparametrizations of the space coordinates.

4.6 Equivalence of Multivalued Scalar andSingle-Valued Vector Potential Representation

In the previous sections we have given examples for the use of multivalued fields indescribing magnetic phenomena. The multivalued gauge transformations by whichwe created line-like nonzero field configurations were shown to be the natural ori-gin of the minimal couplings to the classical actions as well as to the Schrodingerequation. It is interesting to establish the complete equivalence of the multivaluedscalar theory with the usual vector potential theory of magnetism we properly dealwith the freedom in choosing the jumping surfaces S.

To understand this we pose ourselves the problem of setting up an action for-malism for calculating the magnetic energy of a current loop in the gradient repre-sentation of the magnetic field. In this Euclidean field theory, the action is the fieldenergy

H =1

2

d3xB2(x). (4.83)

Inserting the gradient representation (4.35) of the magnetic field, we can write

H =I2

2(4π)2

d3x [∇Ω(x)]2. (4.84)

This holds for the multivalued solid angle Ω(x) which is independent of S. Inorder to perform field theoretic calculations, we must go over to the single-valuedrepresentation (4.30) of the magnetic field for which the energy is

H =I2

2(4π)2

d3x [∇Ω(x;S) + 4π(x;S)]2. (4.85)

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4.6 Equivalence of Multivalued Scalar and Single-Valued Vector Potential Representation123

The δ-function removes the unphysical field energy on the artificial magnetic dipolelayer on S.

The Hamiltonian is extremized by the scalar field (4.24). Moreover, due toinfinite field strength on the surface, all field configurations Ω(x;S ′) with a jumpingsurface S ′ different from S will have an infinite energy. Thus we may omit theargument S in Ω(x;S) and admit an arbitrary field Ω(x) to the Hamiltonian (4.85).Only the field (4.24) will give a finite contribution.

Let us calculate the magnetic field energy of the current loop from the energy(4.85). For this we rewrite the energy (4.85) in terms of an independent auxiliaryvector field B(x) as

H =∫

d3x

−1

2B2(x) − 1

4πB(x) · [∇Ω(x) + 4πI(x;S)]

. (4.86)

A partial integration brings the second term to

d3x1

4π∇· B(x) Ω(x).

Extremizing this in Ω(x) yields the equation

∇· B(x) = 0, (4.87)

implying that the field lines of B(x) form closed loops. This equation may be en-forced identically (as a Bianchi identity) by expressing B(x) as a curl of an auxiliaryvector potential A(x), setting

B(x) ≡ ∇×A(x). (4.88)

With this ansatz, the equation which brings the energy (4.86) to the form

H =∫

d3x

−1

2[∇×A(x)]2 − I [∇× A(x)] · (x;S)

. (4.89)

A further partial integration leads to

H =∫

d3x

−1

2[∇×A(x)]2 − IA(x) · [∇× (x;S)]

, (4.90)

and we identify in the linear term in A(x) the auxiliary current

j(x) ≡ I ∇× (x;S) = I (x;L), (4.91)

due to Stoke’s law (4.23). According to Eq. (4.9), this current is conserved for closedloops L.

The representation (4.90) of the energy is called the dually transformed versionof the original energy (4.85).

By extremizing the energy (4.89), we obtain Ampere’s law (4.1). Thus the aux-iliary quantities B(x), A(x), and j(x) coincide with the usual magnetic quantities

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124 4 Multivalued Gauge Transformations in Magnetostatics

with the same name. If we insert the explicit solution (4.5) of Ampere’s law intothe energy, we obtain the Biot-Savart energy for an arbitrary current distribution

H =1

d3x d3x′ j(x)1

|x − x′| j(x′). (4.92)

If we insert here two current filaments running parallel in thin wires, the energy(4.92) decreases with increasing distance suggesting, for a moment, that the forcebetween them is repulsive. The experimental force, however, is attractive. Thesign change is due to the fact that when increasing the distance of the wires wemust perform work against the inductive forces in order to maintain the constantcurrents. This work is not calculated above and turns out to be exactly twice theenergy gain implied by (4.92). The energy responsible for discussing the forces ofexternal current distributions is the free magnetic energy

F =1

2

d3xB2(x) −∫

d3x j(x) ·A(x). (4.93)

Extremizing this in A(x) yields the vector potential (4.5), and reinserting this into(4.93) we find, indeed, that the free Biot-Savart energy is the opposite of (4.92):

F |ext = − 1

d3x d3x′ j(x)1

|x − x′| j(x′). (4.94)

As a consequence, parallel wires with fixed currents attract each other rather thanrepel.

Note that the energy (4.89) is invariant under two mutually dual gauge transfor-mations, the usual magnetic one in (4.2), by which the vector potential receives agradient of an arbitrary scalar field, and the gauge transformation (4.28), by whichthe irrelevant surface S is moved to another configuration S ′.

Thus we have proved the complete equivalence of the gradient representation ofthe magnetic field to the usual gauge field representation. In the gradient repre-sentation, there exists a new type of gauge invariance which expresses the physicalirrelevance of the jumping surface appearing when using single-valued solid angles.

The energy (4.90) describes magnetism in terms of a double gauge theory [12],in which both the gauge of A(x) and the shape of S can be changed arbitrarily.By setting up a grand-canonical partition function of many fluctuating surfaces it ispossible to describe a large family of phase transitions mediated by the proliferationof line-like defects. Examples are vortex lines in the superfluid-normal transition inhelium, to be discussed in the next chapter, and dislocation and disclination linesin the melting transition of crystals, to be discussed later [9, 4, 10, 11, 2].

4.7 Multivalued Field Theory of Magnetic Monopolesand Electric Currents

Let us now go through the analogous discussion for a gas of monopoles at xn withstrings Lxn inporting their fluxed from infinity, and electric currents along closed

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4.7 Multivalued Field Theory of Magnetic Monopoles and Electric Currents 125

Ln′ . The free energy of fixed currents is given by the energy of the magnetic field(4.69) coupled to the currents as in the action Eq. (4.93):

F =∫

d3x

1

2

[

∇ × A− g∑

n

(x;Lxn)

]2

− IA(x) ·∑

n′

(x, Ln′)

. (4.95)

Extremizing this in A(x) we obtain

A(x) = − 1

∇2

[

g∑

n

∇ × (x;Lxn) + I∑

n′

(x, Ln′)

]

. (4.96)

Reinserting this into (4.95) yields three terms. First, there is an interactionbetween the current lines

HII =−I2

2

d3x∑

n,n′

(x;Ln)1

−∇2 (x;Ln′)=−I

2

2

n,n′

Lndxn

Ln′dxn′

1

|xn − xn′| ,(4.97)

which corresponds to (4.94). Second, there is an interaction between monopolestrings

g2

2

d3x

[

n

(x;Lxn)

]2

+

[

n

∇ × (x;Lxn)

]

1

∇2

[

n

∇ × (x;Lxn)

]

,(4.98)

which can be brought to the form

Hgg =g2

2

d3x

[

n

∇ · (x;Lxn)

]2

=g2

2

d3x

[

n

δ(x − xn)

]2

=g2

n,n′

1

|xn − xn′ | . (4.99)

Finally, there is an interaction between the monopoles and the currents

HgI = −gI∫

d3x∑

n,n′

∇ × (x;Lxn)1

∇2 (x;Ln′). (4.100)

An integration by parts brings this to the form

HgI = −gI∫

d3x∑

n,n′

(x;Lxn)1

∇2∇ × (x;Ln′)

= −gI∫

d3x∑

n,n′

(x;Lxn)1

∇2∇ × [∇ × (x;Sn′)] , (4.101)

which is equal to

HgI = H ′gI + ∆HgI , (4.102)

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126 4 Multivalued Gauge Transformations in Magnetostatics

with

H ′gI = −gI∫

d3x∑

n,n′

(x;Lxn)∇1

∇2 [∇ · (x;Sn′)] . (4.103)

and

∆HgI = gI∫

d3x∑

n,n′

(x;Lxn)(x;Sn′). (4.104)

Each integral in the sum yields an integer number which counts how often the linesLn pierce the surface Sn′ , so that

∆HgI = gIk, k = integer. (4.105)

Recalling (4.31), the interaction (4.103) can be rewritten as

HgI = −gI4π

d3x∑

n,n′

(x;Lxn)∇Ω(x;Sn′). (4.106)

An integration by parts and the relation (4.58) brings this to the form

HgI =gI

n,n′

Ω(xn;Sn′). (4.107)

It is proportional to the sum of the solid angles Ω(xn;Sn′) under which the currentloops Ln′ are seen from the monopoles at xn. The result does not depend on thesurfaces Sn, only on the boundary lines Ln along which the currents flow.

The total interaction is obviously invariant under shape deformations of S, exceptfor the term (4.105). This term, however, is physically irrelevant provided we subjectthe charges Q in the currents to quantization rule

Qg = 2πk, k = integer. (4.108)

This rule was first found by Dirac [7].The quantization rule is a consequence of quantum theory. This is governed by

amplitudes which can be calculated from the classical action by means of a functionalintegral

Amplitude =∑

field configurations

eiA/h, (4.109)

where A is the full four-dimensional action of the system, which for static currentsand monopoles is simply

A = −∫

dtH = −Qg. (4.110)

This shows that ∆H in (4.105) does indeed not contribute to (4.109) if the quanti-zation condition (4.108) is fulfilled since it does not change the amplitude eiA/h.

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Notes and References 127

Notes and References

[1] This gauge freedom is independent of the electromagnetic one. SeeH. Kleinert, Phys. Lett. B 246, 127 (1990) (kl/205); Int. J. Mod. Phys. A 7,4693 (1992) (kl/203); Phys. Lett. B 293, 168 (1992) (kl/211), where kl isshort for the www address http://www.physik.fu-berlin.de/~kleinert.

[2] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, Superflow and VortexLines, World Scientific, Singapore, 1989 (kl/b1).

[3] The theory of multivalued fields developed first in detail in the textbook [2] isso unfamiliar to field theorists that Physical Review Letters found it of broadinterest to publish a rather trivial Comment paper on Eq. (4.40) byC. Hagen, Phys. Rev. Lett. 66, 2681 (1991),and a reply byR. Jackiw and S.-Y. Pi, Phys. Rev. Lett. 66, 2682 (1991).

[4] H. Kleinert, Int. J. Mod. Phys. A 7, 4693 (1992) (kl/203).

[5] H. Kleinert, Phys. Lett. B 246, 127 (1990) (kl/205).

[6] H. Kleinert, Phys. Lett. B 293, 168 (1992) (kl/211).

[7] P.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948),Phys. Rev. 74, 817 (1948).

[8] M.N. Saha, Ind. J. Phys. 10, 145 (1936);J. Schwinger, Particles, Sources and Fields , Vols. 1 and 2, Addison Wesley,Reading, Mass., 1970 and 1973;G. Wentzel, Progr. Theor. Phys. Suppl. 37, 163 (1966);E. Amaldi, in Old and New Problems in Elementary Particles, ed. by G. Puppi,Academic Press, New York (1968);D. Villaroel, Phys. Rev. D 14, 3350 (1972);Yu.D. Usachev, Sov. J. Particles Nuclei 4, 92 (1973);A.O. Barut, J. Phys. A 11, 2037 (1978);J.D. Jackson, Classical Electrodynamics , John Wiley and Sons, New York,1975, Sects. 6.12-6.13.

[9] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects,World Scientific, Singapore, 1989 (kl/b2).

[10] H. Kleinert, Nonholonomic Mapping Principle for Classical and Quantum Me-chanics in Spaces with Curvature and Torsion, Gen. Rel. Grav. 32, 769 (2000)(kl/258); Act. Phys. Pol. B 29, 1033 (1998) (gr-qc/9801003).

[11] H. Kleinert, Theory of Fluctuating Nonholonomic Fields and Applications:Statistical Mechanics of Vortices and Defects and New Physical Laws in Spaces

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128 4 Multivalued Gauge Transformations in Magnetostatics

with Curvature and Torsion, in: Proceedings of NATO Advanced Study Insti-tute on Formation and Interaction of Topological Defects, Plenum Press, NewYork, 1995, pp. 201–232 (kl/227).

[12] H. Kleinert, Double Gauge Theory of Stresses and Defects, Phys. Lett. A 97,51 (1983) (kl/107).

H. Kleinert, MULTIVALUED FIELDS

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Actions lie louder than words

Aristotle (384 - 322) BC

5

Multivalued Fields in Superfluids andSuperconductors

Multivalued fields play an important role in understanding a great variety of phasetransitions. In this chapter we shall discuss two simple but important examples.

5.1 Superfluid Transition

The simplest phase transitions which can be explained by multivalued field theoryis the so-called λ-transition of superfluid helium. The name has its origin in theshape of the peak in the specific heat observed at a critical temperature Tc ≈ 2.18Kshown in Fig. 5.1.

Figure 5.1 Specific heat of superfluid 4He. For very small T , it shows the typical power

behavior ∝ T 3 characteristic for massless excitations in three dimensions in the Debye

theory of specific heat. Here these excitations are phonons of the second sound. The peak

is caused by the proliferation of vortex loops at the superfluid-normal transition.

For temperatures T below Tc, the fluid shows no friction and possesses only mass-less excitations. These are the quanta of the second sound , called phonons. They

129

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130 5 Multivalued Fields in Superfluids and Superconductors

Figure 5.2 Energies of the elementary excitations in superfluid 4He measured by neutron

scattering showing the roton minimum near k ≈ 2 rA (data are taken from Ref. [1]).

cause the typical temperature behavior of the specific heat which in D dimensionsstarts out like

C ∼ TD. (5.1)

This was first explained in 1912 by Debye in his theory of specific heat [2], in whichhe generalized Planck’s theory of black-body radiation to solid bodies.

As the temperature rises, another type of excitations appears in the superfluid.These are the famous rotons whose existence was deduced in 1947 by Landau fromthe thermodynamic properties of the superfluid [3, 4]. Rotons are the excitations ofwavenumber near 2 rAwhere the phonon dispersion curve has a minimum. The fullshape of this curve can be measured by neutron scattering and is displayed in Fig.5.2.

As long as T stays sufficiently far below Tc, the thermodynamic properties of thesuperfluid are dominated by phonons and rotons. If the temperature approaches Tc,

Figure 5.3 Near Tc, more and more rotons join side by side to form surfaces whose

boundary appears as a large vortex loop. The adjacent roton boundaries cancel each

other.

the rotons join side by side and form large surfaces, a shown in Fig. 5.3. The adjacent

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5.1 Superfluid Transition 131

boundaries of the rotons cancel each other, so that the memory of the surfaces is lost,their shape becomes irrelevant, and only the boundaries of the surfaces are physicalobjects, observable as vortex loops. At Tc, the vortex loops become infinitely longand proliferate. The large activation energies for creating single rotons are overcomeby the high configurational entropy of the long vortex loops.

The inside of a vortex line consists of normal fluid since the large rotation velocitydestroys superfluidity. For this reason, the proliferation of the vortex loops fills thesystem with normal fluid, and the fluid looses its superfluid properties. The existenceof such a mechanism for a phase transition was realized more than fifty years agoby Onsager in 1949 [5]. It was re-emphasized by Feynman in 1955 [6], and turnedinto a proper disorder field theory in the 1980’s by the author [7]. The same ideawas advanced in 1952 by Shockley [8] who proposed a proliferation of defect lines insolids to be responsible for the melting transition. His work instigated the authorto develop a detailed theory of melting in the textbook [9].

The disorder field theory of superfluids and superconductors will be derived inSubsection 5.1.10, the melting theory in Chapter 10.

5.1.1 Configuration Entropy

There exits a simple estimate for the temperature of a phase transition based onthe proliferation of line-like excitations. A long line of length l with an energy perlength ε is suppressed strongly by a Boltzmann factor e−εl/T . This suppression is,however, counteracted by configurational entropy. Suppose the line can bend easilyon a length scale ξ which is of the order of the coherence length of the system. Henceit can occur in approximately (2D)l/ξ possible configurations, where D is the spacedimension [10]. A rough approximation to the partition function of a single loop ofarbitrary length is given by the integral

Z1 ≈∮

dl

l(2D)l/ξe−εl/T . (5.2)

The factor 1/l in the integrand accounts for the cyclic invariance of the loop. Byexponentiating this one-loop expression we obtain the partition function of a grand-canonical ensemble of loops of arbitrary length l, whose free energy is thereforeF = −Z1/β.

The integral (5.2) converges only below a critical temperature

T < Tc = εξ/ log(2D). (5.3)

Above Tc, the integral diverges and the ensemble undergoes a phase transitionin which the loops proliferate and become infinitely long. This process will becalled condensation of loops. A Monte-Carlo simulations of this process is shown inFig. 5.4.

From Eq. (5.3) we can immediately deduce a relation between the critical tem-perature and the roton energy in superfluids. The size of a roton will be roughly

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132 5 Multivalued Fields in Superfluids and Superconductors

Figure 5.4 Vortex loops in XY-model with periodic boundary condition for different

values of β = 1/kBT . Close to Tc ≡ 3, the loops proliferate, with some becoming infinitely

long (from Ref. [11]). The plots show the views of left and right eye. To perceive the loops

three-dimensionally, place a sheet of paper vertically between the pictures and point the

eyes parallel until you see only one picture.

πξ. Its energy is therefore Eroton ≈ πξε. Inserting this into Eq. (5.3) we estimatethe critical temperature of a line ensemble as

Tc = clinesErot. (5.4)

It is proportional to the roton energy with a proportionality constant in D = 3dimensions

clines ≈ 1/π log 6 ≈ 1/5.6. (5.5)

This prediction was recently verified experimentally [12].

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5.1 Superfluid Transition 133

5.1.2 Origin of Massless Excitations

The massless excitations in superfluid helium are a consequence of a spontaneousbreakdown of a continuous symmetry of the Hamiltonian. Such massless excitationsare called Nambu-Goldstone modes. These arise as follows. Superfluid 4He is de-scribed by a complex order field φ(x) which is the wave function of the condensate.Near the transition and for smooth spatial variations, the energy density is given bythe Hamiltonian of Landau, Ginzburg, and Pitaevskii [13]

H [φ] =1

2

d3x

|∇φ|2 + τ |φ|2 +λ

2|φ|4

, (5.6)

where τ denotes the relative temperature distance from the critical temperature

τ ≡ T/Tc − 1. (5.7)

Below the critical temperature where τ < 0, the ground state lies at

φ(x) = φ0 =

−τλeiα. (5.8)

This field value is called the order parameter of the superfluid.The ground state is not unique but infinitely degenerate. Only its absolute value

of |φ0| is fixed, the phase α is arbitrary. For this reason, the entropy does notgo to zero at zero temperature. The degeneracy in α is due to the fact that theHamiltonian density (5.6) is invariant under constant U(1) phase transformations

φ(x) → eiαφ(x). (5.9)

The Nambu-Goldstone theorem states that such a degenerate ground state pos-sesses massless excitations, unless there is another massless excitation which preventsthis by mixing with the Goldstone excitation. In the field theory with Hamiltonian(5.6), the appearance of massless excitations is easily understood by decomposingthe order field φ(x) into size and phase variables

φ(x) = ρ(x)eiθ(x), (5.10)

and rewrite (5.6) as

H [ρ, θ] =1

2

d3x

[

(∇ρ)2 + ρ2 (∇θ)2 +τρ2 +λ

2ρ4

]

. (5.11)

If τ is negative, the size of the order field is frozen at the minimum (5.8), imply-

ing ρ0 =√

−τ/λ, and the Hamiltonian (5.6) can be approximated by its so-called

hydrodynamic limit , also called London limit (see Section (7.1.1)).

Hhy[θ] = ρ20

d3x (∇θ)2 . (5.12)

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134 5 Multivalued Fields in Superfluids and Superconductors

We have omitted a constant condensation energy

Hhyc = −

d3xτ 2

2λ. (5.13)

The Hamiltonian density (5.12) shows that the energy of a plane-wave excitation ofthe phase grows with the square of the wave vector k, and goes to zero for k → 0.These are the massless Nambu-Goldstone modes. By rewriting (5.12) as

Hhy[θ] =ρs

2M

d3x (∇θ)2 , (5.14)

we obtain the usual hydrodynamic kinetic energy, and identify

ρs = Mρ20 (5.15)

as the superfluid density .Apart from the constant field φ(x) = φ0, the Hamiltonian can be extremized by

nontrivial field configurations which represent vortex lines. At the center of eachline the size ρ(x) of order field vanishes. The question arises as to what happens tothese solutions in the hydrodynamic limit where ρ(x) is constant everywhere? Thealert reader may have noticed that in going from (5.6) to (5.11) we have made animportant error which for the discussion of the Nambu-Goldstone mechanism wasirrelevant but becomes important for the understanding of the λ-transition. Wehave used the chain rule of differentiation to express

∇φ(x) = i[∇θ(x)]ρ+ ∇ρ(x)eiθ(x), (5.16)

However, this rule cannot be applied here. Since θ(x) is the phase of the complexorder field φ(x), it is a multivalued field . At every point x it is possible to add anarbitrary integer-multiple of 2π without changing eiθ(x).

The correct chain rule is [14]

∇φ(x) = i [∇θ(x) − 2π(x;S)] ρ(x) + ∇ρ(x) eiθ(x) (5.17)

where (x;S) is the δ-functions on the surface S defined in (4.22) across which θ(x)jumps by 2π. With this, we may approximate in the London limit:

|∇φ(x)|2 −→London limit

|φ|2 [∇θ(x) − v(x)]2 (5.18)

where we have introduced the field

v(x) ≡ 2π(x, S). (5.19)

With this notation, the correct version of (5.11) reads

H [ρ, θ] =1

2

d3x

[

(∇ρ)2 + ρ2 (∇θ − v)2 +τρ2 +λ

2ρ4

]

, (5.20)

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5.1 Superfluid Transition 135

wherev(x) ≡ 2π(x, S). (5.21)

In the London limit, the gradient energy density (5.14) must be corrected accord-ingly, so that the hydrodynamic limit of the Ginzburg-Landau Hamiltonian contain-ing phonons and vortex lines reads, from now on in natural units with ρs/M = 1,

Hhyv [θ] =

1

2

d3x (∇θ − v)2 . (5.22)

This Hamiltonian density is obviously gauge invariant under deformations of thesurface, under which v(x) and θ(x) change by

v(x) → v(x) + ∇Λvδ(x), θ(x) → θ(x) + Λv

δ(x), (5.23)

with the gauge functionsΛvδ(x) = 2πδ(x;V ). (5.24)

Thus we encounter again the gauge transformations (4.28) and (4.29) of the gradientrepresentation of magnetic fields. In the present context, the field v(x) is calledvortex gauge field .

In the sequel we shall see that all physical properties of the complex field theorycan be found in the theory of the multivalued field θ(x) with the vortex gauge-invariant Hamiltonian (5.22). Care has to be taken that all observable quantitiesare vortex gauge-invariant.

5.1.3 Vortex Density

As in the magnetic discussion in Section 4.2, the physical content of the vortex gaugefield v(x) appears when forming its curl. By Stokes’ theorem (4.23) we find thevortex density

∇ × v(x) ≡ jv(x) = 2π(x;L). (5.25)

As a consequence of Eq, (4.9), the vortex density satisfies the conservation law

∇ · jv(x) = 0, (5.26)

implying that vortex lines are closed.The conservation law is a trivial consequence of jv being the curl of v. It is

therefore a Bianchi identity associated with the vortex gauge field structure.The expression (5.22) is in general not the complete energy of a vortex config-

uration. It is possible to add a gradient energy in the vortex gauge field, whichintroduces an extra core energy to the vortex line. The extended Hamiltonian ofthe hydrodynamic limit of the Ginzburg-Landau Hamiltonian containing phononsand vortex lines with an extra core energy reads

Hhyvc =

d3x[

1

2(∇θ − v)2 +

εc2

(∇ × v)2]

. (5.27)

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136 5 Multivalued Fields in Superfluids and Superconductors

The extra core energy does not destroy the invariance under vortex gauge transfor-mations (5.23).

The core energy term is proportional to the square of a δ-function which is highlysingular. The singularity is a consequence of the hydrodynamic limit in which thefield ρ(x) in (5.11) is completely frozen at the minimum of (5.20). Moreover, thecoherence length of the ρ-field is zero, and this is the origin of the above δ-functions.With this in mind we may regularize the δ-functions in the core energy physicallyby smearing them out over the actual small coherence length ξ of the superfluid,which is of the order of a few rA. Whatever the size of ξ, the regularized last termyields an energy proportional to the total length of the vortex lines.

5.1.4 Partition Function

The partition function of the Nambu-Goldstone modes and all fluctuating vortexlines may be written as a functional integral

Zhyvc =

S

∫ π

−πDθe−βHhy

vc , (5.28)

where β is the inverse temperature β ≡ 1/T in natural units where the Boltzmannconstant kB is equal to unity. The measure

∫ π−πDθ is defined by discretizing the

space into a fine simple cubic lattice of spacing a and integrating at each latticepoint over all θ ∈ (−π, π), and taking the continuum limit a → 0. The sum overall surface configurations

S is defined on the lattice by setting at each latticepoint x

θvi (x;S) ≡ 2πni(x), (5.29)

where ni(x) is an integer-valued version of the vortex gauge field θvi (x;S), and by

summing over all integer numbers ni(x):

S

≡∑

ni(x)

. (5.30)

The partition function (5.28) is the continuum limit of the lattice partition function

ZV =∑

ni(x)

[

x

∫ π

−πdθ(x)

]

e−βHV , (5.31)

where HV is the lattice version of the Hamiltonian (5.27):

HV =1

2

x

[∇θ(x) − 2πn(x)]2 +1

2εc[∇ × n(x)]2. (5.32)

Here the symbol ∇i denotes lattice derivative which act on an arbitrary functionf(x) as

∇if(x) ≡ a−1[f(x + aei) − f(x)], (5.33)

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5.1 Superfluid Transition 137

where ei are the unit vectors to the nearest neighbors in the plane, and a is thelattice spacing. There exists also a conjugate lattice derivative

∇if(x) ≡ a−1[f(x) − f(x − aei)], (5.34)

which arises in the lattice version of partial integration∑

x

f(x)∇ig(x) = −∑

x

[∇if(x)]g(x), (5.35)

which holds for functions f(x), g(x) vanishing on the surface, or satisfying periodicboundary conditions. In Fourier space, the eigenvalues of ∇i, ∇i are Ki = (eikia −1)/a, Ki = (1 − e−ika)/a, respectively, where ki are the wave numbers in the i-direction.

The lattice version of the Laplacian ∇2 is the lattice Laplacian ∇∇. Its eigen-values are in D dimensions [15]

KK = 2D∑

i=1

(1 − cos kia), (5.36)

where ki ∈ (−π/a, π/a) are the wave numbers in the Brillouin zone of the lattice[15]. In the continuum limit a → 0, both lattice derivatives reduce to the ordinaryderivative ∂i, and KK goes over into k2.

In the Hamiltonian (5.32), the lattice spacing a has been set equal to unity, forsimplicity.

In the partition function (5.31), the integer-valued vortex gauge fields ni(x) aresummed without restriction. Alternatively, we may fix a gauge of ni(x) by somefunctional Φ[n], and obtain [7]

ZV =∑

ni(x)

Φ[n]

[

x

∫ ∞

−∞dθ(x)

]

e−βHV . (5.37)

On the lattice we can always enforce the axial gauge [16]

n3(x) = 0. (5.38)

Note that in contrast to continuum gauge fields it is impossible to choose the Lorentzgauge ∇ · n(x) = 0.

In the formulation (5.37), the gauge freedom has been absorbed into the θ(x)-field which now runs, for each x, from θ = −∞ to ∞ rather than from −π to π in(5.31). This has the advantage that the integrals over all θ(x) can be done yielding

ZV = Det1/2[−∇∇−1]

ni(x)

Φ[n]e−βH′

V , (5.39)

with

βH ′V =∑

x

[

4π2

2

n2(x) − [∇ · n(x)]1

−∇∇[∇ · n(x)]

+1

2εc[∇ × n(x)]2

]

. (5.40)

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138 5 Multivalued Fields in Superfluids and Superconductors

In calculating partition functions we shall always ignore trivial overall factors. If weintroduce lattice curls of the integer-valued jump fields (5.29):

l(x) ≡ ∇ × n(x), (5.41)

we can rewrite the Hamiltonian (5.40) as

βH ′V =∑

x

[

4π2

2l(x)

1

−∇∇l(x) +

εc2l2(x)

]

. (5.42)

Being lattice curls. the fields l(x) satisfy ∇ · l(x) = 0. They are, of course, integer-valued versions of the vortex density jv(x)/2π defined in Eq. (5.25). The energy(5.42) is the interaction energy between the vortex loops.

The inverse lattice Laplacian −∇∇−1 in (5.31) and (5.42) is the lattice version

of the inverse Laplacian −∇−2 whose local matrix elements 〈x2|−∇

−2|x1〉 yield theCoulomb potential of the coordinate difference x = x2 − x1:

V0(x) ≡∫

d3k

(2π)3

eikx

k2=

1

4πr, r ≡ |x|. (5.43)

The corresponding matrix elements on the lattice 〈x2|−∇∇−1|x1〉 are given by

v0(x) =∫

BZ

d3k

(2π)3

eikx

KK=

1

a

[

3∏

i=1

∫ π/a

−π/a

d3(aki)eikixi

(2π)3

]

1

2∑3i=1(1 − cos aki)

. (5.44)

where the subscript BZ of the momentum integral indicates the restriction to theBrillouin zone.

The lattice Coulomb potential (5.44) is the zero-mass limit of the lattice Yukawapotential

vm(x) =1

a

[

3∏

i=1

∫ π/a

−π/a

d3(aki)eikixi

(2π)3

]

1

2∑3i=1(1 − cos aki) +m2a2

, (5.45)

whose continuum limit is the ordinary Yukawa potential

Vm(r) ≡∫ d3k

(2π)3eikx 1

k2 +m2=e−mr

4πr, r ≡ |x|. (5.46)

In terms of the lattice Coulomb potential, we can write the partition function(5.39) for zero extra core energy as

ZV = Det1/2[v0]∑

l,∇·l=0

e−(4π2βa/2)Σx,x′ l(x)v0(x−x′) l(x′), (5.47)

where v0 abbreviates the operator −∇∇−1.

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5.1 Superfluid Transition 139

The momentum integrals over the different lattice directions can be done sep-arately by applying the Schwinger trick to express the denominator as an integralover an exponential

1

a=∫ ∞

0ds e−sa, (5.48)

so that1

2∑3i=1(1 − cos aki) +m2a2

=∫ ∞

0ds e−(6+m2a2)s

3∏

i=1

ecos kia. (5.49)

Using now the integral representation of the modified Bessel functions

In(z) =∫ π

−πdκeiκz+cosκz, (5.50)

we find

vm(x) =1

a

∫ ∞

0ds e−(6+m2a2)sIx1/a(2s)Ix2/a(2s)Ix3/a(2s). (5.51)

In contrast to the continuum version (5.46), the lattice potential vm(x) is finite atthe origin. The values of vm(0) as a function of m2a2 are plotted in Fig. 5.5. TheCoulomb potential has the value v0(0) ≈ 0.2527/a [17].

5 10 15 20

0.05

0.10

0.15

0.20

0.25

m2a2

vm(0)

5 10 15 20

0.02

0.04

0.06

0.08

0.10

0.12

m2a2

−Tr log(−∇∇ + m2)/N + log(6/a2 + m2)

Figure 5.5 Lattice Yukawa potential at the origin and the associated Tracelog. The plot

shows the subtracted expression Tr log(−∇∇ + m2)/N , where N is the number of sites

on the lattice.

The functional determinant of the lattice Laplacian appearing in (5.39) and(5.42) as a prefactor can be calculated easily from the Yukawa potential. We simplyuse the relation

Det−1/2(−∇∇+m2)=Det1/2(vm)=e−12Tr log(−∇∇+m2)=e−

a3

2Σx〈x| log(−∇∇+m2)|x〉, (5.52)

and calculate

Tr log(−∇∇ +m2)=a3

2

dm2∑

x

〈x|(−∇∇ +m2)−1|x〉=Na3

2

dm2 vm(0),(5.53)

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140 5 Multivalued Fields in Superfluids and Superconductors

where N is the number of lattice sites and the constant is the limes m2 → 0 oflogm2. Performing the integral over m2 in (5.51), we obtain

a3∫

dm2 vm(0) = −∫ ∞

0

ds

se−(6+m2a2)sI2

0 (2s). (5.54)

The divergence at s = 0 can be removed by subtracting a similar integral represen-tation

a2∫

dm2 (6 +m2a2)−1 = −∫ ∞

0

ds

se−(6+m2a2)s. (5.55)

Thus we obtain the finite result

1

NTr log(−∇∇ +m2) − log(6/a2 +m2) = −

∫ ∞

0

ds

se−(6+m2a2)s

[

I30 (2s) − 1

]

. (5.56)

The m2-behavior of this expression is displayed in Fig. 5.5.In the form (5.47) it is easy to perform a graphical expansion of the partition

function adding terms of longer and longer loops each term carrying a Boltzmannfactor e−βconst/2. This expansion converges fast for low temperatures. As the tem-perature is raised, fluctuations create more and longer loops. If there is no ex-tra core energy εc, the loops become infinitely long and dense at a critical valueTc = 1/βc = 1/0.33 ≡ 3, where the sum in (5.39) diverges. At that point the systemis filled with vortex loops. Since the inside of each vortex loop consists of normalfluid, this condensation of vortex loops makes the entire fluid normal. See Fig. 5.4for visualizing this condensation process. The successive orders of the associatedspecific heat are plotted in Fig. 5.6.

Without the extra core energy, ZV defines the famous Villain model [18], adiscrete Gaussian approximation to the so-called XY-model whose Hamiltonian is

HXY =∑

x

3∑

i=1

cos[∇iθ(x)]. (5.57)

Both the XY-model and the Villain model can be simulated with Monte Carlotechniques on a computer and displays a second-order phase transition for βc ≈ 0.33.The critical exponents of the two models coincide. The resulting specific heat of theVillain model is shown in Fig. 5.6. It has the typical λ-shape observed in 4He inFig. 5.1.

By analogy with the lattice formulation, we fix the gauge in the continuumpartition function (5.28), with the energy (5.22) or (5.27), by inserting a gauge-fixing functional Φ[v]. The axial gauge is fixed by the δ-functional

Φ[v] = δ[θv3 ]. (5.58)

Note that since the partition function (5.28) contains the sum over the vortexgauge fields v, is describes superfluid 4He not only at zero temperature, where

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5.1 Superfluid Transition 141

β

CV /3N

Figure 5.6 Specific heat of Villain model in three dimensions plotted against β = 1/T

in natural units. The λ-transition is seen as a sharp peak, with properties near Tc similar

to the experimental curve in Fig. 5.1. The solid curves stem from analytic expansion in

powers of T ≡ 1/β (low-temperature or weak-coupling expansion) and in power of T−1 = β

(high-temperature or strong-coupling expansion) (see Ref. [19]).

the Nambu-Goldstone modes were identified, but at any not too large tempera-ture. In particular, the temperature regime around the superfluid phase transitionis included.

The vortex gauge field extends the partition function of fluctuating Nambu-Goldstone modes in the same way as the size of the order field ψ does in a Landaudescription of the phase transition. In fact, it is easy to show that near the transition,the partition function (5.28) can be transformed into a |ψ|4-theory of the Landautype [7].

5.1.5 Continuum Derivation of Interaction Energy

Let us calculate the interaction energy (5.42) between vortex loops once more inthe continuum formulation. Omitting the core energy, for simplicity, the partitionfunction with a fixed vortex gauge is given by

Zhyv =

S

Φ[v]∫ ∞

−∞Dθ e−βHhy

v , (5.59)

where

Hhyv =

1

2

d3x (∇θ − v)2 (5.60)

is the energy (5.27) without core energy. Let us expand this into two parts

Hhyv =

1

2

d3x[

(∇θ)2 + 2θ∇v + v2]

= Hhyv1 +Hhy

v2 , (5.61)

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142 5 Multivalued Fields in Superfluids and Superconductors

where

Hhyv1 =

1

2

d3x(

θ +1

−∇2∇ · v

)

(−∇2)(

θ +1

−∇2∇ · v

)

(5.62)

and

Hhyv2 =

1

2

d3x(

v2 − ∇ · v 1

−∇2∇ · v

)

. (5.63)

Inserting this into (5.59), we can perform the Gaussian integrals over θ(x) at eachx using the generalization of the Gaussian formula

∫ ∞

−∞

2πe−a(θ−c)

2/2 = a−1/2 (5.64)

to fields θ(x) and differential operators O in x-space

∫ ∞

−∞Dθ e−

d3x [θ(x)−c(x)] O [θ(x)−c(x)]/2 = [DetO]−1/2. (5.65)

Applying this formula to (5.59), we obtain

Zhyv =

[

Det(−∇2)]−1/2∑

S

Φ[v] e−βHv , (5.66)

where Hv is the interaction energy of the vortex loops corresponding to (5.42):

Hv =1

2

d3x(∇ × v)1

−∇2 (∇ × v) =

1

2

d3x jv1

−∇2jv

=1

d3xd3x′ jv(x)1

|x− x′| jv(x′). (5.67)

This has the same form as the magnetic Biot-Savart energy (4.92) for current loops,implying that parallel vortex lines repel each other [as currents would do if no workwere required to keep them constant against the inductive forces, which reverses thesign. Recall the discussion of the free magnetic energy (4.94)]. On a lattice, thepartition function (5.66) takes once more, the form (5.39).

The process of removing some variables from a partition function by integrationwill occur frequently in the sequel and will be referred to as integrating out . Itwill be used also in discussions of Hamiltonians without writing always down theassociated partition function in which the integrals are actually performed.

5.1.6 Physical Jumping Surfaces

The invariance of the energy (5.27) under vortex gauge transformations guaranteesthe physical irrelevance of the jumping surfaces S. If we destroy this invariance,the surfaces become physical objects. This may be done by destroying the originalU(1)-symmetry explicitly. This will give a mass to the Nambu-Goldstone modes.

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5.1 Superfluid Transition 143

To lowest approximation, it adds to the energy (5.27) without core energy a massterm m2θ(x)2:

Hhyvm =

1

2

d3x

[∇θ(x) − v(x)]2 +m2θ(x)2

. (5.68)

The mass term gives an energy to the surfaces S. To see this we write the energy as

Hhyvm =

1

2

d3x[

(∇θ)2 +m2θ2]

− θ∇ · v + v2]

, (5.69)

and decompose this into two parts as in (5.61):

Hhyvm1 =

1

2

d3x(

θ +1

−∇2+m2∇ · v

)

(−∇2+m2)

(

θ +1

−∇2+m2∇ · v

)

(5.70)

and

Hhyvm2 =

1

2

d3x(

v2 − ∇ · v 1

−∇2+m2∇ · v

)

. (5.71)

The Gaussian integrals over θ(x) can be done as before, the partition function (5.66)becomes

Zhyvm = Det−1/2[−∇

2 +m2]∑

S

Φ[v] e−βHhyvm2 . (5.72)

The energy Hhyvm2 in the exponent can be rewritten as

Hhyvm2 =

1

2

d3x[

(∇ × v)1

−∇2 +m2

(∇ × v) +m2 v 1

−∇2 +m2

v]

. (5.73)

The first term is a modification of the Biot-Savart-type of energy (5.67):

Hhyvm =

1

2

d3x (∇ × v)1

−∇2 +m2

(∇ × v) =1

2

d3x jv1

−∇2 +m2jv

=1

d3xd3x′ jv(x)e−m|x−x′|

|x− x′| jv(x′). (5.74)

The presence of the mass m changes the long-range Coulomb-like interaction 1/Rin Eq. (5.67) into a finite-range Yukawa-like interaction e−mR/R.

The second term in (5.73),

HSm =m2

2

d3x v 1

−∇2 +m2

v =1

d3xd3x′ v(x)e−m|x−x′|

|x − x′| v(x′), (5.75)

is of a completely new type. It describes a Yukawa-like interaction between thenormal vectors of the surface elements, and gives rise to a field energy within a layerof thickness 1/m around the surfaces S. As a consequence, the surfaces acquiretension. Their shape is no longer irrelevant, but for a given set of vortex loops Lat their boundaries, the surface will span minimal surfaces. For m = 0, the tensiondisappears and the shape of the surface becomes again irrelevant, thus restoringvortex gauge invariance.

This mechanism of generating surface tension will be used in Chapter 8 to con-struct a simple model of quark confinement.

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144 5 Multivalued Fields in Superfluids and Superconductors

5.1.7 Canonical Representation of Superfluid

We can set up an alternative representation of the partition function of the superfluidin which the vortex loops are more directly described by their physical vortex density,instead of their jumping surfaces S. This is possible by eliminating the Nambu-Goldstone modes in favor of a new gauge field. It is canonically conjugate to theangular field θ and called generically stress gauge field [7]. In the particular caseof the superfluid under discussion it is a gauge field of superflow . Recall that thecanonically conjugate momentum variable p(t) in an ordinary path integral [20]

Dx exp(

−M2

∫ tb

tadt x2

)

(5.76)

is introduced by a quadratic completion, rewriting (5.76) as

DxDp exp

[

∫ tb

tadt

(

ipx − p2

2M

)]

. (5.77)

By analogy, we introduce a canonically conjugate vector field b(x) to rewrite thepartition function (5.59) as

Zhyv =

∫ ∞

−∞Db

S

Φ[v]∫ ∞

−∞Dθ e−βHhy

v (5.78)

where [21]

βHhyv =

d3x

1

2βb2(x) − ib [∇θ(x) − v(x)]

. (5.79)

In principle, the gradient energy could contain higher powers of ∂iθ. Then thecanonical representation (5.79) would contain more complicated functions of bi(x).

Note that if we go over to a Minkowski space formulation in which x0 = −ix3

plays the role of time, the integral

∫ ∞

−∞Db0 e−ib0(x)∂0θ(x) (5.80)

creates, on a discretized time axis, a product of δ-functions

〈θn+1|θn〉〈θn|θn−1〉〈θn−1|θn−2〉 (5.81)

with

〈θn|θn−1〉 = δn(θn − θn−1). (5.82)

These can be interpreted as Dirac scalar products in the Hilbert space of the system.On this Hilbert space, there exists an operator bi(x) whose zeroth component is givenby

b0 = −i∂θ (5.83)

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5.1 Superfluid Transition 145

and satisfies the equal-time canonical communication rule[

b0(x⊥, x0), θ(x′⊥, x0)

]

= −iδ(2)(x⊥ − x′⊥), (5.84)

where x⊥ = (x1, x2) denotes the spatial components of the vector (x0, x1, x2). Thecharge associated with b0(x),

Q(x0) =∫

d2x b0(x⊥, x0), (5.85)

generates a constant shift in θ:

e−iαQ(x0)θ(x⊥, x0)eiαQ(x0) = θ(x⊥, x0) + α. (5.86)

Thus it multiplies the original field eiθ(x) by a phase factor eiα. The charge Q(x0) isthe generator of the U(1)-symmetry transformation whose spontaneous breakdownis responsible for the Nambu-Goldstone nature of the fluctuations of θ(x). Since theoriginal theory is invariant under the transformations φ → eiαφ, the energy (5.79)does not depend on θ itself, but only on ∂iθ.

In the partition function (5.78) we may use the formula

∫ ∞

−∞Dθ ei

d3x f(x)θ(x) = δ[f(x)], (5.87)

and obtain from f(x) = ∇ · b(x) the conservation law

∇ · b(x) = 0, (5.88)

implying that Q(x0) is a time-independent charge and eiαQ is a symmetry transfor-mation.

If the energy in (5.78) would depend on θ itself, then the charge Q(x0) wouldno longer be time-independent. However, it would still generate the above U(1)-transformation.

In general, the conjugate variable to the phase angle of a complex field is theparticle number (recall Subsection 3.5.3). This role is played here by Q(x0) whichcounts the number of particles in the superfluid. Thus we may identify the vectorfield b(x) as the particle current density of the superfluid condensate:

js(x) ≡ b(x), (5.89)

also called the supercurrent density of the superfluid.After integrating out the θ-fields in the partition function (5.78), we can also

perform the sum over all surface configurations of the vortex gauge field v(x). Forthis we employ the following useful formula applicable to any function b(x) with∇ · b(x) = 0:

S

e2πi∫

d3x (x;S)b(x) =∑

L

δ [b(x) − (x;L)] . (5.90)

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146 5 Multivalued Fields in Superfluids and Superconductors

This can easily be proved by going on a lattice where this formula reads [recall(5.30)]

ni

e2πiΣxni(x)fi(x) =∑

mi

x,i

δ (fi(x) −mi(x)) , (5.91)

and using for each x, i the Poisson formula [20]∑

n

e2πinf =∑

m

δ(f −m). (5.92)

The we obtain for (5.78) the following alternative representation

Zhyv =

L

e−∫

d3xb2/2β, (5.93)

where b = (x;L). On the lattice, this partition function becomes

Zhyv =

b,∇·b=0

e−∑

x

b2/2β , (5.94)

where b(x) is now an integer-valued divergenceless field representing the closed linesof superflow.

The partition function (5.94) can be evaluated graphically adding terms of longerand longer loops each term carrying a Boltzmann factor e−const/2β. This expansionconverges fast for high temperature. The specific heat following from the lowestapproximations obtained in this way are plotted in Fig. 5.6. For very high temper-ature, there is no loop of superflow. As the temperature is lowered, fluctuationscreate more and longer loops. At a critical value Tc = 1/βc = 1/0.33 ≡ 3, the loopsbecome infinitely long and dense, and the sum in (5.94) diverges. The system hasbecome superfluid.

Note that the superflow partition function (5.94) look quite similar to the vor-tex loop partition function (5.47). Both contain the same type of sum over non-self-backtracking loops. The main difference is the long-range Coulomb interactionbetween the loop elements. Suppose we forget for a moment the non-local parts ofthe Coulomb interaction and approximate the vortex loop partition function (5.47)keeping only the self-energy part of the Coulomb interaction:

ZV app = Det1/2[vm]∑

l,∇·l=0

e−(4π2βa/2)v0(0)Σx l2(x). (5.95)

Apart from a constant overall factor, this approximation coincides with the superflowpartition function (5.94). By comparing the prefactors of the energy we see that(5.95) has a second-order phase transition at

4π2aβv0(0) ≈ Tc ≈ 3. (5.96)

This is solved by β ≈ 3/4π2av0(0) ≈ 0.30, corresponding to a critical temperature

T appc ≈ 4π2av0(0)

3≈ 3.3. (5.97)

This is only 10% larger than the accurate value Tc = 1/βc ≈ 3, so that we concludethat the nonlocal parts of the Coulomb interaction in (5.47) have little effect uponthe transition temperature.

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5.1 Superfluid Transition 147

5.1.8 Yukawa Loop Gas

The above observation allows us to estimate the transition temperature in partitionfunction closely related to (5.47)

ZYV = Det1/2[vm]

l,∇·l=0

e−(4π2βa/2)Σx,x′ l(x)vm(x−x′) l(x′), (5.98)

where vm(x) is the lattice version of the Yukawa potential (5.46), and vm the asso-ciated operator (−∇∇ +m2)−1.

Performing also here the local approximation of the type (5.95),

ZYV app = Det1/2[v0]

l,∇·l=0

e−(4π2βa/2)vm(0)Σx l2(x), (5.99)

we estimate the critical value βm,c of the Yukawa loop gas by the equation corre-sponding to (5.96):

4π2aβm,cvm(0) ≈ Tc ≈ 3. (5.100)

Since the Yukawa potential becomes more and more local for increasing m, thelocal approximation (5.99) becomes exact. Thus we conclude that the error in theestimating of the critical temperature Tm,c = 1/βm,c from Eq. (5.100) drops from10% at m = 0 to zero as m goes to infinity. We have plotted the resulting criticalvalues of Tm,c = 1/βm,c in Fig. 5.7.

5 10 15 20

1

2

3

m2a2

Tm,c = 4π2avm(0)/3

Figure 5.7 Critical temperature Tm,c = 1/βm,c of a loop gas with Yukawa interactions

between line elements, estimated by Eq. (5.100). The error is with 10% the largest at m =

0, and decreases to zero for increasing m. The dashed curve is the analytic approximation

(5.104).

We conclude that the Yukawa loop gas (5.98) has a second-order phase transitionas the Villain- and the XY-models. The critical exponents of the Yukawa loop gasare all of the same as those of the Villain-model, and thus also of the XY-model. Inthe terminology of the theory of critical phenomena, the Yukawa loop gases lie forall m in the same universality class as the XY-model.

It is possible to find a simple analytic approximation for the critical temperatureplotted in Fig. 5.7. For this we use the so-called hopping expansion [7] of the lattice

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148 5 Multivalued Fields in Superfluids and Superconductors

Yukawa potential (5.1.8). It is found by expanding the modified Bessel functionIxi/a(2s) in Eq. (5.51) in powers of s using the series representation

In(2s) =∞∑

k=0

s2k

k!Γ(n + k + 1). (5.101)

At the origin x = 0, the integral over s yields the expansion

vm(0) =1

a

n=0,2,4

Hn

(m2a2 + 6)n+1, H0 = 1, H2 = 6, . . . . (5.102)

To lowest order, this implies the approximate ratio vm(0)/v0(0) ≡ 1/(m2a2/6 + 1).A somewhat more accurate fit to the ratio

vm(0)

v0(0)≈ 1

σm2a2/6 + 1, with σ ≈ 1.6. (5.103)

Together with (5.97) this leads to the analytic approximation

Tm,c =1

βm,c≈ 4π2avm(0)

3≈ 4π2av0(0)

3

1

σm2a2/6 + 1. (5.104)

A comparison with the numerical evaluation of (5.100) is shown in Fig. 5.7. The fithas only a 10% error for m = 0 and become accurate for large m.

5.1.9 Gauge Field of Superflow

The current conservation law ∇·b(x) = 0 can be ensured automatically as a Bianchiidentity, if we represent b(x) as a curl of a gauge field of superflow

b(x) = ∇ × a(x). (5.105)

With respect to the gauge transformations (5.107), the current conservation law∇ · b(x) = 0 is a Bianchi identity.

The energy (5.79), with the core energy reinserted, goes over into what is calledthe dual representation:

βHavc =∫

d3x

[

1

2β(∇ × a)2 + ia · (∇ × v) +

βεc2

(∇ × v)2

]

. (5.106)

The energy is now double-gauge invariant. Apart from the invariance under thevortex gauge transformation (5.23), there is now the additional invariance underthe gauge transformations of superflow

a(x) → a(x) + ∇Λ(x), (5.107)

with arbitrary functions Λ(x).

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5.1 Superfluid Transition 149

The energy (5.106) can be expressed in terms of the vortex density of Eq. (5.25)as

βH ′avc =∫

d3x

[

1

2β(∇ × a)2 + ia · jv +

βεc2

jv2

]

. (5.108)

In this expression, the freely deformable jumping surfaces have disappeared and theenergy depends only on the vortex lines. For a fixed set of vortex lines along L,the action (5.108) has a similar form as the free magnetic energy of a given currentdistribution in Eq. (5.108). The only difference is a factor i. Around a vortex line,the field b(x) = ∇ × a(x) looks precisely like a magnetic field B(x) = ∇ × A(x)around a current line, except for the factor i. Extremizing the energy in a andreinserting the extremum yields once more the Biot-Savart interaction energy of theform Eq. (5.67) which is of the form (4.92) [not (4.92) due to the factor i).

If we want to express the partition function (5.78) in terms of the gauge field ofsuperflow a(x), we must fix its gauge. Here we may choose the transverse gauge:

ΦT [a] = δ[∇ · a], (5.109)

and the partition function (5.78) becomes

Zhyv =

∫ ∞

−∞DaΦT [a]

S

Φ[v]e−βHavc . (5.110)

In terms of the Hamiltonian (5.108), the partition function becomes a sum overvortex lines L:

Zhyv =

∫ ∞

−∞DaΦT [a]

L

ΦT [jv]e−βH′

avc , (5.111)

where

ΦT [jv] = δ[∇ · jv] (5.112)

ensures the closure of the vortex lines.

Note that if the energies Hhyvc or Hhy

v in (5.27) and (5.60) contain an explicitθ-dependent term, such as the mass term in the Hamiltonian (5.68), there exits noreformulation of the θ-fluctuations in terms of a gauge field a. For a mass term, theformula (5.87) turns into

∫ ∞

−∞Dθ e−

d3x [βm2θ2(x)/2−if(x)θ(x)] = δm[f(x)], (5.113)

where δm[f(x)] denotes the softened δ-functional

δm[f(x)] ∝ e−∫

d3x f2(x)/2βm2

. (5.114)

For f(x) = ∇ · b(x) this implies that b(x) is no longer purely transverse, as in(5.88). Hence it no longer possesses a curl representation (5.105).

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150 5 Multivalued Fields in Superfluids and Superconductors

5.1.10 Disorder Field Theory

Let us discuss the sum over all vortex line configurations in the partition function(5.111) separately. For this we define an a-dependent vortex partition function

Zv[a] =∑

L

δ[∇ · jv] exp

[

−∫

d3x

(

βεc2

jv2 − i a · jv)]

. (5.115)

It is possible to express this with the help of an auxiliary fluctuating vortex gaugefield v

(x) singular on surfaces S as a sum over auxiliary surface configurations Sas follows

Zv[a]=∑

S

Djvδ[∇ · jv] exp

−∫

d3x

[

βεc2

jv2 − i jv ·(

v+ a

)

]

. (5.116)

In this expression, jv is an ordinary field. The sum over all S configuration ensuresvia formula (5.91) that jv is a superposition of δ-functions on lines L, so that (5.116)agrees with (5.115).

Next we introduce an auxiliary field θ, and rewrite the δ functional of the diver-gence of jv as a functional Fourier integral, so that we obtain the identity

Zv[a]=∑

S

Djv∫

Dθ exp

−∫

d3x

[

βεc2

jv2 + i jv ·(

∇θ − v− a)

]

. (5.117)

Now jv is a completely unrestricted ordinary field. It can therefore be integrated toyield

Zv[a] =∑

L

Dθ exp

[

− 1

2βεc

d3x(∇θ − v− a)2

]

. (5.118)

Remembering the derivation of the Hamiltonian (5.22) from the hydrodynamic limitof the Ginzburg-Landau |φ|4 theory (5.6), we may interprete (5.118) as the parti-tion function of the hydrodynamic limit of another U(1)-invariant Ginzburg-Landautheory with partition function

Zv[a] =∫

DψDψ∗ exp

− 1

d3x[

|(∇ − ia)ψ|2 +m2 |ψ|2 +g

2|ψ|4

]

,

(5.119)

where ψ(x) is another complex field ψ with a |ψ|4 interaction. Inserting this into(5.111) we obtain the combined partition function

Zhyv =

∫ ∞

−∞DaΦT [a]Zv[a] exp

−∫

d3x

[

1

2β(∇ × a)2

]

, (5.120)

which defines the desired disorder field theory.

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5.1 Superfluid Transition 151

The representation of ensembles of lines in terms of a single disorder field is theEuclidean version of what is known as second quantization in the quantum mechanicsof many-particle systems

At high temperature, the mass term m2 of the ψ-field is negative and the disorder

field acquires a nonzero expectation value ψ0 =√

−m2/g. Setting, as in (5.10),

ψ(x) = ρ(x)eiθ(x) (5.121)

and freezing out the fluctuations of ρ leads directly to the partition function (5.118).The disorder field theory possesses similar vortex lines as the original Ginzburg-

Landau theory with Hamiltonian (5.6), or its hydrodynamic limit (5.27). But incontrast to it, the fluctuations of the disorder field are “frozen out” at high temper-ature, as we can see from the prefactor 1/β in the exponents of (5.118) and (5.119),and the partition function (5.119) reduces to (5.118) in the hydrodynamic limit. Asbefore in (5.59) we may perform the functional integral over θ. Here this removesthe longitudinal part of v− a, and (5.118) becomes

Zv[a] = exp

[

−m2a

d3x(

v − a)2

T

]

(5.122)

where

m2a =

1

εc, (5.123)

and vT denotes transverse part of the vector field v. This and the longitudinal partvL are defined by

vT i ≡(

δij −∇i∇j

∇2

)

vj , vLi ≡∇i∇j

∇2vj . (5.124)

At high temperatures, where the disorder field ψ has no vortex lines L (while theorder field φ has many vortex lines L), the partition function (5.122) becomes

Zv[a] ≈ exp

(

−m2a

d3x a2T

)

, (5.125)

and the exponent gives a mass to the transverse part aT of the gauge field of super-flow. Recalling the gradient term (1/β)(∇× a)2 of the a-field in (5.120) we see themass has the value ma.

Having obtained this result we go once more back to the expression (5.115) andrealize that the same mass can also be obtained from Zv[a] by simply ignoring theδ-function nature of jv(x) = 2π(x, L) and integrating jv(x) out using the Gaussianformula (5.65). With such an approximate treatment, the partition function (5.115)yields for the vortex density the simple correlation function

〈jiv(x)jjv(x′)〉 =

1

εc

(

δij −∇i∇j

∇2

)

δ(3)(x − x′). (5.126)

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152 5 Multivalued Fields in Superfluids and Superconductors

The reason why this simplification is applicable in the high-temperature phaseis easy to understand. On a lattice, the sums over lines L in (5.115) correspond toGaussian sums of the type

∑∞ni=−∞

e−βεc4πn2i /2 at each x, i. At high temperatures

where β is small, the sum over ni can obviously be replaced by 1/√β times an in-

tegral over the quasi-continuous variable νi ≡√βni. In general, if lines or surfaces

of volumes are prolific, the statistical mechanics of fields proportional to the corre-sponding δ-functions (x;L), (x;S), δ(x;V ) can be treated as if they were ordinaryfields. The sums of the geometric configurations turn into functional integrals.

The same mass generation can, of course, be observed in the complex disorderfield theory (5.119). At high temperature, the mass termm2 of the ψ-field is negative

and the disorder field acquires a nonzero expectation value ψ0 =√

−m2/g. This

produces again the mass term (5.122) with m2a = ψ2

0.Let us now look at the low-temperature phase. There the δ-function nature of

the density jv(x) = 2π(x;L) cannot be ignored in the partition function (5.115).At low temperatures, vortex lines appear only as small loops. An infinitesimal loopgives a simple curl contribution [22]

Zv[a] ∼ exp

[

− 1

d3x (∇ × a)2

]

, (5.127)

whereas larger loops contribute

Zv[a] ∼ exp[

−1

2

d3x (∇ × a)f(−i∇)(∇ × a)]

, (5.128)

where f(k) is some smooth function of k starting out with a constant, the so-calledstiffness of the a-field. Hence the contributions of small vortex loops change onlythe dispersion of the gauge fields of superflow. Infinitely long vortex lines in v arenecessary to produce a mass term. These appear when the temperature is raisedabove the critical point, in particular at high temperatures, where the correlationfunction of the vortex densities is approximately given by (5.126), and (5.115) leadsdirectly to the mass term in (5.125).

With the help of the disorder partition function Zv[a], the partition function(5.28) can be replaced by the completely equivalent dual partition function

Zhyv =

∫ ∞

−∞DaΦT [a]

S

Φ[v]∫ ∞

−∞Dθ e−βHhy

v (5.129)

with the exponent

βHhyv =

1

d3x[

(∇ × a)2 +m2a

(

∇θ − v − a)2]

. (5.130)

This energy is invariant under the following two gauge transformations. First, thereis invariance under the gauge transformations of superflow (5.107), if it is accompa-nied by a compensating transformation of the angular field θ:

a(x) → a(x) + ∇Λ(x), θ(x) → θ(x) + 2πΛ(x). (5.131)

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5.2 Phase Transition in Superconductor 153

Second, there is gauge invariance under the joint vortex gauge transformationsEq. (5.23) with (5.24) and the phase transformation of the disorder field,

v(x) → v

(x) + ∇Λvδ(x), θ(x) → θ(x) + Λv

δ(x), (5.132)

with gauge functionsΛv(x) = 2πδ(x; V ). (5.133)

At high temperatures, the vortex lines in vare frozen out and the energy (5.130)

shows again the mass term (5.122).The mass term implies that at high temperatures, the gauge field of super-

flow possesses a finite range. At some critical temperature superfluidity has beendestroyed. This is the disorder analog of the famous Meissner effect in superconduc-tors [23], to be discussed in Section 5.2.1. Without the gauge field of superflow a, thefield θ would be of long range, i.e., massless. The gauge field of superflow absorbsthis massless mode and the system has only short-range excitations. More pre-cisely, it can be shown that all correlation functions involving local gauge-invariantobservable quantities must be of short range in the high-temperature phase.

Take, for instance, the local gauge-invariant current operator of the disorder field

js ≡ ∇θ − a. (5.134)

Choosing θ to absorb the longitudinal part of a, only the transverse part of a re-mains in (5.134), which becomes js = −aT . [24]. From the Hamiltonian (5.130) weimmediately find the free correlation function of superflow:

〈jis(x1)jjs(x2)〉 ∝

d3k

(2π)3

δij − kikj/m2a

k2 +m2a

eik(x1−x2), (5.135)

which has no zero-mass pole.

5.2 Phase Transition in Superconductor

The specific heat of a superconductor looks quite different from that of helium asshown in Fig. 5.8 (compare Fig. 5.1 on p. 129). It starts out with a behavior typicalfor an activation process, which is governed by a Boltzmann factor cs ∝ e−∆(0)/kBT ,where kB is the Boltzmann constant. The activation energy ∆(0) shows the energygap in the electron spectrum at T = 0. It is equal to the binding energy of theCooper pairs formed from electrons of opposite momentum near the Fermi sphere.At the critical temperature Tc, the specific heat drops down to the specific heat ofa free electron gas

cn =2

3π2N(0)T, (5.136)

where

N(0) =3ne4εF

(5.137)

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154 5 Multivalued Fields in Superfluids and Superconductors

T (K)

cs

c (ml/Mol K)

cn

Figure 5.8 Specific heat of superconducting aluminum [N.E. Phillips, Phys. Rev. 114,

676 (1959)]. For very small T , it shows the typical power behavior e−∆(0)/kBT instead

of the power behavior in superfluid helium. At the critical temperature Tc ≈ ∆(0)/1.76,

there is a jump down to the linear behavior characteristic for a free electron gas. The

ratio ∆c/cn = 1.43 agrees well with the BCS theory [25]. A normal metal shows only the

linear behavior labeled by cs.

is the density of electron states at the surface of the Fermi sphere of energy εF , andne the electron density.

According to the theory of Bardeen, Cooper, and Schrieffer (BCS) [25], the jumpis given by the universal law

cs − cncn

≡ ∆c

cn=

3

2

8

7ζ(3)≡ 1.4261 . . . , (5.138)

where ζ(3) is Riemann’s zeta function ζ(z) ≡ ∑∞n=1 n

−z, with ζ(3) = 1.202057 . . ..This jump agrees perfectly with the experiment in Fig. 5.8.

In the BCS-theory there exists also a universal ratio between the gap ∆(0) andTc:

∆(0)

Tc= πe−γ ≈ 1.76, (5.139)

where γ ≈ 0.577 . . . is the Euler-Mascheroni constant . This ratio is also observed inFig. 5.8.

5.2.1 Ginzburg-Landau Theory

The BCS theory can be used to derive the Ginzburg-Landau Hamiltonian for thesuperconducting phase transition

HHL[ψ, ψ∗,A] =1

2

d3x

|(∇ − iqA)ψ|2 + τ |ψ|2 +g

2|ψ|4 + (∇ ×A)2

(5.140)

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5.2 Phase Transition in Superconductor 155

governing the neighborhood of the critical point. The parameter q is the chargeof the ψ-field, and τ may be identified with T /TMF

c − 1, the relative temperaturedistance from the critical point. It is positive in the normal state and negative in thesuperconducting state. The field ψ(x) is a so-called collective field describing theCooper pairs of electrons of opposite momenta slightly above and below the Fermisphere [27]. The Cooper pairs carry a charge twice the electron charge, q = 2e ,and are coupled in (5.140) minimally to the vector potential A(x). For simplicity,we have set the light velocity c equal to unity, c = 1. The size of ψ is equal to theenergy gap in the electron spectrum, which is caused by the binding of electrons toCooper pairs.

Ginzburg and Landau [26] found the Hamiltonian (5.140) by a formal expan-sion of the energy in powers of the energy gap which they considered as an orderparameter. They convinced themselves that for small τ only the terms up to ψ4

would be important. To this truncated expansion they added a gradient term toallow for spatial variations of the order parameter, making it an order field denotedby ψ(x). There exists an elegant derivation of the Ginzburg-Landau Hamiltonian(5.140) from the BCS theory via functional integration [27].

In the critical regime, the Ginzburg-Landau provides us with a simple explana-tion of many features of superconductors. In most applications, one may neglectfluctuations of the Ginzburg-Landau order field φ(x), which is why one speaks ofmean-field results, an why one attaches the superscript to the critical tempera-ture TMF

c in the above definition of τ . Close to the transition, the properties of asuperconductor are well described by the Ginzburg-Landau Hamiltonian [compare(5.119)].

The Ginzburg-Landau Hamiltonian (5.140) possesses a conserved supercurrentwhich is found by applying Noether’s rule (3.116) to (5.140). The current density is

j(x, t) ≡ i

2ψ∗(x, t)

↔∇ψ(x, t) − qAψ∗(x, t)ψ(x, t). (5.141)

This differs from the Schrodinger current density (5.141) by the use of natural unitsm = 1, c = 1.

Let us now proceed as in (5.10) and (5.121) and decompose the field ψ as

ψ(x) = ρ(x) eiθ(x). (5.142)

Inserting this into (5.140 and remembering (5.17), we find

HGL[ρ, θ, v,A] =

d3x

[

ρ2

2

(

∇θ− v−qA)2

+1

2(∇ρ)2+V (ρ)+

1

2(∇ × A)2

]

, (5.143)

where V (ρ) is the potential of the ρ-field:

V (ρ) =τ

2ρ2 +

g

4ρ4. (5.144)

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156 5 Multivalued Fields in Superfluids and Superconductors

In the low-temperature phase we go to the hydrodynamic limit by setting ρ(x)

equal to its value ρ0 =√

−τ/g at the minimum of the energy (5.140). The resultinghydrodynamic or London energy of the superconductor is

HhySC[θ, v

,A] =∫

d3x

[

m2A

2(∇θ − v− qA)2 +

1

2(∇ ×A)2

]

(5.145)

where we have introduced the density of superfluid particles

n0 = ρ20. (5.146)

At very low temperatures where vortices are absent, the first term in (5.145) givesa mass

mA =√n0q (5.147)

to the transverse part of the vector field. This causes a finite penetration depthλ = 1/mA of the magnetic field in a superconductor, thus explaining the famousMeissner effect of superconductivity.

This mechanism is imitated in the standard model of electromagnetic and weakinteractions to give the vector mesons W+,0,− and Z a finite mass, thereby explainingthe strong suppression of weak with respect to electromagnetic interactions. Therethe Meissner effect is called Higgs effect .

In the same limit, the current density of superfluid particles becomes

js = n0(∇θ − v− qA). (5.148)

The partition function reads

ZhySC =

DAΦT [A]∑

S

Φ[v]∫ ∞

−∞Dθ e−βHhy

SC[θ,˜

v,A]. (5.149)

To distinguish this discussion from the previous one of superfluid helium we call thetemperature of the superconductor T and its inverse β.

The energy (5.145) has the same form as the energy in the disorder representation(5.130) of superfluid 4He. The role of the gauge field of superflow is now played bythe vector potential A of magnetism. The energy has the following two types ofgauge symmetries: the magnetic invariance

A(x) → A(x) + q−1∇Λ(x), θ(x) → θ(x) + Λ(x), (5.150)

and the vortex gauge invariance

v(x) → v

(x) + ∂iΛδ(x), θ(x) → θ(x) + Λδ(x), (5.151)

with gauge functionsΛδ(x) = 2πδ(x; V ). (5.152)

As in the description of superfluid 4He with the partition function (5.28), thepartition function (5.149) gives us the statistical behavior of the superconductor notonly at zero temperature, where the energy (5.145) was constructed, but at all nottoo large temperatures. The fluctuating vortex gauge field v

ensures the validitythrough the phase transition.

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5.2 Phase Transition in Superconductor 157

5.2.2 Disorder Theory of Superconductor

We shall now derive the disorder representation of this partition function in whichthe vortex lines of the superconductor play a central role in describing the phasetransition [23].

At low temperatures, the vortices are frozen, and the θ-fluctuations in the par-tition function can be integrated out. This reduces the energy to

Hhy ∼ m2A

2

d3xA2T , (5.153)

i.e., to a simple transverse mass term for the vector potential A. This is the famousMeissner effect in a superconductor, which limits the range of a magnetic field to afinite penetration depth λ = 1/mA. The effect is completely analogous to the oneobserved previously in the disorder description of the superfluid where the superfluidacquired a finite range in the normal phase.

To derive the disorder theory of the partition function (5.149), we supplementthe energy by a core energy of the vortex lines

Hc =εc2

d3x (∇ × v)2. (5.154)

As in the partition function (5.78), an auxiliary bi field can be introduced to bringthe exponent in (5.145) to the canonical form

βHhySC =

d3x

[

1

2βm2A

b2+ ib (∇θ− v−qA) +β

2(∇ ×A)2+

βεc2

(∇ × v)2

]

.(5.155)

By integrating out the θ-fields in the associated partition function, we obtain theconservation law

∇ · b = 0, (5.156)

which is fulfilled by expressing b as a curl of the gauge field a of superflow in thesuperconductor

b = ∇ × a. (5.157)

This brings the energy to the form

βHhySC =

d3x

[

1

2βm2A

(∇ × a)2 −iq a · (∇ ×A)+β

2(∇ × A)2−ia · jv+

βεc2

jv2

]

,

(5.158)

wherejv = ∇ × v

(5.159)

is the vortex density in the superconductor. At low temperatures where β is largeand the vortex lines are frozen out, the two last terms in the Hamiltonian can beneglected. Integrating out the a-field we re-obtain the transverse mass term (5.153)

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158 5 Multivalued Fields in Superfluids and Superconductors

of the Meissner effect. At high temperatures, on the other hand, the vortex lines areprolific and the vortex density jv can be integrated out in the associated partitionfunction like an ordinary field using the analog to the correlation function (5.126).This produces the transverse mass term

1

2βm2A

d3xm2a a2

T (5.160)

wherem2a = q2m2

A/εc. (5.161)

This can immediately be seen to destroy the Meissner effect in the superconductorat high temperature. Indeed, inserting the curl (5.157) into the energy (5.155), andusing the result (5.160), we obtain at high T :

βHhySC =

d3x

[

1

2βm2A

[

(∇ × a)2 +m2a a

2]

−ia · (∇ ×A)+β

2(∇ × A)2

]

. (5.162)

If we integrate out the massive a-field, the Hamiltonian of the vector potentialbecomes

HA =1

2

d3x ∇ ×A

(

1 +m2A

−∇2 +m2a

)

∇ ×A. (5.163)

Expanding the denominator in powers of −∇2 we see that only gradient energies

appear, but no mass term. Thus, the vector potential A maintains its long rangeand yields Coulomb-like forces at large distances. Only the dispersion is modifiedto a more complicated k-dependence of the energy.

In the low-temperature phase, on the other hand, the mass ma is zero, and them2A-term in (5.163) produces again the transverse mass Hamiltonian (5.153) which

is responsible for the Meissner effect.We can represent the fluctuating vortices in the superconductor by a disorder

field theory in the same way as we did for the vortices in the superfluid, by repeatingthe transformations from (5.115) to (5.118). The angular field variable of disorderwill now be denoted by d θ, the vortex lines in the disorder theory by v. Thedisorder action reads

βHhySC =

d3x[

1

2βm2A

(∇ × a)2 − iq a · (∇ × A) +β

2(∇ ×A)2

+m2a

2βm2A

(∇θ − v − a)2]

. (5.164)

Near the phase transition, this is equivalent to a disorder field energy

βHhySC ∼

d2x[

1

2βm2A

(∇ × a)2 − iq a · (∇ × A) +β

2(∇ × A)2

+1

2[(∇ − ia)φ]2 +

τ

2|φ|2 +

g

4|φ|4

]

. (5.165)

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5.3 Order versus Disorder Parameter 159

The vector potential A fluctuates harmonically in such a way that the associatedmagnetic field is on the average equal to a/β. Integrating A out we obtain from(5.165)

βHhySC ∼

d2x[

1

2βm2A

[(∇ × a)2 +m2aa

2T ] +

1

2[(∇ − ia)φ]2 +

τ

2|φ|2 +

g

4|φ|4

]

.

(5.166)

This Hamiltonian is invariant under the gauge transformations

φ(x) → eiΛ(x)φ(x), a(x) → a(x) + ∇Λ(x). (5.167)

The partition function is

ZdualSC =

Dφ∫

Dφ∗DaΦ[a] e−βHhySC, (5.168)

where Φ[a] is some gauge-fixing functional.This partition function can be evaluated perturbatively as a power series in

g. The terms of order gn consist of Feynman integrals which can be pictured byFeynman diagrams with n + 1 loops [28]. These loops are pictures of the topologyof vortex loops in the superconductor.

The disorder field theory for the superconductor was for a long time the onlyformulation which has led to a determination of the critical and tricritical proper-ties of the superconducting phase transition [23, 29]. Within the Ginzburg-Landautheory, an explanation was found only recently [30].

In the hydrodynamic Hamiltonian (5.164), the elimination of A(x) leads to theHamiltonian

βHhySC =

d3x[

1

2βm2A

[(∇ × a)2 +m2aa

2T ] +

m2a

2βm2A

(∇θ − v − a)2]

, (5.169)

which is gauge-invariant under

θ(x) → θ(x) + Λ(x), a(x) → a(x) + ∇Λ(x). (5.170)

5.3 Order versus Disorder Parameter

Since Landau’s 1947 work [3], phase transitions are characterized by an order pa-rameter which is nonzero in the low-temperature, ordered phase, and zero in thehigh-temperature, disordered phase. In the 1980s, this characterization has beenenriched by the dual disorder field theory of various phase transitions [7]. The ex-pectation value of the disorder field is the disorder parameter which has the oppositetemperature behavior, being nonzero in the high-temperature and zero in the low-temperature phase. Let us identify the order and disorder fields in superfluids andsuperconductors, and study their expectation values in the hydrodynamic theoriesof the two systems.

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160 5 Multivalued Fields in Superfluids and Superconductors

5.3.1 Superfluid 4He

In Landau’s original description of the superfluid phase transition with the Hamil-tonian (5.6), the role of the order parameter O is played by the expectation valueof the complex order field O(x) = φ(x):

O ≡ 〈O(x)〉 = 〈φ(x)〉. (5.171)

Its behavior can be extracted from the large-distance limit of the correlation functionof two order fields O(x)

GO(x2,x1) ≡ 〈O(x2)O∗(x1)〉 = 〈φ(x2)φ∗(x1)〉. (5.172)

This is done by taking advantage of the cluster property of the correlation functionsof arbitrary local operators

〈O1(x2)O2(x1)〉 −→|x2−x1|→∞

〈O1(x2)〉〈O2(x1)〉. (5.173)

Hence we obtain the large-distance limit of the correlation function (5.172)

GO(x2,x1) −→|x2−x1|→∞

|O|2. (5.174)

If we go to the hydrodynamic limit of the theory where the size of φ(x) is frozenand the order field reduces to O(x) = eiθ(x), the order parameter becomes

O ≡ 〈O(x)〉 = 〈eiθ(x)〉. (5.175)

This is extracted from the large-distance limit of the correlation function

GO(x2,x1) = 〈eiθ(x2)e−iθ(x1)〉. (5.176)

If we want to use (5.175) as an order parameter to replace (5.171), it is importantthat the correlation function (5.176) is vortex-gauge-invariant under the transfor-mations (5.23). This is not immediately obvious. A quantity where the invarianceis obvious is the expectation value

GO(x2,x1) =⟨

ei∫

x2

x1dx[∇θ(x)−

v(x)]⟩

. (5.177)

The transformations (5.23) do not change the exponent. We have, however, achievedvortex gauge invariance at the price of an apparent dependence of (5.177) on theshape of the path from x1 to x2. Fortunately it is possible to show that this shapedependence is not really there, so that the vortex gauge-invariant correlation func-tion is uniquely defined, and that it is in fact the same as (5.176), thus ensuring thevortex gauge invariance of (5.176).

For the proof let us rewrite (5.177) in the form

GO(x2,x1) =⟨

ei∫

d3xbm(x)[∇θ(x)−v(x)]

, (5.178)

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5.3 Order versus Disorder Parameter 161

where the fieldbm(x) = (x; Lx2

x1) (5.179)

is a δ-function on an arbitrary line Lx2x1

running from x1 to x2. This field satisfies[recall (4.10) and (4.11)]

∇ · bm(x) = q(x), (5.180)

whereq(x) = δ(3)(x − x1) − δ(3)(x − x2). (5.181)

It is now easy to see that the expression (5.178) is invariant under deformations ofLx2

x1. Indeed, let L′x2

x1be a different path running from x1 to x2. Then the difference

between the two is a closed path L, and the exponents in (5.178) differ by an integral

i∫

d3x (x; L) [∇θ(x) − v(x)] . (5.182)

The first term vanishes after a partial integration due to Eq. (4.9). The second termbecomes, after inserting (5.21),

−2πi∫

d3x (x; L) (x;S) = −2πik, k = integer. (5.183)

The integer k counts how many times the line L pierces the surface S. Since −2πikappears in the exponential it does not contribute to the correlation function (5.178).Thus we have proved that the expectation value (5.177) is independent of the pathalong which the integral runs from x1 to x2.

We recognize the analogy to the discussion of magnetic monopoles in Section 4.4.For this reason we shall speak of q(x) as a charge density of a monopole-antimonopolepair located at x2 and x1, respectively. In the description of monopoles in Sec-tion 4.4, a monopole at x2 is accompanied by a Dirac string Lx2 along which theflux is imported from infinity, the antimonopole at x1 is accompanied by a Diracstring Lx1

along which the flux is exported to infinity. Since the shape of the twolines is irrelevant, we may distort them into a single line Lx2

x1connecting x1 with

x2 along an arbitrary path, which in the present context becomes the line Lx2x1

in(5.179).

The field bm(x) is a gauge field with the same properties as the monopole gaugefield in Eq. [31]. A change of the shape of the line Lx2

x1is achieved by a monopole

gauge transformation transformation [recall (4.63)]

bm(x) → bm(x) + ∇ × (x; S). (5.184)

Note that the invariant field strength of this gauge field is the divergence (5.180)rather than a curl [compare (5.25) for a vortex gauge field].

In this way, the independence of the manifestly vortex gauge-invariant correlationfunction (5.178) on the shape of the line connecting x1 with x2 is expressed asan additional invariance under monopole gauge transformations. The correlationfunction is thus a double-gauge-invariant object.

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162 5 Multivalued Fields in Superfluids and Superconductors

After this discussion we are able to define a manifestly vortex gauge-invariantformulation of the order parameter (5.175). It is given by the expectation value

O = O(x) =⟨

ei∫

x

dx′[∇θ(x′)−v(x′)]

=⟨

ei∫

d3x′ (x′;Lx)[∇θ(x′)−v(x′)]

, (5.185)

where (x;Lx) is the δ-function on an arbitrary line as defined in Eq. (4.57). Itcomes from infinity along an arbitrary path ending at x.

Let us now study the large-distance behavior (5.173) of the correlation function(5.177) at low and high temperatures. At low temperature where vortices are rare,the θ(x)-field fluctuates almost harmonically. By Wick’s theorem, according towhich harmonically fluctuating variables θ1, θ2 satisfy the equation [32]

〈eiθ1eiθ2〉 = e−12〈θ1θ2〉, (5.186)

we can approximate

GO(x2,x1) ≈ e−12〈[θ(x2)−θ(x1)]2〉 = e〈[θ(x2)θ(x1)− 1

2θ2(x1)− 1

2θ2(x2)]〉. (5.187)

The correlation function of two θ(x)-fields is

〈θ(x2)θ(x1)〉 ≈ Tv0(|x2 − x1|), (5.188)

where v0(r) is the Coulomb potential (5.43) which goes to zero for r → ∞. Thecorrelation function (5.187) is then equal to

GO(x2,x1) ≈ e−Tv0(0)eTv0(|x2−x1|). (5.189)

This is finite only after remembering that we are studying the superfluid in thehydrodynamic limit which is correct only for length scales larger than the coherencelength ξ. In He this is of the order of a few rA. Thus we should perform all wavevector integrals only for |k| ≤ Λ ≡ 1/ξ, which makes v0(0) as finite quantity

v0(0) = 1/2ξπ2. (5.190)

As a result, the correlation function (5.187) has a nonzero large-distance limit

GO(x2,x1) −→|x2−x1|→∞

const, (5.191)

implying via Eq. (5.173) that the order parameter O = 〈eiθ(x)〉 is nonzero.Let us now calculate the large-distance behavior in the high-temperature phase.

To find the correlation function GO(x2,x1), we insert into the partition function(5.28) the extra source term

eiθ(x2)e−iθ(x1) = e−i∫

d3x q(x)θ(x). (5.192)

This term enters the canonical representation (5.79) of the energy as follows:

βH =∫

d3x

[

1

2βb2 − ib (∇θ − v) +

βεc2

(∇ × v)2 + iq(x)θ(x)

]

, (5.193)

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5.3 Order versus Disorder Parameter 163

where we have allowed for an extra core energy, for the sake of generality. Integratingout the θ-field in the partition function gives the constraint

∇ · b(x) = −q(x). (5.194)

The constraint is solved by the negative of the monopole gauge field (5.179), andhas the general solution

b(x) = ∇ × a(x) − bm(x), (5.195)

so that the energy (5.193) can be replaced by [using once more (5.183)]

βH =∫

d3x

[

1

2β(∇ × a − bm)2 − ia · jv +

βεc2

jvc2

]

. (5.196)

Under a monopole gauge transformation (5.184), this remains invariant if the gaugefield of superflow is simultaneously transformed as

a(x) → a(x) + (x; S). (5.197)

The correlation function (5.178) is now calculated from the functional integralover the Boltzmann factor with the Hamiltonian (5.196).

In the transformed energy (5.196), the presence of the source term (5.192) in thefunctional integral is accounted for by the bm-dependent integrand

e−iΣxq(x)θ(x) = e−1β

d3x 12bm(x)2−bm(x)[∇×a(x)]. (5.198)

It is instructive to calculate the large-distance behavior (5.191) of the correlationfunction in the low-temperature phase once more in this canonical formulation. Atlow temperatures, the vortex lines are frozen out and we can omit the last two termsin (5.196). We integrate out the gauge field a of superflow in the associated partitionfunction and find that the partition function contains bm in the form of a factor

e− 1

d3x

bm(x)2−[∇×bm(x)] 1

−∇2 [∇×bm(x)]

= e− 1

d3x∇·bm(x) 1

−∇2 ∇·bm(x). (5.199)

From this we obtain the correlation function

GO(x1,x2) = e− 1

d3x q(x) 1

−∇2 q(x)= e−

12β

d3xd3x′ q(x)v0(x−x′)q(x′). (5.200)

Inserting (5.181), this becomes

GO(x1,x2) = e−v0(0)/βev0(x1−x2)/β (5.201)

in agreement with the previous result (5.189).The canonical formulation (5.196) of the energy enables us to calculate the large-

distance behavior of the correlation function in the high-temperature phase. The

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164 5 Multivalued Fields in Superfluids and Superconductors

prolific vortex fluctuations produce a transverse mass term m2a a

2 which changes(5.199) to (see also Ref. [33])

e− 1

d3x

bm(x)2−[∇×bm(x)] 1

−∇2+m2a

[∇×bm(x)]

= e− 1

d3x

[

∇·bm(x) 1

−∇2+m2a

∇·bm+bm m2a

−∇2+m2abm(x)

]

. (5.202)

Using (5.180), we factorize this as

e− 1

d3x q(x) 1

−∇2+m2aq(x) × e

− 12β

d3xbm(x)m2a

−∇2+m2abm(x)

. (5.203)

The first exponent contains the Yukawa potential

vma(r) ≡∫

d3k

(2π)3eikx 1

k2 +m2a

=e−mar

4πr(5.204)

between the monopole-antimonopole pair at x2 and x1, respectively, in the sameform as in (5.201), e−vma (0)/βevma (|x1−x2|)/β , and goes to zero for large distances, i.e.,the exponential tends towards a constant. The second factor, on the other hand,has the form [recall (5.179)]

e−12β

d3xd3x′(x;Lx2x1

)vma (|x−x′|)(x′;Lx2x1

). (5.205)

This is the Yukawa self-energy of the line Lx2x1

connecting x1 and x2. For |x1 − x2|much larger than the range of the Yukawa potential 1/ma, this is proportional to|x1−x2|. Hence the second exponential in (5.203) vanishes in this limit, and so doesthe correlation function:

GO(x1,x2) ∼ e−const·|x1−x2| −→|x1−x2|→∞

0. (5.206)

Due to the cluster property (5.173) of correlation functions, this shows that at hightemperatures, the expectation value O = 〈O(x)〉 = 〈eiθ(x)〉 vanishes, so that O isindeed a good order parameter.

The mechanism which gives an energy to the initially irrelevant line Lx2x1

) con-necting monopole and antimonopole is completely analogous to the generation ofsurface energy in the previous Eq. (5.75). There the energy arose from a mass ofthe θ-fluctuations, here from a mass of the a-field fluctuations which was caused bythe proliferation of infinitely long vortex lines in the high-temperature phase.

Note that an exponential falloff is also found within Landau’s complex order fieldtheory where

〈ψ(x1)ψ(x2)〉 ∝∫

d3k

(2π)3eik(x1−x2)

1

k2 +m2=

1

e−m|x1−x2|

|x1 − x2|. (5.207)

However, here the finite range arises in a different way. In calculating (5.207), thesize fluctuations of the order field play an essential role. In the partition function(5.28), their role is taken over by the fluctuations of the vortex gauge field v(x),as pointed out at the end of Section 5.1.4. The proliferation of the vortex linesproduces the finite range 1/ma of the Yukawa potential and the exponential falloff(5.206).

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5.3 Order versus Disorder Parameter 165

5.3.2 Superconductor

In contrast to the expectation value (5.175) for superfluid helium, the expectationvalue of the order field ψ(x) of the Ginzburg-Landau Hamiltonian (5.140) cannotbe used as an order parameter since it is not invariant under the ordinary magneticgauge transformations (5.150). The expectation of all non-gauge-invariant quantitiesvanishes for all temperatures. This intuitively obvious fact is known as Elitzur’stheorem [34]. The theorem applies also to the hydrodynamic limit of ψ(x), so that

the expectation value of the exponential eiθ(x) cannot serve as an order parameter.Let us search for other possible candidates to be extracted from the large-distancelimit of various gauge-invariant correlation functions.

a) Schwinger Candidate for Order Parameter

As a first possible candidate, consider the following gauge-invariant version ofthe expectation value of 〈eiθ(x2)e−iθ(x1)〉:

GO(x2,x1) = 〈eiθ(x2)e−i∫

x2

x1dxA(x)

e−iθ(x1)〉, (5.208)

which can also be written as

GO(x2,x1) = 〈eiθ(x2)e−i∫

d3xbm(x)A(x)e−iθ(x1)〉, (5.209)

where bm(x) is the δ-function (5.179) along the line Lx2x1

connecting x1 with x2. Thisexpression is obviously invariant under magnetic gauge transformations (5.150), dueto Eqs. (5.180) and (5.181).

As before, we must make the correlation function (5.209) manifestly invariantunder vortex gauge transformations (5.151). This can be done by adding, as in(5.192), a vortex gauge field:

GO(x2,x1) = 〈ei∫

d3xbm(x)[∇θ(x)−A(x)−˜v(x)]〉. (5.210)

The associated order parameter would be [compare (5.185)]

O ≡ 〈O(x)〉 = 〈ei∫

d3x′ (x;Lx)[∇θ(x′)−A(x′)−˜v(x′)]〉. (5.211)

We now observe that in contrast to the correlation function in the superfluid(5.176), this is not invariant under deformations of the shape of the line Lx2

x1con-

necting the points x1 and x2. Indeed, if we apply the associated monopole gaugetransformation (5.184) to (5.209), we see that

e−i∫

d3xbm(x)A(x)→e−i∫

d3xbm(x)A(x)+[∇× (x;S)]A(x)=e−i∫

d3xbm(x)A(x)+B(x)(x;S),(5.212)

where S is the surface over which Lx2x1

has swept. Thus, the correlation function(5.209) changes under monopole gauge transformations by a phase

GO(x2,x1) → e−i∫

d3xB(x)(x;S)GO(x2,x1), (5.213)

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166 5 Multivalued Fields in Superfluids and Superconductors

which depends on the fluctuating magnetic flux through the surface S. For thisreason, we must first remove the freedom of choosing the shape of Lx2

x1which connects

x1 with x2. The simplest choice made by Schwinger [35] is the straight path betweenfrom x1 to x2.

Still, the correlation function (5.210) does not supply us with an order parameterwhen taking the limit of large |x2−x1|. In order to verify this, we go to the partitionfunction with the canonical representation (5.155) of the Hamiltonian, and insertthe expression (5.210). Then we change field variables from b to b − bm, and use(5.157) to obtain (5.158) with (∇ × a)2 replaced by (∇ × a − bm)2:

βHhySC =

d3x

[

1

2βm2A

(∇ × a−bm)2−ia · (∇×A)+β

2(∇× A)2−ia · jv+

βεc2

jvc2

]

.

(5.214)This is quadratic in the magnetic vector potential A which can be integrated out inthe associated partition function, leading to a Hamiltonian

βHhySC =

d3x1

2βm2A

[

(∇ × a − bm)2 +m2A a2 −ia · jv+

βεc2

jvc2

]

. (5.215)

With this Hamiltonian, the correlation function (5.209) can be calculated from theexpectation value [compare (5.198)]:

GO(x2,x1) =

e− 1

βm2A

d3x 12bm(x)2−bm(x)[∇× a(x)]⟩

. (5.216)

Consider first the low-temperature phase where the vortices in the superconduc-tor are frozen out, and we may omit the last two terms in (5.214). Then the massivefield a can be integrated out trivially in the partition function, leading to

GO(x1,x2)∼e−βm2

A2

d3x

[

bm2−(∇×bm) 1

−∇2+m2A

(∇×bm)

]

. (5.217)

This is the same expression as in the high-temperature phase of the superfluid inEq. (5.202), except that the relevant mass is now the Meissner mass mA of thesuperconductor rather than ma. The mass mA makes the line Lx2

x1between x1 and

x2 in bm(x) energetic, and leads to the same type of exponential long-distance falloffof the correlation function as in Eq. (5.206):

GO(x1,x2)∼e−const·|x1−x2| −→|x1−x2|→∞

0. (5.218)

This implies a vanishing of the candidate (5.211) for the order parameter:

O = 〈O(x)〉 = 0. (5.219)

Thus O fails to indicate the order of the low-temperature phase.

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5.3 Order versus Disorder Parameter 167

Could O be a disorder parameter? To see this we go to the high-temperaturephase where the vortex lines are prolific. In the Hamiltonian (5.214), this corre-sponds to integrating out jvc like an ordinary Gaussian variable. This produces aHamiltonian

βHhySC =

d3x

[

1

2βm2A

[

(∇ × a − bm)2 +m2a a2

]

−ia · (∇ × A)+β

2(∇ ×A)2

]

.

(5.220)If we now integrate out the magnetic vector potential A, the mass term changesfrom m2

a to m2a +m2

A, causing the correlation function to fall off even faster than in(5.218). Hence O is again zero and does not distinguish the different phases.

b) Dirac Candidate for Order Parameter

An alternative to Schwinger’s choice of a straight line connection from x1 to x2

in Eq. (5.208) we may choose a different monopole gauge field in Eq. (5.210) whichpossesses the same divergence ∇ · bm(x) = q(x) as bm(x) in Eq. (5.210), but has alongitudinal gauge [36, 37, 38]:

∇ × bm(x) = 0. (5.221)

Such a choice exists. We simply take

bm(x) = ∇1

∇2 q(x) = − 1

4π∇

[

1

|x − x1|− 1

|x − x2|

]

. (5.222)

The monopole gauge field (5.222) is the associated Coulomb field which is longitu-dinal. Now the exponent in (5.217) simplifies and we obtain the limit

GO(x1,x2)∼e−βm2

A2

d3xbm2 ∼e−βm2

A2

d3x q 1

−∇2 q = e−βm

2A/8π|x1−x2| −→

|x1−x2|→∞1. (5.223)

Actually, this result could have been deduced directly from the energy (5.215). Inthe longitudinal gauge, bm is orthogonal to the purely transversal field ∇ × a, sothat it decouples:

(∇ × a − bm)2 = (∇ × a)2 + bm 2, (5.224)

and leads directly to (5.223).The nonzero long-distance limit (5.223) is what we expect in the ordered phase,

giving rise to the hope that (5.211), with (x′;Lx) replaced by the field (5.222) of asingle monopole at x:

bmx (x′) ≡ −∇

′ 1

∇′2 δ

(3)(x′ − x), (5.225)

can supply us with an order parameter:

O =⟨

O(x)⟩

=⟨

exp

iθ(x) −∫

d3x′ bmx (x′) ·

[

A(x′) − v(x′)

]

.⟩

. (5.226)

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168 5 Multivalued Fields in Superfluids and Superconductors

The important question is whether this is zero in the high-temperature, disorderedphase of the superconductor [37, 38]. The answer is, unfortunately, negative. Wehave observed before that the vortex lines merely change the mass square in (5.217)from m2

A to m2A + m2

a. This does not modify the expression (5.223). Hence thecorrelation function has the same type of large-distance limit as in (5.223) as be-fore, implying that (5.226) is again nonzero and thus capable of distinguishing thedisordered from the ordered phase.

The reason why (5.223) is the same in both phases is very simple: It lies in thedecoupling of the transverse ∇× a from the longitudinal field bm in Eq. (5.223), sothat the asymptotic behavior is unaffected by a change in the mass of a.

c) Disorder Parameter

The only way to judge the order of the superconductor is to use the disorderfield theory and define a disorder parameter whose expectation value is zero for thelow-temperature, ordered phase and nonzero for the high-temperature, disorderedphase. For a superconductor, the disorder Hamiltonian was written down in (5.165).Recalling (5.172) we might at first consider extracting the disorder parameter froma large-distance limit of the correlation function

GD(x2,x1) = 〈φ(x2)φ∗(x1)〉. (5.227)

This, however, would not possess the gauge invariance (5.167) of the disorder Hamil-tonian (5.165). An invariant expression is obtained by inserting a factor of the typeused in (5.209)

GD(x2,x1) = 〈φ(x2)e−i∫

d3xbm(x)a(x)φ∗(x1)〉, (5.228)

where bm(x) is again the δ-function (5.179) along the line Lx2x1

connecting x1 with x2.The phase factor ensures the gauge invariance under (5.167). In the hydrodynamiclimit, (5.228) becomes

GD(x2,x1) = 〈eiθ(x2)e−i∫

d3xbm(x)a(x)e−iθ(x1)〉, (5.229)

where bm(x) is again the δ-function (5.179) along the line Lx2x1

connecting x1 withx2. This can be rewritten as similar to (5.210) as

GD(x2,x1) = 〈ei∫

d3xbm(x)[∇θ(x)−a(x)−v(x)]〉. (5.230)

which now defines a disorder parameter of the superconductor [compare (5.185)]

D ≡ 〈D(x)〉 = 〈ei∫

d3x′ (x′;Lx)[∇θ(x2)−a(x)−v(x)]〉, (5.231)

where the line L imports the flux from infinity to x.Thus we must study the energy

βHhy,DSC =

d3x

1

2βm2A

(∇ × a)2 − ia · (∇ × A) +β

2(∇ ×A)2

+m2a

2βm2A

(∇θ − v − a)2 + bm · (∇θ − a − v)

. (5.232)

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5.3 Order versus Disorder Parameter 169

Integrating out the A-field in (5.232) makes the a-field massive and the Hamiltonianbecomes

βHhy,DSC =

d3x

1

2βm2A

[

(∇ × a)2 +m2Aa

2]

+m2a

2βm2A

(∇θ − v − a)2 + bm · (∇θ − a− v)

, (5.233)

where ma is the mass parameter in Eq. (5.161), although it does not coincide withthe mass of the a-field as it did there.

As usual, we introduce an auxiliary field b to rewrite the last two terms of (5.233)in the form

d3x

[

βm2A

2m2a

(b − bm)2 + ib · (∇θ − a− v)

]

, (5.234)

and further as

d3x

[

βm2A

2m2a

(∇ × a − bm)2 + i a · (∇ × a + jv) +βεc2

jv2

]

. (5.235)

We have added a core energy to simplify the following discussion.In the low-temperature phase, there are no vortices in the superconductor but

prolific vortices in the dual formulation whose vortex density is jv, so that we canintegrate out jv in (5.235) as if it were an ordinary Gaussian field. This gives riseto a mass term for the a-field, so that the Hamiltonian (5.233) becomes

βHhy,D′

SC =∫

d3x

1

2βm2A

[

(∇ × a)2 +m2Aa

2]

+βm2

A

2m2a

[

(∇ × a − bm)2 +m2aa

2]

+ i a · ∇ × a

. (5.236)

Upon integrating out the a-field, we obtain

βHhy,D′

SC =∫

d3x

1

2βm2A

[

(∇ × a− bm)2 +m2aa

2]

+ ∆H, (5.237)

where

∆H =βm2

A

2

d3x∇ × a1

−∇2 +m2

A

∇ × a. (5.238)

If we forget this term for a moment we derive from the Hamiltonian (5.237) thecorrelation function

GD(x1,x2)∼e−βm2

A2

d3x

[

bm2−(∇×bm) 1

−∇2+m2a

(∇×bm)

]

. (5.239)

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170 5 Multivalued Fields in Superfluids and Superconductors

As in Eq. (5.202), the mass of a gives the line Lx2x1

in bm(x) an energy proportionalto its length, so that the disorder correlation function (5.230) (5.239) goes to zeroat large distances.

This result is unchanged by the omitted term (5.238). By expanding this inpowers of ∇

2, it becomes

∆H =β

2

d3x∇ × a

[

1 +∞∑

1=0

(

∇2

m2A

)n]

∇ × a, (5.240)

we see that this term changes only the dispersion of the a-field, but not its mass.

In the high-temperature phase, there are no dual vortices so that the a-fieldremains massless, and the correlation function is given by an expression like (5.199):

GD(x2,x1) ≈ e−βm2

a2

d3x

[

bm2−(∇×bm) 1

−∇2 (∇×bm)

]

= e−βm2

a2

d3x∇·bm 1

−∇2 ∇·bm

. (5.241)

This has the same constant large-distance behavior as (5.200) which is independentof the shape of Lx2

x1, implying a nonzero disorder parameter (5.231). The monopole

gauge invariance is unbroken in this phase.

Thus (5.231) is a good disorder parameter for the superconducting phase tran-sition.

c) Another Disorder Parameter

At this point we are reminded of the discussion of the behavior of the right-hand side of (5.217) and realize that the same large-distance behaviors as in (5.239)and (5.241) would arise if the Meissner mass mA of the vector potential in (5.217)would not be replaced, in the high-temperature, disordered phase, by the mass ma

of the a-field, but by the zero mass of the magnetic vector potential A(x). Then thecorrelation function (5.217) in the disordered phase would be the same as in (5.241)

GD(x1,x2)∼e−βm2

A2

d3x

[

bm2−(∇×bm) 1

−∇2 (∇×bm)

]

= e−βm2

A2

d3x∇·bm 1

−∇2 ∇·bm

(5.242)

and thus have the same long-distance behavior as (5.239), i.e., go to a nonzeroconstant for |x1−x2| → ∞. This behavior would be found if the correlation functionis defined by an expression like (5.216), but with the field a(x) is replaced by themagnetic vector potential A(x):

GO(x2,x1) =

e− 1

βm2A

d3x 12bm(x)2−bm(x)[∇×A(x)]⟩

. (5.243)

The singular line Lx2x1

in bm(x) [recall (5.179)] is taken to be the straight line con-necting x1 with x2, as in (5.216).

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5.4 Order of Superconducting Phase Transition and Tricritical Point 171

The corresponding Hamiltonian looks like (5.214), but with the magnetic gaugefield bm(x) inserted into the magnetic gradient term rather than the gradient termof the field a(x):

βHhySC =

d3x

[

1

2βm2A

(∇ × a)2−ia · (∇× A)+β

2(∇×A −bm)2−ia · jv+

βεc2

jvc2

]

.

(5.244)The correlation function (5.245) defines a disorder parameter as the expectation

value

D = 〈D(x)〉 =

e− 1

βm2A

d3x 12bmx

(x)2−bmx

(x)[∇×A(x)]⟩, (5.245)

wherebm

x (x) = (x; Lx) (5.246)

is singular on a straight line Lx from x to infinity.

5.4 Order of Superconducting Phase Transition andTricritical Point

Most experimental data obtained for the superconducting phase transition since itsdiscovery by Kamerlingh Onnes in 1908 are fitted very well by the BCS theory (recallFig. 5.8). In the neighborhood of the critical point, this is approximated by theGinzburg-Landau Hamiltonian (5.143) [39]. One may usually neglect fluctuationsof the Ginzburg-Landau order field φ(x), which is why one speaks of mean-fieldresults. The reason why these are so accurate was first explained by Ginzburg [40]who estimated the temperature interval ∆TG around Tc for which fluctuations can beimportant. Actually, his criterion cannot be applied to superconductors, as has oftenbeen done, but only to systems with a real order parameter. Since superconductorshave a complex order parameter, one must apply a different criterion which has onlybeen found recently [41]. If the order parameter has a symmetry O(N), the truefluctuation interval ∆TGK is by a factor N2 larger than Ginzburg’s estimate TG.The fluctuations cause a divergence in the specific heat at Tc very similar to thedivergence observed in the λ-transition of superfluid helium (recall Fig. 5.1). Thisinterval is in all transitions of traditional superconductors too small to be resolved[40, 41], so that it was not astonishing that no fluctuations were observed [42, 43].

In 1972, however, the order of the superconducting phase transition became amatter of controversy after a theoretical paper by Halperin, Lubensky, and Ma [44]predicted that the transition should really be of first order. The argument was basedon an application of renormalization group methods [45] to the partition function

ZGL =∫

DψDψ∗DAΦT [A] e−HGL[ψ,ψ∗,A] (5.247)

associated with the Ginzburg-Landau Hamiltonian (5.140) in 4 − ε. The technicalsignal for the first-order transition was the nonexistence of an infrared-stable fixed

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172 5 Multivalued Fields in Superfluids and Superconductors

point in the renormalization group flow [46] of the coupling constants e and g as afunction of the renormalization scale. The fact that all experimental observationsindicated a second-order transition was explained by the fact that the fluctuationinterval ∆TGK was too small to be detected. Since then there has been much work[47] trying to find an infrared-stable fixed point by going to higher loop orders or bydifferent resummations of the divergent perturbations expansions, with little success.This controversy was resolved only 10 years later in 1982 by the author [48] whodemonstrated that superconductors can have first- and second-order transitions,separated by a tricritical point .

With the advent of modern high-Tc superconductors, the experimental situationhas been improved. The temperature interval of large fluctuations is now broadenough to observe critical properties beyond the mean-field approximation. Severalexperiments have found a critical point of the XY universality class [50]. In addition,there seems to be recent evidence for an additional critical behavior associated withthe so-called charged fixed point [51]. In future experiments it will be important tounderstand the precise nature of the critical fluctuations.

Starting point of the theoretical discussion is the Ginzburg-Landau Hamiltonian(5.140). It contains the field ψ(x) describing the Cooper pairs, and the vector po-tential A(x). Near the critical temperature, but outside the narrow interval ∆TGK

of large fluctuations, the energy (5.140) describes well the second-order phase tran-sition of the superconductor. It takes place when τ drops below zero where the

pair field ψ(x) acquires the nonzero expectation value ρ0 =√

−τ/g. The properties

of the superconducting phase are approximated well by the energy (5.145). TheMeissner-Higgs mass term in (5.145) gives rise to a finite penetration depth of themagnetic field λ = 1/mA = 1/ρ0q.

By expanding the Hamiltonian (5.143) around (5.145) in powers of the fluctua-tions δρ ≡ ρ− ρ0, we find that the ρ-fluctuations have a quadratic energy

Hδρ =1

2

d3x[

(δρ)2 − 2τ(δρ)2]

, (5.248)

implying that these have a finite coherence length ξ = 1/√−2τ .

The ratio of the two length scales

κ ≡ λ/√

2ξ, (5.249)

which for historic reasons [52] carries a factor√

2, is the Ginzburg parameter whose

mean field value is κMF ≡√

g/q2. Type I superconductors have small values of κ,type II superconductors have large values. At the mean-field level, the dividing lineis lies at κ = 1/

√2.

The higher operating temperatures in the new high-Tc superconductors makefield fluctuations important. These can be taken into account by calculating thepartition function and field correlation functions from the functional integral [com-pare (5.149)]or, after the field decomposition (5.142),

ZGL =∫

Dρ ρ DAΦT [A]∑

S

Φ[v]∫

Dθe−HGL[ρ,θ,A], (5.250)

H. Kleinert, MULTIVALUED FIELDS

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5.4 Order of Superconducting Phase Transition and Tricritical Point 173

This can be approximated by the hydrodynamic formulation (5.149). From now onwe use natural temperature units where kBT = 1 and omit all tildes on top of ρ, θ,T , etc., for brevity, so that we shall rewrite (5.149):

ZhySC =

DAΦT [A]∑

S

Φ[v]∫ ∞

−∞Dθ e−βHhy

SC. (5.251)

As described above, all analytic approximations to ZGL investigated since theinitial work [44] have had difficulties in accounting for the order of the supercon-ducting phase transition. Let us recall the simplest argument suggesting a first-ordertransition. One performs a mean-field approximation in the pair field ρ and ignoresthe effect of vortex fluctuations, setting v ≡ 0 in the Hamiltonian (5.143), so thatit becomes (written without wiggles on top of ρ, θ, and v)

HappGL ≈

d3x

[

ρ2

2(∇θ − qAL)

2 +1

2(∇ρ)2+V (ρ)+

1

2(∇ × A)2 +

ρ2q2

2A2T

]

. (5.252)

The approximate sign has the following reason. We have found it useful to performanother approximation: separate A into longitudinal and transverse parts AL andAT as defined in Eq. (5.124). If ρ were a constant and not a field this separationwould be exactly possible. Due to the x-dependence, however, there will be correc-tions proportional to the gradient of ρ(x) which we shall ignore, assuming sufficientlysmooth fields ρ(x).

After these approximations we can integrate out the Gaussian phase fluctuationsθ(x) in the partition function (5.247) and obtain

Zapp′

GL = Det−1/2[−∇2]∫

Dρ DAΦT [A] e−Happ′

GL , (5.253)

with

Happ′

GL =∫

d3x

[

1

2(∇ρ)2+V (ρ)+

1

2(∇ × A)2 +

ρ2q2

2A2T

]

. (5.254)

The fluctuations of the vector potential are also Gaussian and can be integrated outin (5.253) yielding

Zapp′

GL = Det−1/2[−∇2] Det−1[−∇

2 + ρ2q2]∫

Dρ e−Happ′

GL , (5.255)

where

Happ′

GL =∫

d3x[

1

2(∇ρ)2+V (ρ)

]

. (5.256)

Assuming again that ρ is smooth, the functional determinant Det−1[−∇2 +ρ2q2]

may be done in the Thomas-Fermi approximation [53] where it yields

Det−1[−∇2 + ρ2q2]= e−Tr log[−∇

2+ρ2q2] ≈ e−V∫

[d3k/(2π)3](k2+ρ2q2) = eρ3q3/6π. (5.257)

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174 5 Multivalued Fields in Superfluids and Superconductors

Thus the A-fluctuations contribute simply a cubic term to the potential V (ρ) inEq. (5.144), changing it to

V (ρ) =τ

2ρ2 +

g

4ρ4 − c

3ρ3, c ≡ q3

2π. (5.258)

The cubic term generates, for τ < c2/4g, a second minimum in the potential V (ρ)at

ρ0 =c

2g

1 +

1 − 4τg

c2

, (5.259)

as illustrated in Fig. 5.9.

V (ρ)

ρ

ρ1

Figure 5.9 Potential for the order parameter ρ with cubic term. A new minimum

develops around ρ1 causing a first-order transition for τ = τ1.

If τ decreases belowτ1 = 2c2/9g, (5.260)

the new minimum drops below the minimum at the origin, so that the order param-eter jumps from ρ = 0 to

ρ1 = 2c/3g (5.261)

in a phase transition. At this point, the coherence length of the ρ-fluctuationsξ = 1/

√τ + 3gρ2 − 2cρ has the finite value (the same as the fluctuations around

ρ = 0)

ξ1 =3

c

g

2. (5.262)

The fact that the transition occurs at a finite ξ = ξ1 6= 0 indicates that the phasetransition is of first order. In a second-order transition, ξ would go to infinity as Tapproaches Tc.

This conclusion is reliable only if the jump of ρ0 is sufficiently large. For smalljumps, the mean-field discussion of the energy density (5.258) cannot be trusted.At a certain small ρ0, the transition becomes second-order. The change of the orderis caused by the neglected vortex fluctuations in (5.264). We must calculate thepartition function (5.253) including the sum over vortex gauge fields v(x), with aHamiltonian equal to (5.143) but with omitted wiggles:

HGL =∫

d3x

[

ρ2

2(∇θ − v− qA)2 +

1

2(∇ρ)2+V (ρ)+

1

2(∇ × A)2

]

. (5.263)

H. Kleinert, MULTIVALUED FIELDS

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5.4 Order of Superconducting Phase Transition and Tricritical Point 175

If we now integrate out the θ-fluctuations, and assume smooth ρ-fields, we obtainthe partition function (5.253) extended by the sum over vortex gauge fields v(x),and with the Hamiltonian

Happ′

GL =∫

d3x

[

1

2(∇ρ)2+V (ρ)+

1

2(∇ × A)2 +

ρ2

2(qA− v)2

T

]

. (5.264)

We may now study the vortex fluctuations separately by defining a partition functionof vortex lines in the presence of a fluctuating A-field for smooth ρ(x):

Zv,A[ρ]≡∫

DvTDAT exp

−1

2

d3x

[

(∇ ×A)2 +ρ2

2(qA − v)2

T

]

. (5.265)

The transverse part of the vortex gauge field v is defined as in (5.124). We haveabbreviated the sum over the jumping surfaces S with vortex gauge fixing

SΦ[v]defined in (5.30) as

∫ DvT . In addition, we have fixed of the vector potential to be

transverse and indicated this by the functional integration symbol DAT .The important observation is now that for smooth ρ-fields, this partial partition

function possesses a second-order transition of the XY-model type if the averagevalue of ρ drops below a critical value ρc. To see this we integrate out the A-fieldand obtain

Zv,A[ρ] = exp

[

d3xq3ρ3

]

DvT exp

[

ρ2

2

d3x

(

1

2vT

2 − vT

ρ2q2

−∇2 + ρ2q2

vT

)]

.

(5.266)

The first factor yields, once more, the cubic term of the potential (5.258). Thesecond factor accounts for the vortex loops. The integral in the exponent can berewritten as

ρ2

2

d3x

(

vT

−∇2

−∇2 + ρ2q2

vT

)

. (5.267)

Integrating this by parts, and using identity

d3x∇iA∇iB =∫

d3x [(∇ × A)(∇ ×B) + (∇ · A)(∇ ·B)], (5.268)

together with the transversality property ∇·vT = 0 and the curl relation ∇×v

T = jv

of Eq. (5.25), the partition function (5.266) without the prefactor takes the form

Zv,A[ρ] =∫

DvT exp

[

−ρ2

2

d3x

(

jv1

−∇2 + ρ2q2

jv)]

. (5.269)

This is the partition function of a grand-canonical ensemble of closed fluctuatingvortex lines. The interaction between them is of the Yukawa type with a finiterange equal to the penetration depth λ = 1/ρq.

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176 5 Multivalued Fields in Superfluids and Superconductors

It is well-known how to compute pair and magnetic fields of the Ginzburg-Landautheory for a single straight vortex line from the extrema of the energy density [42].In an external magnetic field, there exist triangular and various other regular arraysof vortex lines, such as vortex lattices. In the presence of impurities, there are vortexglasses, etc.. The study of such phases and the transitions between them is an activefield of research [54].

In the core of each vortex line, the pair field ρ goes to zero over a distanceξ. If we want to sum over a grand-canonical ensemble of fluctuating vortex linesof any shape in the partition function (5.269), the space dependence of ρ causescomplications. These can be avoided by an approximation, in which the system isplaced on a simple-cubic lattice of spacing a = α ξ, with α of the order of unity,and a fixed value ρ = ρ0 given by Eq. (5.259). Thus we replace the partial partitionfunction (5.269) approximately by

Zv,A[ρ0] ≈∑

l;∇·l=0

exp

[

−4π2ρ20a

2

x

l(x)vρ0q(x − x′)l(x′)

]

. (5.270)

The sum runs over the discrete versions of the vortex density jv in (5.269). Re-calling (5.29) and (5.41), these are 2π times the integer-valued vectors l(x) =(l1(x), l2(x), l3(x)) = ∇× n(x), where ∇ denotes the lattice derivative (5.33). Be-ing lattice curls of the integer vector field n(x) = (n1(x), n2(x), n3(x)), they satisfy∇ · l(x) = 0, This condition restricts the sum over l(x)-configurations in (5.270) toall non-selfbacktracking integer-valued closed loops. The partition function (5.265)has precisely the form discussed before in Eq. (5.47) with ρ0q playing the role ofthe Yukawa mass m in (5.47). The lattice partition function (5.270) has there-fore a second-oder phase transition in the universality class of the XY-model. Thetransition temperature was plotted in Fig. 5.7.

Comparing (5.270) with the partition function (5.98) of the Yukawa loop gas, weconclude that there is a second-order phase transition when [compare (5.100)

4π2aρ20vρs0q(0) ≈ Tc ≈ 3. (5.271)

Using the analytic approximation (5.103), we may write this as

4π2av0(0)ρ2

0

σ a2ρ20q

2/6 + 1≈ Tc ≈ 3, (5.272)

orρ2

0a

σ a2ρ20q

2/6 + 1≈ r

3, (5.273)

where r = 9/4π2v0(0) ≡ 0.90. The solution is

ρ0 ≈1√3a

r

1 − σrae2/18. (5.274)

H. Kleinert, MULTIVALUED FIELDS

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5.4 Order of Superconducting Phase Transition and Tricritical Point 177

Replacing here a by αξ1 = α(3/c)√

g/2 of Eq. (5.262), and ρ0 by ρ1 = 2c/3g of

Eq. (5.261). Inserting further c = q3/2π of Eq. (5.258), we find the equation for the

mean-field Ginzburg parameter κMF =√

g/q2 [recall (5.249)]:

κ3MF + α2σ

κMF

3−

√2α

πr= 0. (5.275)

For the best value σ ≈ 1.6 in the approximation (5.103), and r ≈ 0.9, and the roughestimate α ≈ 1, the solution of this equation yields the tricritical value

κtricMF ≈ 0.82/

√2. (5.276)

In spite of the roughness of the approximations, this result is very close to the value

κtricMF =

3√

3

1 − 4

9

(

π

3

)

≈ 0.80√2

(5.277)

derived from the dual theory in [23]. The approximation (5.276) has three uncer-tainties. First, the identification of the effective lattice spacing a = αξ with α ≈ 1;second the associated neglect of the x-dependence of ρ and its fluctuations, and thirdthe localization of the critical point of the XY-model type transition in Eqs. (5.104)and the ensuing (5.273).

5.4.1 Disorder Theory

In the disorder theory (5.166) it is much easier to prove that superconductors canhave a first- and a second-order phase transition, depending on the size of theGinzburg parameter κ defined in Eq. (5.249). Before we start let us rewrite thedisorder theory in a more convenient way. As before, we decompose the complexdisorder field φ as φ = ρeiθ. In the partition function (5.168), this changes themeasure of functional integration from

∫ Dφ ∫ Dφ∗ to∫ Dρρ ∫ Dθ. Now we fix the

gauge by absorbing the phase θ of the field into a by a gauge transformation (5.167).This brings the Hamiltonian (5.166) to the form

βHhySC ∼

d3x[

1

2βm2A

[(∇ × a)2 +m2Aa

2T ] +

ρ2

2

(

a2T + a2

L

)

+1

2(∇ρ)2 +

τ

2ρ2 +

g

4ρ4]

,

(5.278)

where we have again assumed a smooth ρ-field to separate ρ2a2 into ρ2a2T + ρ2a2

L.The partition function (5.168) reads now

ZdualSC =

DρρDa e−βHhySC. (5.279)

We may integrate out aL to obtain a factor Det[ρ2]−1/2 which removes the factor ρin the measure of path integration Dρρ.

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178 5 Multivalued Fields in Superfluids and Superconductors

Next we integrate out aL and obtain

ZdualSC =

DρDet[−∇2 +m2

A(1 + βρ2)]e−βHhySC, (5.280)

with

βHhySC =

d3x[

1

2(∇ρ)2 +

τ

2ρ2 +

g

4ρ4]

+ Tr log[−∇2 +m2

A(1 + βρ2)] . (5.281)

In the superconducting phase, there are only a few vortex lines and the disorderfield ρ of vortex lines fluctuates around zero. In this phase we may expand theTracelog into powers of ρ2. The first expansion term is proportional to ρ2 andrenormalizes τ in the Hamiltonian (5.281), corresponding to a shift in the criticaltemperature.

The second expansion term is approximately given by

−β2m4A

d3x∫

d3k

(2π)3

1

(k2 +m2A)2

ρ4 ∝ −m3A

d3x ρ4. (5.282)

This term lowers the interaction term (g/4)ρ4 in the Hamiltonian (5.281). Anincrease in mA corresponds to a decrease of the penetration depth in the supercon-ductor, i.e. to materials moving towards the type-I regime. At some larger value ofmA, the ρ4-term vanishes and the disorder field theory requires a ρ6-term to stabilizethe fluctuations of the vortex lines. In such materials, the superconducting phasetransition turns from second to first order.

A more quantitative version of this argument was used in Ref. [48] to show theexistence of the tricritical point and its location at the Ginzburg parameter κ ≡ g/q2

in Eq. (5.277), which agrees well with a recent Monte Carlo value (0.76± 0.04)/√

2of Ref. [49].

5.5 Vortex Lattices

The model action (5.22) represents the gradient energy in superfluid 4He correctlyonly in the long-wavelength limit. The neutron scattering data yield the energyspectrum ω = ε(k) shown in Fig. 5.2.

To account for this, the energy should be taken as follows:

HNG =1

2

d3x(∇ θ − v)ε2(−i∇)

−∇2(∇ θ − v). (5.283)

The roton peak near 2rA−1 gives rise to a repulsion between opposite vortex lineelements at the corresponding distance. If a layer of superfluid 4He is diluted with3He, the core energy of the vortices decreases, the fugacity y and the average vortexnumber increases. For a sufficiently high average spacing, a vortex lattice forms. Inthis regime, the superfluid has three transitions when passing from zero temperatureto the normal phase. There is first a condensation process to a vortex lattice, then

H. Kleinert, MULTIVALUED FIELDS

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5.5 Vortex Lattices 179

Figure 5.10 Phase diagram of a two-dimensional layer of superfluid 4He. At a higher

fugacity y > y∗, an increase in temperature causes the vortices to first condense to a lattice

and to undergo a Kosterlitz-Thouless vortex unbinding transition only after a melting

transition.

a melting transition of this lattice into a fluid of bound vortex-antivortex pairs, andfinally a pair-unbinding transitions of the Kosterlitz-Thouless type [55, 56]. Thelatter two transitions have apparently been seen experimentally [57] (see Figs. 5.10and (5.11).

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180 5 Multivalued Fields in Superfluids and Superconductors

Figure 5.11 Experimental phase diagram of a two-dimensional layer of superfluid 4He

diluted by 3He which decreases the fugacity and separates the vortex melting transition

from the Kosterlitz-Thouless transition.

Notes and References

[1] R.A. Cowley and A.D. Woods, Can. J. Phys 49, 177 (1971).

[2] P. Debye, Zur Theorie der spezifischen Warmen, Annalen der Physik 39(4),789 (1912).

[3] L.D. Landau, J. Phys. U.S.S.R. 11, 91 (1947) [see also Phys. Rev. 75, 884(1949)].

[4] C.A. Jones and P.H. Roberts, J. Phys. A: Math. Gen. 15, 2599 (1982).

[5] L. Onsager, Nuovo Cimento Suppl. 6, 249 (1949).

[6] R.P. Feynman, in Progress in Low Temperature Physics, ed. by C. J. Gorter,North-Holland, Amsterdam, 1955.

[7] H. Kleinert, Gauge fields in Condensed Matter , Vol. I: Superflow and VortexLines, Disorder Fields, Phase Transitions , World Scientific, Singapore, 1989(kl/b1), where kl is short for the www address http://www.physik.fu-ber-lin.de/~kleinert.

[8] W. Shockley, in L’Etat Solid, Proc. of Neuvieme Conseil de Physique, Brussels,ed. R. Stoops (Inst. de Physique, Solvay, Brussels, 1952).

[9] H. Kleinert, Gauge fields in Condensed Matter , Vol. II: Stresses and De-fects, Differential Geometry, Crystal Defects, World Scientific, Singapore, 1989(kl/b2).

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 181

[10] Note that this configurational entropy cannot be properly accounted for by amodel restricted only to circular vortex lines proposed byG. Williams, Phys. Rev. Lett. 59, 1926 (1987)

[11] See pp. 529-530 in the textbook [7] (kl/b1/gifs/v1-530s.html).

[12] This relation was first stated on p. 517 of the textbook [7] and confirmedexperimentally inY.J. Uemura, Physica B 374, 1 (2006) (cond-mat/0512075).

[13] L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys.-JETP 13, 451(1961)].

[14] H. Kleinert, Theory of Fluctuating Nonholonomic Fields and Applications:Statistical Mechanics of Vortices and Defects and New Physical Laws in Spaceswith Curvature and Torsion, in: Proceedings of NATO Advanced Study Insti-tute on Formation and Interaction of Topological Defects, Plenum Press, NewYork, 1995, pp. 201–232 (kl/227).

[15] See pp. 136–145 in the textbook [7] (kl/b1/gifs/v1-136s.html).

[16] See p. 468 in the textbook [7] (kl/b1/gifs/v1-468s.html).

[17] See p. 241 in the textbook [7] (kl/b1/gifs/v1-241s.html).

[18] J. Villain, J. Phys. (Paris) 36, 581 (1977). See also the textbook discussion atkl/b1/gifs/v1-489s.html.

[19] See p. 503 in the textbook [7] (kl/b1/gifs/v1-503s.html). The high-temperature expansions of the partition function (5.31) and the associatedfree energy are given in Eqs. (7.42a) and (7.42b), the low-temperature expan-sion of the free energy in Eq. (7.43).

[20] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets, 4th ed., World Scientific, Singapore 2006(kl/b5).

[21] Similar canonical representations for the defect ensembles in a variety of phys-ical systems are given inH. Kleinert, J. Phys. 44, 353 (1983) (Paris) (kl/102).

[22] See the textbook [7], Part 2, Sections 9.7 and 11.9, in particular Eq. (11.133);H. Kleinert and W. Miller, Phys. Rev. Lett. 56, 11 (1986) (kl/130);Phys. Rev. D 38, 1239(1988) (kl/156).

[23] H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982) (kl/97). The tricritical valueκ ≈ 0.80/

√2 derived in this paper was confirmed only recently by Monte Carlo

simulations [49].

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182 5 Multivalued Fields in Superfluids and Superconductors

[24] The equation js = −aT is the disorder version of the famous first Londonequation for the superconductor js = −(q2n0/Mc)AT to be discussed furtherin Section 7.1.1.

[25] J. Bardeen, L.N. Cooper, Schrieffer, Phys. Rev. 108, 1175 (1957);M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York,1975.

[26] L.D. Landau, Zh. Elsp. Teor. Fiz. 7, 627 (1937);V.L. Ginzburg and L.D. Landau, ibid. 20, 1064 (1950).

[27] H. Kleinert, Collective Quantum Fields, Lectures presented at the First EriceSummer School on Low-Temperature Physics, 1977, in Fortschr. Physik 26,565-671 (1978) (kl/55). See Eq. (4,118).

[28] This property of the disorder theory was demonstrated in detail in Ref. [7].

[29] M. Kiometzis, H. Kleinert, and A.M.J. Schakel, Phys. Rev. Lett. 73, 1975(1994) (cond-mat/9503019).

[30] H. Kleinert, Europh. Letters 74, 889 (2006) (cond-mat/0509430).

[31] See Ref. [9], Vol. I, Part 2, Chapter 10.

[32] For a derivation see Section 3.10 of the textbook [20].

[33] This is an analog of the Meissner effect in the dual description of superfluidhelium.

[34] S. Elitzur, Phys. Rev. D 12, 3978 (1975).

[35] J. Schwinger, Phys. Rev. 115, 721 (1959); 127, 324 (1962).

[36] P.A.M. Dirac, Principles of Quantum Mechanics, 4th ed., Clarendon, Cam-bridge 1981, Section 80; Gauge-Invariant Formulation of Quantum Electrody-namics , Can. J. of Physics 33, 650 (1955). See, in particular, his Eqs. (16)and (19).

[37] T. Kennedy and C. King, Phys. Rev. Lett. 55, 776 (1985); Comm. Math.Phys. 104, 327 (1986).

[38] M. Kiometzis and A.M.J. Schakel, Int. J. Mod. Phys. B 7, 4271 (1993).

[39] H. Kleinert, Collective Quantum Fields , Lectures presented at the First EriceSummer School on Low-Temperature Physics, 1977, in Fortschr. Physik 26,565 (1978) (kl/55).

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 183

[40] V.L. Ginzburg, Fiz. Twerd. Tela 2, 2031 (1960) [Sov. Phys. Solid State 2, 1824(1961)]. See also the detailed discussion in Chapter 13 of the textbook L.D.Landau and E.M. Lifshitz, Statistical Physics , 3rd edition, Pergamon Press,London, 1968.

[41] H. Kleinert, Criterion for Dominance of Directional over Size Fluctuations inDestroying Order , Phys. Rev. Lett. 84, 286 (2000) (cond-mat/9908239).The Ginzburg criterion estimates the energy necessary for hopping over anenergy barrier for a real order parameter. For a superconductor, however,the size of directional fluctuations is relevant which gives rise to vortex loopproliferation. In general, fluctuations with symmetry O(N) are important inan N2-times larger temperature interval than Ginzburg’s. See also pp. 18–23in the textbook [46].

[42] D. Saint-James, G. Sarma, and E.J. Thomas, Type II Superconductivity , Perga-mon, Oxford, 1969; M. Tinkham, Introduction to Superconductivity , McGraw-Hill, New York, 1996.

[43] M. Tinkham, Introduction to Superconductivity , 2nd ed., Dover, New York,1996.

[44] B.I. Halperin, T.C. Lubensky, and S. Ma, Phys. Rev. Lett. 32, 292 (1972).

[45] L.P. Kadanoff, Physics 2, 263 (1966); K.G. Wilson, Phys. Rev. B 4, 3174,3184 (1971); K.G. Wilson and M.E. Fisher, Phys. Rev. Lett. 28, 240 (1972);and references therein.

[46] For a general treatment of the renormalization group see the textbooksJ. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., Ox-ford Science Publications, Oxford 2002;H. Kleinert and V. Schulte-Frohlinde, Critical Phenomena in φ4-Theory ,World Scientific, Singapore, 2001 (kl/b8).

[47] A small selection of papers on this subject is:J. Tessmann, Two Loop Renormalization of Scalar Electrodynamics, MS thesis1984 (the pdf file is available on the internet at (kl/MS-Tessmann.pdf), wherekl is short for www.physik.fu-berlin.de/~kleinert;S. Kolnberger and R. Folk, Critical fluctuations in superconductors, Phys. Rev.B 41, 4083 (1990);R. Folk and Y. Holovatch, On the critical fluctuations in superconductors, J.Phys. A 29, 3409 (1996);I.F. Herbut and Z. Tesanovic, Critical Fluctuations in Superconductors andthe Magnetic Field Penetration Depth, Phys. Rev. Lett. 76, 4588 (1996);H. Kleinert and F.S. Nogueira, Charged fixed point in the Ginzburg-Landausuperconductor and the role of the Ginzburg parameter κ, Nucl. Phys. B 651,361 (2003) (cond-mat/0104573).

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184 5 Multivalued Fields in Superfluids and Superconductors

[48] H. Kleinert, Disorder Version of the Abelian Higgs Model and the Su-perconductive Phase Transition, Lett. Nuovo Cimento 35, 405 (1982).See also the more detailed discussion in Ref. [7], Part 2, Chapter 13(kl/b1/gifs/v1-716s.html), where the final disorder theory was derived [see,in particular, Eq. (13.30)].

[49] J. Hove, S. Mo, and A. Sudbo, Vortex interactions and thermally inducedcrossover from type-I to type-II superconductivity , Phys. Rev. B 66, 64524(2002) (cond-mat/0202215);S. Mo, J. Hove, and A. Sudbo, Order of the metal to superconductor transition,Phys. Rev. B 65, 104501 (2002) (cond-mat/0109260).The Monte Carlo simulations of these authors yield the tricritical value (0.76±0.04)/

√2 for the Ginzburg parameter κ =

g/q2.

[50] T. Schneider, J. M. Singer, A Phase Transition Approach to High TemperatureSuperconductivity: Universal Properties of Cuprate Superconductors, WorldScientific, Singapore, 2000; Evidence for 3D-xy critical properties in under-doped YBa2Cu3O7+x , (cond-mat/0610289).

[51] T. Schneider, R. Khasanov, and H. Keller, Phys. Rev. Lett. 94, 77002 (2005);T. Schneider, R. Khasanov, K. Conder, E. Pomjakushina, R. Bruetsch, andH. Keller, J. Phys. Condens. Matter 16, L1 (2004) (cond-mat/0406691).

[52] There is also a good physical reason for the factor√

2: In the high-temperature,disordered phase the fluctuations of real and imaginary part of the order fieldψ(x) have equal range for κ = 1/

√2.

[53] See Section 4.10 of the textbook [32].

[54] For a theoretical discussion and a detailed list of references seeJ. Dietel and H. Kleinert, Defect-Induced Melting of Vortices in High-Tc Su-perconductors: A model based on continuum elasticity theory , Phys. Rev. B74, 024515 (2006) (cond-mat/0511710); Phase Diagram of Vortices in High-TcSuperconductors from Lattice Defect Model with Pinning (cond-mat/0612042)

[55] M. Gabay and A. Kapitulnik, Phys. Rev. Lett. 71, 2138 (1993).

[56] S.-C. Zhang, Phys. Rev. Lett. 71, 2142 (1993).

[57] M.T. Chen, J.M. Roessler, and J.M. Mochel, J. Low Temp. Phys. 89, 125(1992).

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There is no excellent beauty

that hath not some strangeness in the proportion.

Francis Bacon (1561 - 1626)

6Dynamics of Superfluids

It has been argued by Feynman [1] that at zero temperature, the time dependenceof the φ-field in the Hamiltonian (5.6) is governed by the action

A =∫

dt∫

d3xL =∫

dt∫

d3x ihφ∗∂tφ−H [φ]

, (6.1)

so that the Lagrangian density is

L = ihφ∗∂tφ− 1

2

|∇φ|2 + τ |φ|2 +λ

2|φ|4

. (6.2)

In the superfluid phase where τ < 0 and φ = φ0 =√

−τ/λeiα, this can be writtenmore explicitly, using proper physical units rather than natural units, i.e., includingthe Planck constant h and the mass M of the superfluid particles, as

L = ihφ∗∂tφ− h2

2M|∇φ|2 − c20M

2n0

(φ∗φ− n0)2 +

c20Mn0

2, (6.3)

where n0 = |φ0|2 = −τ/λ is the density φ∗φ of the superfluid particles in the groundstate, i.e., the superfluid density (5.15) which we name n(x) to avoid confusionwith the field size ρ(x) = |φ(x)| in Eq. (5.10)–(5.20). The last term is the negativecondensation energy density −c20Mn0/2 in the superfluid phase. The interactionstrength λ in (5.6) has been reparametrized as 2c20M/n0 and τ as −2c20M for reasonsto be understood below.

The equation of motion of the time-dependent field φ(t,x) ≡ φ(x) is

ih∂tφ(x)=

[

− h

2M∇

2− c20M +c20M

n0φ∗(x)φ(x)

]

φ(x). (6.4)

6.1 Hydrodynamic Description

After substituting φ(x) by ρ(x)eiθ(x) as in Eq. (5.10), and further ρ(x) by√

n(x),

the Lagrangian density in (6.3) becomes

L=n(x)

−h[∂tθ(x)+θvt (x)]−

h2

2M[∇θ(x)−v(x)]2 − e∇n(x) − en(x)

(6.5)

185

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186 6 Dynamics of Superfluids

where

en(x) ≡c20M

2n0n(x)

[n(x) − n0]2 − n2

0

(6.6)

is the energy per particle associated with the density of the fluctuating condensate,and

e∇n(x) ≡h2

8M

[∇n(x)]2

n2(x)(6.7)

the gradient energy of the condensate. This energy may be also be written with

e∇n(x) =pos2(x)

2M(6.8)

where

pos(x) ≡Mvos(x) ≡ h

2

∇n(x)

n(x)(6.9)

is i times the quantum-mechanical momentum associated with the expansion of thecondensate, the so-called osmotic momentum. The vector vos(x) is the associatedosmotic velocity.

If the particles move in an external trap potential V (x), this is simply added toe(x), so that the two last terms in (6.5) are replaced by

etot(x) = e∇n(x) + en(x) + V (x). (6.10)

The field θvt (x) is the time component of the vortex gauge field. Together with

v(x) it forms the four-vector

θµv(x) = (cθvt (x), v(x)), (µ = 0, 1, 2, 3), (6.11)

which is the spacetime extension of the vortex gauge field (5.21). If the jumpingsurface S in Eq. (5.17) moves along the time axis, it becomes a volume V , for whichwe define a δ-function as follows:

δabc(x;V ) ≡∫

dσdτdλ

P (abc)

εP∂xb∂σ

∂xc∂τ

∂xd∂λ

δ(4) (x− x(σ, τ, λ)) , (6.12)

where the sum runs over all 6 permutations P of the indices and εP denotes theirparity (εP = +1 for even and −1 for odd permutation P ). From this we form thedual δ-function

δa(x;V ) ≡ εabcdδabc(x;V ). (6.13)

We may conveniently chose the axial gauge of the vortex gauge field where the timecomponent θv

t (x) vanishes and only the spatial part v(x) is nonzero. Then thespatial components of δa(x;V ) can be written as (x;S(t)).

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6.1 Hydrodynamic Description 187

After gauge fixing, the field θ(x) runs from −∞ to ∞ rather than −π to π [recallthe steps leading from the partition function (5.31) to (5.39)].

We now introduce the velocity field with vortices

v(x) ≡ h[∇θ(x) − v(x)]/M, (6.14)

and the local deviation of the particle density from the ground-state value δn(x) ≡n(x) − n0, so that (6.5) can be written as

L = −n(x)[

h∂tθ(x) +M

2v2(x) + etot(x)

]

. (6.15)

The Lagrangian density (6.15) is invariant under changed of θ(x) by an additiveconstant Λ. According to Noether’s theorem, this implies the existence of conservedcurrent density density. We can calculate the charge and particle current densitiesfrom the rule (3.100) as

n(x) = −1

h

∂L∂∂tθ(x)

, j(x) = −1

h

∂L∂∇θ(x)

= n(x)v(x). (6.16)

To find the second expression we must remember (6.14). The prefactor 1/h is chosento have the correct physical dimensions. The associated conservation law reads

∂tn(x) = −∇ · [n(x)v(x)], (6.17)

which is the continuity equation of hydrodynamics. This equation is found from theLagrangian density (6.15) by extremizing the associated action with respect to θ(x),

Functional extremization with respect to δn(x) yields

h∂tθ(x) +M

2v2(x) + V (x) + h∇n(x) + hn(x) = 0, (6.18)

where we have included a possible external potential V (x) as in Eq. (6.10). Thelast term is the enthalpy per particle associated with the energy density en(x). It isdefined by

hn(x) ≡∂[n(x)en(x)]

∂n(x)= en(x) + n(x)

∂en(x)

∂n(x)= en(x) +

pn(x)

n(x), (6.19)

where pn(x) is the pressure due to the energy en(x):

pn(x) ≡ n2(x)∂

∂nen(x) =

(

n∂

∂n− 1

)

[n(x)en(x)] . (6.20)

For en(x) from Eq. (6.6), and allowing for an external potential V (x) as in (6.10),wefind

hn(x) =c20M

n0

δn(x), pn(x) =c20M

2n0

n2(x). (6.21)

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188 6 Dynamics of Superfluids

The term h∇n(x) is the so called quantum enthalpy . It is obtained from the energydensity e∇n(x) as a contribution from the Euler-Lagrange equation:

h∇n(x) ≡∂[n(x)e∇n(x)]

∂n(x)− ∇

∂[n(x)e∇n(x)]

∂∇n(x)(6.22)

This can be written as

h∇n(x) = e∇n(x) +p∇n(x)

n(x), (6.23)

where

p∇n(x)=n2(x)

[

∂n−∇

∂∇n

]

e∇n(x)=

n(x)

[

∂n−∇

∂∇n

]

−1

[n(x)e∇n(x)].(6.24)

is the so called quantum pressure.Inserting (6.7) yields

h∇n(x) =h2

8M

[∇n(x)]2

n(x)− 2∇2n(x)

, p∇n(x) = − h2

4M∇

2n(x). (6.25)

The two equations (6.17) and (6.18) were found by Madelung in 1926 [2].The gradient of (6.18) yields the equation of motion

M∂tv(x) + h∂tv +M

2∇v2(x) = −∇Vtot(x) − ∇h∇n(x) − ∇hn(x), (6.26)

whereVtot(x) ≡ V (x) ≡ +h∇n(x) + ∇hn(x). (6.27)

We now use the vector identity

1

2∇v2(x) = v(x) × [∇× v(x)] + [v(x) · ∇]v(x), (6.28)

and rewrite Eq. (6.26) as

M∂tv(x) +M [v(x) · ∇]v(x) = −∇Vtot(x) + fv(x),

(6.29)

where

fv(x) ≡ −h∂tv(x)−Mv(x)×[∇× v(x)] = −h∂tv(x)+hv(x)×[∇× v(x)] (6.30)

is a force due to the vortices. The classical contribution to the second term is theimportant Magnus force [3] acting upon a rotating fluid:

fvMagnus(x) ≡ −Mv(x) × [∇× v(x)] . (6.31)

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6.1 Hydrodynamic Description 189

The important observation is now that this force is in fact zero for the vortices inthe superfluid. Let us prove this. Consider first the two-dimensional situation witha point-like vortex which lies at the origin at a given time t. This we can describedby a vortex gauge field

θv1(x) = 0, θv

2(x) = 2πΘ(x1)δ(x2), (6.32)

where Θ(x1) is the Heaviside step function which is zero for negative and unity forpositive x1. The curl of (6.32) is the vortex density, which is proportional to aδ-function at the origin:

∇ × v(x) = ∇1θv2(x) −∇2θ

v1(x) = 2πδ(2)(x), (6.33)

in agreement with the general relation (5.25). Suppose that the vortex moves, aftera short time ∆t, to the point x + ∆x = (∆x1, 0), where

θvx(x) = 0, θvy(x) = Θ(x1 + ∆x1)δ(x2), ∇ × v(x) = 2πδ(2)(x + ∆x). (6.34)

Since Θ(x1 + ∆x1) = Θ(x1) + ∆x1δ(x1), we see that ∆v(x) = ∆x × [∇ × v(x)]which becomes

∂tv(x) = v(x) × [∇× v(x)] (6.35)

after dividing it by ∆t and taking the limit ∆t → 0, thus proving the vanishing offv(x).

The result can easily be generalized to a line with wiggles by approximating itas a sequence of points in closely stacked planes orthogonal to the line elements. Aslong as the line is smooth, the change in the direction is of higher order in ∆x anddoes not influence the result in the limit ∆t → 0. Thus we can omit the last termin (6.29).

Equation (6.35) is the equation of motion for the vortex gauge field. The timedependence of this field is governed by quantum analog of the Magnus force (6.31).

Note that for a vanishing force fv(x) and quantum pressure p∇n(x), Eq. (6.29)coincides with the classical Euler equation of motion for an ideal fluid

Md

dtv(x)=M∂tv(x)+M [v(x) · ∇]v(x) = −∇V(x) −∇pn(x)

n(x). (6.36)

The last term is initially equal to −∇hn(x). However, since en(x) depends only onn(x) (such systems are referred to as barytropic), we see that (6.19) implies

∇hn(x) =∇en(x)−pn(x)

n2(x)∇n(x)+

∇pn(x)

n(x)=

[

∂en(x)

∂n(x)− pn(x)

n2(x)

]

∇n(x) +∇pn(x)

n(x)

=∇pn(x)

n(x). (6.37)

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190 6 Dynamics of Superfluids

There are only two differences between (6.29) with fv(x) = 0 and the classicalequation (6.37). One is the extra quantum part −∇h∇n(x) in (6.29). The other liesin the nature of the vortex structure. In a classical fluid, the vorticity1

w(x) ≡ ∇× v(x) (6.38)

can be an arbitrary function of x. For instance, a velocity field v(x) = (0, x1, 0) hasthe constant vorticity ∇×v(x) = 1. In a superfluid, such vorticities do not exist. Ifone performs the integral over any closed contour M

dx ·v(x), one must always findan integer multiple of h to ensure the uniqueness of the wave function around thevortex line. This corresponds to the Sommerfeld quantization condition

dx·p(x) =hn. In a superfluid, there exists no continuous regions of nonzero vorticity, onlyinfinitesimally thin lines. This leaves only vorticities which are superpositions ofδ-functions 2πhδ(x;L), which is guaranteed here by the expression (6.14) for thevelocity.

By taking the curl of the the right-hand side of the vanishing force fv(x), weobtain an equation for the time derivative of the vortex density

∂t[∇ × v(x)] = ∇ × v(x) × [∇× v(x)] . (6.39)

Such an equation was first found in 1942 by Ertel [4] for the vorticity w(x) of aclassical fluid, rather than ∇ × v(x). Using the vector identity

∇×v(x)×w(x) = −w(x)[∇ · v(x)] − [v(x) · ∇]w(x)

+v(x)[∇ · w(x)] + [w(x) · ∇]v(x), (6.40)

and the identity ∇ ·w(x) = ∇ · [∇×v(x)] ≡ 0, we can rewrite the classical versionof (6.39) in the form

d

dtw(x) = ∂tw(x) + [v(x) · ∇]w(x) = −w(x)[∇ · v(x)] + [w(x) · ∇]v(x), (6.41)

which is the form stated by Ertel. This and Eq. (6.36) are the basis for deriving thefamous Helmholtz-Thomson theorem of a ideal perfect classical fluid which statesthat the vorticity is constant along a vortex line if the forces possess a potential.

Equation (6.39) is the quantum version of Ertel’s equation where the vorticityoccurs only in infinitesimally thin lines satisfying the quantization condition

dx ·p(x) = hn.

Inserting the vortex density (5.25) into Eq. (6.39), we obtain for a line L(t)moving in a fluid with a velocity field v(x) the equation

∂tδ(x;L(t)) = ∇ × v(x) × δ(x;L(t)). (6.42)

It has been argued by L. Morati [5, 6] on the basis of a stochastic approachto quantum theory by E. Nelson [7] that the force fv(x) is not zero but equal toquantum force

fqu(x) ≡ − h2

[

∇n(x)

n(x)+ ∇

]

× [∇ × v(x)]. (6.43)

1The letter w stems from the German word for vorticity=“Wirbelstarke”.

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6.2 Velocity of Second Sound 191

Our direct derivation from the superfluid Lagrangian density (6.2) and the equivalent(6.2) does not produce such a term.

6.2 Velocity of Second Sound

Consider the Lagrangian density (6.15) and omit the trivial constant condensationenergy density −c20Mn0/2 as well as external potential V (x). The result is

L = −[n0 + δn(x)][

h∂tθ(x) +M

2v2(x)

]

− h2

8M

[∇δn(x)]2

n(x)− c20M

2n0[δn(x)]2. (6.44)

For small δn(x) n0, this is extremal at

δn(x) =n0

c20M

1

1 − ξ2∇2

[

h∂tθ(x) +M

2v2(x)

]

, (6.45)

where

ξ ≡ 1

2

h

c0M=

1

2

c

c0λM (6.46)

is the range of the δn(x)-fluctuations, i.e., the coherence length of the superfluid,and λM = h/Mc the Compton wavelength of the particles of mass M .

Reinserting (6.45) into (6.44) leads to the alternative Lagrange density

L = − n0

[

h∂tθ(x) +M

2v2(x)

]

+n0

2c20M

[

h∂tθ(x) +M

2v2(x)

]

1

1 − ξ2∇2

[

h∂tθ(x) +M

2v2(x)

]

. (6.47)

The first term is an irrelevant surface term and can be omitted. The quadraticfluctuations of θ(x) are governed by the Lagrange density

L0 =n0h

2

2M

1

c20[∂tθ(x) − θv

t (x)]1

1 − ξ∇2 [∂tθ(x) − θvt (x)] − [∇θ(x) − v(x)]2

,

(6.48)

For the sake of manifest vortex gauge invariance we have reinserted the time-component θv

t (x) of the vortex gauge field which was omitted in (6.48) where weused the axial gauge.

In the absence of vortices, and in the long-wavelength limit, the Lagrangiandensity (6.48) leads to the equation of motion

(−∂2t + c20∇

2)θ(x) = 0. (6.49)

This is a Klein-Gordon equation for θ(x) which shows that the parameter c0 isthe propagating velocity of phase fluctuations, which form the second sound in thesuperfluid.

Note the remarkable fact that although the initial equation of motion (6.4) isnonrelativistic, the sound waves follow a Lorentz-invariant equation in which thesound velocity c0 playing the role of the light velocity. If there is a potential, thevelocity of second sound will no longer be a constant but depend on the position.

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192 6 Dynamics of Superfluids

6.3 Vortex Electric and Magnetic Fields

It is useful to carry the analogy between the gauge fields of electromagnetism furtherand define the vortex electric and magnetic fields as

Ev(x) ≡ − [∇θvt (x) + ∂tv(x)] , Bv(x) ≡ ∇ × v(x). (6.50)

These are the analogs to the definitions (2.73) and (2.74) where θvt (x) corresponds

to At(x) ≡ cφ(x) ≡ A0(x) defined so that dxµAµ ≡ dx0A0 − dx ·A = dtAt− dx ·A.The Bv-field has dimension cm/sec2, the Ev-field 1/sec. They satisfy same type ofBianchi identities as the electromagnetic fields in (1.182) and (1.183):

∇ · Bv(x) = 0, (6.51)

∇ ×Ev(x) + ∂tBv(x) = 0. (6.52)

With these fields, the vortex force (6.31) becomes

fv(x) = Ev(x) + v(x) × Bv(x), (6.53)

which has the same form as the electromagnetic force upon a moving particle of unitcharge. The corresponding vortex force vanishes.

Recall that the vanishing of this force implies that the time dependence of thevortex gauge field is driven by the Magnus force [see Eq. (6.35)].

By substituting Eq. (6.50) for Bv(x) into (6.39), we find the following equationof motion for the vortex magnetic field:

∂tBv(x) = ∇× v(x)×Bv(x). (6.54)

Note that due to Eq. (6.14),

∇ × v(x) = − h

M∇ × v(x) = − h

MBv(x). (6.55)

6.4 Simple Example

As a simple example illustrating the above extension of Madelung’s theory considera harmonic oscillator in two dimensions with the Schrodinger equation in cylindricalcoordinates (r, ϕ) with r ∈ (0,∞) and ϕ ∈ (0, 2π):

(

−1

2∇

2 +1

2r2)

ψnm(r, ϕ) = Enmψnm(r, ϕ), (6.56)

where n, m are the principal quantum number the azimuthal quantum numbers,respectively. For simplicity, we have set M = 1 and h = 1. In particular, we shallfocus on the state

ψ11(r, θ) = π−1/2 r e−r2/2eiϕ. (6.57)

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6.4 Simple Example 193

The Hamiltonian of the two-dimensional oscillator corresponding to the field formu-lation (5.20) reads

H [φ] =1

2

d2xφ∗(−∇2 + x2)φ, (6.58)

where we have done a nabla integration to replace |∇φ|2 by −φ∗∇2φ. The wavefunction (6.57) corresponds to the specific field configuration

ρ(r) = π−1/2 r, θ = arctan(x2/x1). (6.59)

Thus we calculate the energy (6.61) in cylindrical coordinates, where φ∗(−∇2)φ =

−φ∗(r−1∂rr∂r − r−2∂2ϕ)φ becomes for φ(x) = ψ11(r, ϕ):

−φ∗(−∇2)φ =

1

π

(

4 − r2)

r2e−r2

, (6.60)

so that we find

E11 = π∫ ∞

0dr r

[

1

π(4 − r2)e−r

2

+1

πr4e−r

2]

= 1 + 1 = 2. (6.61)

Let us now calculate the same energy from the hydrodynamic expression for theenergy which we read directly off the Lagrangian density (6.5) as

H =∫

d2xH=∫

d2xn(x)

1

2[∇θ(x)−v(x)]2 +

pos2(x)

2+

x2

2

. (6.62)

The gradient of θ(x) = arctan(x2/x1) has the jump at the cut of arctan(x2/x1),which runs here from zero to infinity in the x1, x2-plane:

∇1 arctan(x2/x1) = −x2/r2, ∇2 arctan(x2/x1) = x1/r

2 + 2πΘ(x1)δ(x2). (6.63)

The vortex gauge field is the same as in the example (6.32).When forming the superflow velocity (6.14), the second term in ∇2 arctan(x2/x1)

is removed by the vortex gauge field (6.32), and we obtain simply

v1(x) = −x2/r2, v2(x) = x1/r

2. (6.64)

Since the wave function has n(x) = r2e−r2

/π, the osmotic momentum (6.9) is

pos =1

2

∇n(x)

n(x)=

1

2

∇(r2e−r2

)

r2e−r2x

r=(

1

r− r

)

x

r. (6.65)

Inserting this into (6.62) yields the energy

H = π∫ ∞

0dr r

r2

πe−r

2

[

1

r2+(

1

r− r

)2

+ r2

]

, (6.66)

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194 6 Dynamics of Superfluids

which gives the same value 2 as in the calculation (6.61).Let us check the Ertel equation in the form (6.54). The vortex magnetic field

Bv(x) is according to Eq. (6.50) in the present natural units

Bv(x) = 2πδ(2)(x). (6.67)

This is time-independent, so that the right-hand side of Eq. (6.54) must vanish.Indeed, from (6.64) we see that

v(x) × Bv(x) = 2π(

x1

r,x2

r

)

δ(2)(x), (6.68)

so that its curl gives

2π∇ ×(

x1

r,x2

r

)

δ(2)(x) = 2π(

∇1x2

r−∇2

x1

r

)

δ(2)(x) (6.69)

This vanishes identically due to the rotational symmetry of the δ-function in twodimensions

δ(2)(x) =1

2πrδ(r). (6.70)

After applying the chain rule of differentiation to (6.69) one obtains zero.It is interesting to note that the extra quantum force (6.43) happens to vanish

as well in this atomic state. In terms of the vortex magnetic field it reads

fqu(x) ≡ h2

2M

[

∇n(x)

n(x)+ ∇

]

× Bv(x). (6.71)

The Bv(x)-field (6.67) has a curl

∇×Bv = 2π(∇2δ(2)(x),−∇1δ

(2)(x)). (6.72)

If we insert here the rotationally symmetric expression (6.70) for δ(2)(x) and rewrite(6.72) and rewrite it as

∇×Bv =(

x2

r,−x1

r

) [

1

rδ(r)

]′

= −(

x2

r,−x1

r

) [

1

r2δ(r) − 1

rδ′(r)

]

. (6.73)

The osmotic term adds to this

∇n(x)

n(x)×Bv(x) = 2

(

1

r− r

)

x

r×Bv(x) = 2

(

1

r− r

)

2π(

x2

r,−x1

r

)

1

rδ(r). (6.74)

The two distributions (6.73) and (6.74) are easily shown to cancel each other. Sincethey both point in the same direction, we remove the unit vectors (x2,−x1) /r andcompare the two contributions in the force (6.71) which are proportional to

−[

1

r2δ(r) − 1

rδ′(r)

]

, 2(

1

r2− 1

)

δ(r). (6.75)

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6.5 Eckart Theory of Ideal Quantum Fluids 195

Multiplying both expressions by an arbitrary smooth rotation-symmetric test func-tion f(r) and integrating we obtain

2π∫ ∞

0dr r f(r)

−[

1

r2δ(r) − 1

rδ′(r)

]

, 2π∫ ∞

0dr r f(r) 2

(

1

r2− 1

)

δ(r)

. (6.76)

These integrals are finite only if f(0) = 0, f ′(0) = 0, so that f(r) must have thesmall-r behavior f ′′(0)r2/2!+f (3)(0)r3/3!+ . . . . Inserting this into the two integralsand using the formula

∫∞0 dr rn δ′(r) = −δn,1 for n ≥ 1, we the values −2πf ′′(0)/2

and 2πf ′′(0)/2, respectively, so that the force (6.71) is indeed equal to zero.

6.5 Eckart Theory of Ideal Quantum Fluids

It is instructive to compare these equations with those for an ideal isentropic quan-tum fluid without vortices which is described by a Lagrange density due to Eckart[8]:

L = n(x)M

2v2(x) + Θ(x)M∂tn(x) + ∇ · [n(x)v(x)] − n(x)etot(x), (6.77)

where etot(x) is the internal energy (6.10) per particle.If we extremize the action (6.77) with respect to Θ(x) yields once more continuity

equation (6.17). Extremizing (6.77) with respect to v(x), we see that the velocityfield is given by the gradient of the Lagrange multiplyer:

v(x) = ∇Θ(x). (6.78)

Reinserting this into (6.77), the Lagrange density of the fluid becomes

L = n(x)M

2[∇Θ(x)]2−n(x)e(x)+Θ(x)∂tMn(x)+M∇·[n(x)∇Θ(x)]−n(x)e(x),

(6.79)or, after a partial integration in the associated action,

L = −n(x)

M∂tΘ(x) +M

2[∇Θ(x)]2 + etot(x)

. (6.80)

Since the gradient of a scalar field has no curl, this implies that these actions describeonly a vortexless flow.

Comparing (6.78) with (6.14) in the absence of vortices, we identify the velocitypotential as

Θ(x) ≡ hθ(x)/M. (6.81)

6.6 Rotating Superfluid

If we want to study a superfluid in a vessel which rotates with a constant angularvelocity

, we must add to the Lagrangian density (6.3) a source term

L = −x × j(x) · =i

2h φ∗(x)[x×

∇]φ(x) · , (6.82)

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196 6 Dynamics of Superfluids

where j(x) is the current density (2.70), and x× j(x) the density of angular momen-tum of the fluid. After substituting φ(x) by ρ(x)eiθ(x), this becomes

L = −ih∇ϕn(x) −Mn(x)v(x) · v , (6.83)

where ∇ϕn(x) denotes the azimuthal derivative of the density around the directionof the rotation axis

, and

v(x) ≡ × x (6.84)

is the velocity which the particles at x would have if the fluid would rotate as awhole like a solid. The action associated with the first term vanishes by partialintegration since n(x) is periodic around the axis

. Adding this to the hydrody-

namic Lagrangian density (6.5) and performing a quadratic completion in v(x),we obtain

L=n(x)

−h[∂tθ(x)+θvt (x)]−

h2

2M

[

∇θ(x)−v(x)−M

hv (x)

]2

−etot(x)−n(x)V (x)

,

(6.85)

where V(x) is the harmonic potential

V(x) ≡ −M2

v2(x) = −M

2Ω2r2

⊥, (6.86)

depending quadratically on the distance r⊥ from the rotation axis.

The velocity v (x) has a constant curl

∇ × v (x) = 2. (6.87)

It can therefore not be absorbed into the wave function by a phase transformationφ(x) → eiα(x)φ(x), since this would make the wave function multivalued. The energyof the rotating superfluid can be minimized only by a triangular lattice of vortexlines. Their total number N is such that the total circulation equals that of a solidbody rotation with

. Thus, if we integrate along a circle C of radius R around the

rotation axis, the number of vortices enclosed is given by to

M∮

Cdx · v = 2πhN. (6.88)

In this way the average of the vortex gauge field v(x) cancels the rotation fieldv (x) of constant vorticity.

Triangular vortex lattices have been observed in rotating superfluid 4He [11],and recently in Bose-Einstein condensates [12]. The theory of these lattices wasdeveloped in the 1960’s by Tkachenko and others for superfluid 4He [13], and recentlyby various authors for Bose-Einstein condensates [14].

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 197

Notes and References

[1] R.P. Feynman, Statistical Mechanics, Addison Wesley, New York, 1972,Sec. 10.12.

[2] E. Madelung, Z. Phys. 40,322 (1926).See alsoT.C. Wallstrom, Phys. Rev. A 49, 1613 (1994).

[3] The Magnus force is named after the German physicist Heinrich Magnus whodescribed it in 1853. According toJ. Gleick, Isaac Newton, Harper Fourth Estate, London (2004),Newton observed the effect 180 years earlier when watching tennis players inhis Cambridge college. The Magnus force makes airplanes fly due to a circula-tion of air around the wings. The circulation forms at takeoff, leaving behindan equal opposite circulation at the airport. The latter has caused crashes ofsmall planes starting too close to a jumbo jet. The effect was used by the Ger-man engineer Anton Flettner in the 1920’s to drive ships by a rotor rather thana sail. His ship Baden-Baden crossed the Atlantic in 1926. Presently, only theFrench research ship Alcyone built in 1985 uses such a drive with two rotorsshaped like an airplane wing.

[4] H. Ertel, Ein neuer hydrodynamischer Wirbelsatz , Meteorol. Z. 59, 277 (1942);Naturwissenschaften 30, 543 (1942); Uber hydrodynamische Wirbelsatze,Physik. Z. 43, 526 (1942); Uber das Verhaltnis des neuen hydrodynamis-chen Wirbelsatzes zum Zirkulationssatz von V. Bjerknes, Meteorol. Z. 59, 385(1942).

[5] M. Caliari, G. Inverso, and L.M. Morato, Dissipation caused by a vorticity fieldand generation of singularities in Madelung fluid , New J. Phys. 6, 69 (2004).

[6] M.I. Ioffredo and L.M. Morato, Lagrangian Variational Principle in StochasticMechanics: Gauge Structure and Stability , J. Math. Phys. 30, 354 (1988).

[7] E. Nelson, Quantum Fluctuations , Princeton University Press, Princeton, NJ,1985.

[8] C. Eckart, Phys. Rev. 54, 920 (1938); W. Yourgrau and S. Mandelstam, Vari-ational Principles in Dynamics and Quantum Theory, Pitman, London, 1968;A.M.J. Schakel, Boulevard of Broken Symmetries, (cond-mat/9805152).

[9] For detailed reviews and references seeC.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases , Cam-bridge University Press, Cambridge, 2001;L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Oxford UniversityPress (Intern. Ser. Monog. on Physics), Oxford, 2003.

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198 6 Dynamics of Superfluids

[10] For a detailed review and references seeI. Bloch, Ultracold Quantum Gases in Optical Lattices, Nat. Phys. 1, 23 (2005).

[11] E.J. Yarmchuk, M.J.V. Gordon, and R.E. Packard , Phys. Rev. Lett. 43, 214(1979);E.J. Yarmchuk and R.E. Packard , J. Low Temp. Phys. 46, 479 (1982).

[12] M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E. Wieman, andE.A. Cornell, Vortices in a Bose-Einstein Condensate, Phys. Rev. Lett. 83,2498 (1999);J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Science 292, 476(2001);V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Fast Rotation of a Bose-Einstein Condensate, Phys. Rev. Lett. 92, 050403 (2004);S. Stock, V. Bretin, F. Chevy, and J. Dalibard, Shape Oscillation of a RotatingBose-Einstein Condensate, Europhys. Lett. 65, 594 (2004).

[13] V.K. Tkachenko, Zh. Eksp. Teor. Fiz. 49, 1875 (1965) [Sov. Phys.–JETP 22,1282 (1966)]; Zh. Eksp. Teor. Fiz. 50, 1573 (1966) [Sov. Phys.–JETP 23,1049 (1966)]; Zh. Eksp. Teor. Fiz. 56, 1763 (1969) [Sov. Phys.–JETP 29, 945(1969)].D. Stauffer and A.L. Fetter, Distribution of Vortices in Rotating Helium II,Phys. Rev. 168, 156 (1968);G. Baym, Stability of the Vortex Lattice in a Rotating Superfluid , Phys. Rev.B 51, 11697 (1995).

[14] A.L. Fetter, Rotating Vortex Lattice in a Condensate Trapped in CombinedQuadratic and Quartic Radial Potentials, Phys. Rev. A 64, 3608 (2001);A.L. Fetter and A.A. Svidzinsky, Vortices in a Trapped Dilute Bose-EinsteinCondensate, J. Phys.: Condensed Matter 13, R135 (2001);A.L. Fetter, B. Jackson, and S. Stringari, Rapid Rotation of a Bose-EinsteinCondensate in a Harmonic Plus Quartic Trap, Phys. Rev. A 71, 013605 (2005);A.A. Svidzinsky and A.L. Fetter, Normal Modes of a Vortex in a Trapped Bose-Einstein Condensate, Phys. Rev. A 58, 3168 (1998);K. Kasamatsu, M. Tsubota, and M. Ueda, Vortex lattice formation in a ro-tating Bose-Einstein condensate Phys. Rev. A 65, 023603 (2002); Nonlineardynamics of vortex lattice formation in a rotating Bose-Einstein condensate,Phys. Rev. A 67, 033610 (2003).

H. Kleinert, MULTIVALUED FIELDS

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There is no excellent beauty

that hath not some strangeness in the proportion.

Francis Bacon (1561 - 1626)

7Dynamics of Charged Superfluid andSuperconductor

In the presence of electromagnetism, we simply extend the vortex-covariant deriva-tive in the Lagrangian density (6.3) by a minimally coupled vector potentialAµ(x) = (A0(x),A(x)), thus forming the fully covariant derivatives

Dtφ(x) ≡ [∂t − θvt (x) − eφ(x)]φ(x) (7.1)

This amounts to replacing

θvµ(x) → θv

µ(x) +e

cAµ(x), (7.2)

or with At(x) = cA0(x), θvt (x) = cθv

0(x),

θvt (x) → θv

t (x) +q

hcAt(x), v(x) → v(x) +

q

hcA(x), (7.3)

so that we obtain from (6.5):

L=n(x)

−h[

∂tθ(x)+θvt (x)+

q

hcAt(x)

]

− h2

2M

[

∇θ(x)−v(x)− q

hcA(x)

]2

−etot(x)

.

(7.4)

This has to be supplemented by Maxwell’s electromagnetic Lagrangian density(2.84).

Conversely, we may simply take the electromagnetically coupled equation, andreplace the vector potential by

At(x) → At(x) ≡ At(x) + qmθvt (x), A(x) → A(x) ≡ A(x) + qmv(x), (7.5)

where we have introduced a magnetic charge associated with the electric charge q:

qm =hc

q. (7.6)

199

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200 7 Dynamics of Charged Superfluid and Superconductor

Then we can rewrite (7.4) in the shorter form

L=n(x)

−[

h∂tθ(x) +q

cAt(x)

]

− 1

2M

[

h∇θ(x)− q

cA(x)

]2

−etot(x)

. (7.7)

The equation of motion of the time-dependent field φ(t,x) ≡ φ(x) is

i[

h∂t +q

cAt(x)

]

φ(x)=

− 1

2M

[

h∇θ(x) − q

cA

]2

− c20M +c20M

n0φ∗(x)φ(x)

φ(x).(7.8)

7.1 Hydrodynamic Description of Charged Superfluid

For a charged superfluid, the velocity field is given by

v(x) ≡ 1

M

[

h∇θ(x) − q

cA(x)

]

=h

M

[

∇θ(x) − v(x) − q

hcA(x)

]

. (7.9)

It is invariant under both magnetic and vortex gauge transformations. In termsof the local deviation of the particle density from the ground-state value δn(x) ≡n(x) − n0, the hydrodynamic Lagrangian density (7.7) can be written as

L=−n(x)[

h∂tθ(x)+ hθvt (x) +

q

cqAt(x)+

M

2v2(x)+etot(x)

]

. (7.10)

The electric charge and current densities are simply q times the particle andcurrent densities (6.16). They can now be derived alternatively from the Noetherrule (3.117):

ρ(x)qn(x) = −1

c

∂L∂At(x)

= qn(x), J(x) =1

c

∂L∂A(x)

= qn(x)v(x). (7.11)

This satisfies the continuity equation

q∂t n(x) = −∇ · J(x) = 0, (7.12)

which can again be found by extremizing the associated action with respect to θ(x).Functional extremization of the action with respect to n(x) yields

h∂tθ(x) + hθvt (x) + qA0(x) +

M

2v2(x) + htot(x) = 0, (7.13)

where pqu(x) is the quantum pressure defined in Eq. (6.24). These are the exten-sions of the Madelung equations (6.17) and (6.18) by vortices and electromagnetism.The last term may, incidentally, be replaced by the enthalpy per particle h(x) ofEq. (6.19).

The gradient of the second equation yields the equation of motion

M∂tv(x)+q[

1

c∂tA(x) + ∇A0(x)

]

+h [∂tv(x)+∇θvt (x)] +

M

2∇v2(x)=−∇htot(x).

(7.14)

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7.1 Hydrodynamic Description of Charged Superfluid 201

Inserting here the identity (6.28) we obtain

M∂tv(x) +M [v(x) · ∇]v(x) = −∇htot(x) + f em(x) + fv(x), (7.15)

where the sum of the two forces on the right-hand side is

f em(x)+fv(x)=−q[

1

c∂tA(x) + ∇A0(x)

]

−h [∂tv(x)+∇θvt (x)]−hv(x)×[∇×v(x)].

(7.16)

Inserting here the velocity (7.9), further the defining equation (2.73) and (2.79) forthe electromagnetic fields, and finally Eqs. (6.50) for the vortex electromagneticfields, we see that f em(x) + fv(x) is the sum of the vortex force fv(x) of Eq. (6.53),which was shown to vanish in Eq. (6.35), and the electromagnetic Lorentz forceacting upon a charged moving particle

f em(x) = Ev(x) +v(x)

c2× Bv(x). (7.17)

The classical limit of Eq. (7.16) is the well-known equation of motion of magneto-hydrodynamic [4].

The additional Maxwell action adds equations for the electromagnetic field

∇ · E = ρ (Coulomb’s law), (7.18)

∇ ×B − 1

c

∂E

∂t=

1

cJ (Ampe law), (7.19)

∇ · B = 0 (absence of magnetic monopoles), (7.20)

∇ × E +1

c

∂B

∂t= 0 (Faraday’s law). (7.21)

7.1.1 London Theory of Charged Superfluid

If we ignore the vortex gauge field in Eq. (7.9), the current density (7.11) is

J(x) ≡ qn(x)v(x) =hqn(x)

M∇θ(x) − q2n(x)

McA(x). (7.22)

In our convention the charge q is is equal to −2e since the charge carriers in the thesuperconductor are Cooper pairs of electrons.

The brothers Heinz and Fritz London [1] considered superconductors with con-stant density n(x) ≡ n0 which is the London-limit introduced before in Eq. (5.12).Then they absorbed the phase variable θ(x) into the vector potential A(x) by agauge transformation

Aµ(x) → A′µ(x) = Aµ(x) −c

q∂µθ, (7.23)

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202 7 Dynamics of Charged Superfluid and Superconductor

so that

J(x) ≡ qn0v(x) = −q2n0

cMA′(x). (7.24)

where A′(x) satisfies the gauge condition

∇ ·A′(x) = 0, (7.25)

to make (7.24) compatible with the current conservation law (7.12).Taking the time derivative of this and using the defining equation (2.73) for the

electric field in terms of the vector potential, they obtained

∂tJ(x) =q2n0

M[E(x) + ∇A′0(x)]. (7.26)

At this place they postulated that the electric potential A′0(x) vanishes in a super-conductor, which led them to their famous first London equation:

∂tJ(x) =q2n0

ME(x). (7.27)

In a second step they formed, at constant n(x) = n0, the curl of the current(7.22) , at constant n(x) = n0, and obtained the second London equation:

∇ × J(x) +q2n0

McB(x) = 0. (7.28)

To check the compatibility of the two London equations one may take the curl of(7.26) and use Faraday’s law of induction (7.21) to find

∂t

[

∇ × J(x) +q2n0

McB(x)

]

= 0, (7.29)

in agreement with (7.28).It should be noted that the penetration depth of the electric potentialFrom the second London equation (7.28) one derives immediately the Meissner

effect. First one recalls how the electromagnetic waves are derived from the combi-nation of Ampere’s and Faraday’s laws (1.181) and (1.183), and the magnetic sourcecondition (1.182):

∇ × ∇ × B(x) +1

c2∂2tB(x) = −∇

2B(x) +1

c2∂2tB(x) =

1

c∇ × J(x). (7.30)

In the absence of currents, this equation describes electromagnetic waves propagat-ing with light velocity c. In a superconductor, the right-hand side is replaced by thesecond London equation (7.28), and leads to

[

1

c2∂2tB(x) − ∇

2 + λ−2L

]

B(x) = 0. (7.31)

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7.2 London Equations of Charged Superfluid with Vortices 203

with

λL =

Mc2

n0q2=

1

2

Mc2

n0e2=

1

2√n0λM4πα

, (7.32)

where α ≈ 1/137.0359 . . . is the fine-structure constant (1.189), and λM = h/Mcthe Compton wavelength of the particles of mass M .

Equation (7.31) shows that inside a superconductor, the magnetic field has afinite penetration depth λL found by London.

7.2 London Equations of Charged Superfluid with Vortices

The development in the last section allows us to correct the London equations. Firstwe add the vortex gauge field, so that (7.22) becomes

J(x) ≡ qn(x)v(x) =hqn(x)

M[∇θ(x) − v(x)] − q2n(x)

cMA(x). (7.33)

In the London limit where n(x) ≈ n0, we take again the time derivative of (7.33)and recalling Eq. (6.50), we obtain

∂tJ(x) =hqn0

M[∇∂tθ(x) + Ev(x) + ∇θv

t (x)] +q2n0

M[E(x) + ∇A0(x)]. (7.34)

As before, we fix the vortex gauge to have θvt (x) = 0, and absorb the phase variable

θ(x) in the vector potential A by a gauge transformation (7.23). Thus we remainwith the vortex-corrected first London equation

∂tJ(x) =q2n0

M[E(x) + qmEv(x) + ∇A0(x)], (7.35)

where qm is the magnetic charge (7.6) associated with the electric charge q.Taking at the curl of Eq. (7.33) in the London limit with the same fixing of

vortex and electromagnetic gauge, we obtain the vortex-corrected second Londonequation

∇ × J(x) +q2n0

Mc[B(x) + qmBv(x)] = 0. (7.36)

The compatibility with (7.35) is checked by forming the curl of (7.35) and usingthe Faraday law of induction (1.183) and its vortex analog (6.52). The result is thestatement that the time derivative of (7.36) vanishes, which is certainly true.

Inserting (7.36) into the combined Maxwell equation (7.30) yields the vortex-corrected Eq. (7.31):

[

1

c2∂2t − ∇

2 + λ−2L

]

B(x) = −λ−2L qmBv(x), (7.37)

From this we can directly deduce the interaction between vortex lines

Aint = −q2m

2

d4xd4x′Bv(x)GRλL

(x− x′)Bv(x), (7.38)

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204 7 Dynamics of Charged Superfluid and Superconductor

where GRλL

(x− x′) is the retarded Yukawa Green functions,

GRλL

(x− x′) =1

−c−2∂2t + ∇

2 + λ−2L

(x, x′) = −Θ(x0 − x′0)e−R/λL

4πc2Rδ(t− t′ −R/c), (7.39)

in which R denotes the spatial distance R ≡ |x − x′|.In the limit λL → ∞ this goes over to the Coulomb version which is the origin

of the well-known Lienard-Wiechert potential of electrodynamics.For slowly moving vortices, the retardation can be neglected and, after inserting

qm from (7.6) and Bv(x) from (6.50), and performing the time derivatives in (7.40),we find

Aint = − h2c2

2q2

dt∫

d3x jv(x, t)1

−∇2 + λ−2

L

jv(x, t). (7.40)

This agrees with the previous static interaction energy in the partition function(5.269), if we go to natural units h = c = M = 1.

7.3 Hydrodynamic Description of Superconductor

For a superconductor, the above theory of a charged superfluid is not applicablesince the initial Ginzburg-Landau Lagrangian density can be derived [3] only nearthe phase transition where it has, moreover, a purely damped temporal behavior.Hence the time derivative term in Eq. (6.2) is incorrect. At zero temperature,however, we can still derive from the BCS theory [2] a Lagrangian of the type (6.47)in the harmonic approximation

L0 =−n0h∂tθ(x) +n0h

2

2M

1

c20[∂tθ(x) + θv

t (x)]2 − [∇θ(x) − v(x)]2

+ . . . . (7.41)

with the second sound velocity [3]

c0 =vF√

3, (7.42)

where vF = pF/M =√

2MEF is the velocity of electrons on the surface of the Fermisphere, which is calculated from the density of electrons (which is twice as big asthe density of Cooper pairs n0):

nel = 2∫ d3p

(2πh)3=

p3F

3h3π2=

v3F

3h3M3π2. (7.43)

The dots in (7.41) indicate terms which can be ignored in the long-wavelength limit.We now add the electromagnetic fields by minimal coupling [recall (7.2) and (7.3)]and find

L0 =−n0h[

∂tθ(x)+q

hcAt(x)

]

+n0h

2

2M

1

c20

[

∂tθ(x)+q

hcAt(x)

]2

− [∇θ(x)− q

hcA(x)]2

.

(7.44)

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7.3 Hydrodynamic Description of Superconductor 205

to be supplemented by the Maxwell Lagrangian density (2.84).The derivative of L0 with respect to −A(x)/c yields the current density [recall

(3.117)]:

J(x) = qn0v(x) =qn0h

M[∇θ(x) − v(x)] − q2n0

cMA(x). (7.45)

From the derivative of L0 with respect to −At(x)/c we obtain the charge density:

q[n(x) − n0] =qn0h

M

1

c20[∂tθ(x) + θv

t (x)] −q2n0

c20cMAt(x). (7.46)

If we absorb the field θ(x) in the vector potential, we find the same supercurrentas in (7.22):

J(x) = −q2n0

cMA(x) (7.47)

whereas the charge density becomes

qn(x) = − q2n0

c20cMAt(x) (7.48)

The current conservation law implies that

∇ ·A(x) +c2

c20

1

c2∂tAt = 0. (7.49)

Note the difference by the large factor c2/c20 of the time derivative term with respectto the Lorentz gauge (2.105):

∂aAa(x) = ∇ · A(x) + ∂0A

0(x) = ∇ · A(x) +1

c2∂tA

t(x) = 0. (7.50)

Since the velocity c0 = vF/√

3 is much smaller than the light velocity c, typicallyby a factor 1/100, the ratio c2/c20 is of the order of 104.

At this point we recall that according to the definition (5.249), the ratio of thepenetration depth λL of Eq. (7.32) and the coherence length ξ of Eq. (6.46) definethe Ginzburg parameter κ ≡ λL/

√2ξ. This allows us to express the ratio c0/c in

terms of κ as follows:c0c

=κ√2

n0λ3Mq

2. (7.51)

If the current density is inserted into the combined Maxwell equation (7.30) toobtain Eq. (7.31) for the screened magnetic field B and its vortex-corrected version(7.31).

The field equation for A0, however, has quite different wave propagation proper-ties, due to the factor c2/c20. It is obtained by varying the action A =

dtd3x [Lem]+L0, with respect to −A0(x) = −At(x)/c, which yields

∇ ·E(x) = qn(x). (7.52)

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206 7 Dynamics of Charged Superfluid and Superconductor

Inserting E(x) from (2.73), and qn(x) from Eq. (7.48) in the axial vortex gauge, wefind

−∇2A0(x) − 1

c∂t∇ · A(x) = −q

2n0

c20MA0(x). (7.53)

Eliminating ∇ · A(x) with the help of Eq (7.49), we obtain

(

−∇2 + λ−2

L0

)

A0(x) − c2

c20

1

c2∂2tA

0 = 0. (7.54)

This equation shows that the field A0(x) penetrates a superconductor over the dis-tance

λL0 =c0cλL =

c0c

1√n0λMq2

, (7.55)

which is typically two orders of magnitude smaller than the penetration depth λL

of the magnetic field. Moreover, the propagation velocity of A0(x) is not the lightvelocity c but the mcuh smaller velocity c0 = vF/

√3.

Note that Eq. (7.54) for A0(x) has no gauge freedom left, the gauge being fixedby Eq. (7.49). This is best seen by expressing A0(x) in terms of the charge densityqn(x) via Eq. (7.48) which yields

[

− 1

c2∂2t +

c20c2

(

−∇2 + λ−2

L 0

)

]

n(x) = 0. (7.56)

Using the vanishing of fv(x) of Eq. (6.53), and Eq. (6.55), we can rewrite the diver-gence of Ev(x) as

∇ ·Ev(x) = −∇ · [v(x) × Bv(x)] = −[∇ × v(x)]Bv(x) + v(x) · [∇ ×Bv(x)]

=h

M[Bv(x)]2 + v(x) · [∇ × Bv(x)]. (7.57)

Notes and References

[1] F. London and H. London, Proc. R. Soc. London, A 149, 71 (1935);; PhysicaA 2, 341 (1935);H. London, Proc. R. Soc. A 155, 102 (1936);F. London, Superfluids, Dover, New York, 1961.

[2] J. Bardeen, L.N. Cooper, Schrieffer, Phys. Rev. 108, 1175 (1957);M. Tinkham, Introduction to Superconductivity, McGraw-Hill, New York,1975.

[3] H. Kleinert, Collective Quantum Fields, Lectures presented at the First EriceSummer School on Low-Temperature Physics, 1977, in Fortschr. Physik 26,565-671 (1978) (kl/55). See Eq. (4,118).

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 207

[4] The classical magnetohydrodynamic equations are discussed inJ.D. Jackson, Classical Electrodynamics , John Wiley and Sons, New York,1975, Sects. 6.12-6.13;W.F. Hughes and F.J. Young, The Electromagnetodynamics of Fluids , Wiley,New York, 1966.

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There is no excellent beauty

that hath not some strangeness in the proportion.

Francis Bacon (1561 - 1626)

8Relativistic Magnetic Monopoles and Electric

Charge Confinement

The theory of multivalued fields in magnetism in Chapter 4 can easily be extendedto a full relativistic theory of charges and monopoles [1, 2]. For this we go over tofour spacetime dimensions, which are assumed to be Euclidean with a fourth spatialcomponent dx4 = icdt, to avoid factors of i.

8.1 Monopole Gauge Invariance

The covariant version of the Maxwell equation (4.53) reads

1

2εabcd∂aFcd = −1

cb, (8.1)

where a is the magnetic current density

a = (cρm, jm) . (8.2)

Equation (8.1) implies that the magnetic current density is conserved:

∂aa = 0. (8.3)

The zeroth component of (8.1) reproduces Eq. (4.53) for the monopole charge density[recall the identification of the field components (1.167)], and the spatial componentsyield the modified Faraday law (1.18):

∇ × E +1

c

∂B

∂t= −1

cjm (modified Faraday law). (8.4)

For a single monopole of strength g moving along a world line qa(σ), the magneticcurrent density a can be expressed in terms of a δ-function on the world line,

δa(x;L) ≡∫

dσdxa(σ)

dσδ(4)(x− x(σ)), (8.5)

208

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8.1 Monopole Gauge Invariance 209

as followsa = g c δa(x;L). (8.6)

This satisfies the conservation law (8.3) as a consequence of the identity

∂aδa(x;L) = 0, (8.7)

the four-dimensional version of (4.10) applied to closed worldlines. The spacetimecomponents of the magnetic current density are [compare (1.205) and (1.206)]

cρm (x, t) = g c∫ ∞

−∞dτ γc δ(4) (x− x(τ)) , (8.8)

jm (x, t) = g c∫ ∞

−∞dτ γv δ(4) (x− x(τ)) . (8.9)

Note that with this notation, the electric current density (1.207) of a particle onthe worldline L reads

ja = e c δa(x;L), (8.10)

with the same conservation law ∂aj = 0.Equation (8.1) shows that Fab cannot be represented as a curl of a single-valued

vector potential Aa, since left-hand side is equal to εabcd(∂a∂c−∂c∂a)Ad)/2] implyinga violation of Schwarz’s integrability condition. As in the magnetostatic discussionin Section 4.6 the simplest way to incorporate the monopole worldline into theelectromagnetic field theory is via an extra monopole gauge field. In four spacetimedimensions, this is defined by

FMab ≡ g δab(x;S), (8.11)

where δab(x;S) is the dual tensor

δab(x;S) ≡ 1

2εabcdδcd(x;S), (8.12)

of the δ-function δcd(x;S) which is singular on the world surface S:

δab(x;S) ≡∫

dσdτ

[

∂xa(σ, τ)

∂σ

dxb(σ, τ)

∂τ− (a↔ b)

]

δ(4)(x− x(σ, τ)). (8.13)

This δ-function has the obvious property

∂aδab(x;S) = aδb(x;L), (8.14)

where L is the boundary line of the surface. This follows directly from the simplecalculation:

∂aδab(x;S) =∫

[

dxb(σb, τ)

∂τδ(4)(x− x(σb, τ)) −

dxb(σa, τ)

∂τδ(4)(x− x(σa, τ))

]

−∫

[

dxb(σ, τb)

∂τδ(4)(x− x(σ, τb)) +

dxb(σ, τa)

∂τδ(4)(x− x(σ, τa))

]

,

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210 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

where σa,b and τa,b are the lower and upper values of the surface parameters, respec-tively, so that x(σa, τ), x(σ, τa), x(σb, τ), x(σ, τb), run along the boundary line of thesurface. The dual δ-function (8.12) satisfies

1

2εabcd∂bδcd(x;S) = δa(x;L), (8.15)

due to identity (1A.24). Equation (8.15) is the four-dimensional version of the localformulation (4.23) of Stokes’ theorem.

For the monopole gauge field (8.11) this implies

1

2εabcd∂aF

Mcd =

1

cb. (8.16)

The surface S is the worldsheet of the Dirac string. For any given line L, thereare many possible surfaces S. We can go over from one S to another, say S ′, atfixed boundary L as follows

δcd(x;S) → δcd(x;S′) = δcd(x;S) + ∂aδb(x;V ) − ∂bδa(x;V ), (8.17)

where δa(x;V ) is the δ-function (6.13) which is singular on the three-dimensionalvolume V in four-space swept out when the surface S moves through four-space.The transformation law (8.17) is the obvious generalization of (4.28).

Many monopoles are, of course, represented by a gauge field (8.11) with a super-position of many different surfaces S.

We are now ready to set up the electromagnetic action in the presence of anarbitrary number of monopoles. By analogy with Eq. (4.85) and (5.22) it dependsonly on the difference between the total field strength Fab = ∂aAb − ∂bAa of theintegrable vector potential Aa and the monopole gauge field FM

ab of (8.11), i.e., it isgiven by [3, 4, 5, 6]

A0 + Amg ≡ A0,mg =∫

d4x1

4c

(

Fab − FMab

)2. (8.18)

The subtraction of FMab is essential in avoiding an infinite energy density in the

Maxwell action

A0 ≡∫

d4x1

4cF 2ab , (8.19)

that would arise from the flux tube in Fab inside the Dirac string. The difference

F obsab ≡ Fab − FM

ab (8.20)

is the nonsingular observable field strength. Since only fields with finite action arephysical, the action contains no contributions from squares of δ-functions as it mightinitially appear.

The action (8.18) exhibits two types of gauge invariances. First, the originalelectromagnetic one under [compare (2.103)]

Aa(x) −→ A′a(x) = Aa(x) + ∂aΛ(x), (8.21)

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8.1 Monopole Gauge Invariance 211

where Λ(x) is any smooth field which satisfies the integrability condition

(∂a∂b − ∂b∂a)Λ(x) = 0. (8.22)

under which FMab is trivially invariant. Second, there is gauge invariance under

monopole gauge transformations

FMab → FM

ab + ∂aΛMb − ∂bΛ

Ma , (8.23)

with integrable vector functions ΛMa (x), which by (8.17) have the general form

ΛMa (x) = g δa(x;V ), (8.24)

with arbitrary choices of three-volumes V . If the monopole gauge field (8.11) con-tains many jumping surfaces S, the function ΛM

a (x) will contain a superpositionmany volumes V .

To have invariance of the action (8.18), the transformation (8.23) must be ac-companied by a shift in the electromagnetic gauge field [3, 4, 5, 6]

Aa → Aa + ΛMa . (8.25)

From Eqs. (8.11), (8.15), and (8.17) we see that the physical significance of thepart (8.23) of the monopole gauge transformation is to change the Dirac worldsurface without changing its boundary, the monopole world line. An exception arevortex gauge transformations (8.25) of the gradient type, in which ΛM

a is g timesthe gradient ∂aΛ

M of the δ-function on the four-volume V4:

δ(x;V4) ≡ εabcd

dσdτdλdκ∂xa∂σ

∂xb∂τ

∂xc∂λ

∂xd∂κ

δ(4) (x− x(σ, τ, λ, κ)) , (8.26)

i.e.,Aa → Aa + g∂aδ(x;V4). (8.27)

These do not give any change in FMab since they are particular forms of the original

electromagnetic gauge transformations (8.21).The field strength Fab is , of course, changed by moving the Dirac string through

space, only the observable field strength F obsab = Fab − FM

ab remains invariant.The part (8.25) of the monopole gauge transformations expresses the fact that

in the presence of monopoles the gauge field Aa is necessarily a cyclic variable forwhich Aa(x) and Aa(x) + gn are identical at each point x for any integer n.

The partition function of magnetic monopoles and their electromagnetic inter-actions is given by the functional integral

Z =∫

DATa∫

DFMab e

−A0,mg . (8.28)

Here we have used the same short notation for the measure as in (5.265), indicatingthe gauge-fixed sum

SΦ[FMab ] over fluctuating jumping surfaces S, the world

sheet of the Dirac strings, by the symbol∫ DFM T

ab , and indicate the gauge fixedfunctional integral over Aa in the Lorentz gauge ∂aAa = 0 by the symbol

∫ DATa .

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212 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

8.2 Charge Quantization

Let us now introduce electrically charged particles into the action (8.18). This isdone via the current interaction

Ael =i

c2

d4x ja(x)Aa(x), (8.29)

where ja(x) is the electric current of the world line of a charged particle

ja = e δa(x;L). (8.30)

This is the Euclidean version of the current interaction (2.83) in Minkowski space-time.

Due to ordinary current conservation

∂aja = 0, (8.31)

the action (8.29) is trivially invariant under electromagnetic gauge transformations(8.21). In contrast, it can remain invariant under monopole gauge transformations(8.23), (8.25) only if the monopole charge satisfies the famous Dirac quantizationcondition derived before in Eq. (4.108). Let us see how this comes about in thepresent four-dimensional theory.

Under the monopole gauge transformation (8.25), only the part A0 +Amg of thetotal action

Atot ≡ A0 + Amg + Ael =∫

d4x1

4c

(

Fab − FMab

)2+

i

c2

d4x ja(x)Aa(x) (8.32)

is manifestly invariant. The electric part Ael, and thus the total action, changes by

∆Atot = ∆Ael = ieg

cI, (8.33)

where I denotes the integral

I ≡∫

d4x δa(L)δa(V ). (8.34)

This is an integer number if L passes through V and zero if it misses V . In theformer case, the string in the operation (8.17) sweeps across L, in the other case itdoes not. To prove this we let L run along the first axis and let V be the entirevolume in 234-subspace. Then δa(x;L) and δa(x;V ) have nonzero components onlyin the 1-direction; δ1(x;L) = δ(x2)δ(x3)δ(x4), and δ1(x;V ) = δ(x1). Inserting theseinto the integral (8.17) yields I = 1.

The Dirac charge quantization condition follows now from the rules of quantummechanics that all amplitudes are found from a functional integral over eiA/h [recall(4.109)]. Hence physics is invariant under jumps of the action by 2πh× integer since

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8.3 Electric and Magnetic Current-Current Interactions 213

these do not contribute to any quantum-mechanical amplitude. From (8.49) thisimplies that

eg

hc= 2π × integer. (8.35)

The result may be stated in a dimensionless way by expressing e in terms of thefine-structure constant α ≈ 1/137.0359 . . . of Eq. (1.189) as e2 = 4π hc α, so thatthe charge quantization condition becomes1

g/e = integer/2α. (8.36)

It must be emphasized that the above derivation of (8.36) requires much lessquantum mechanical input than most derivations found in the literature which in-volve the wave functions for the charged particle in a monopole field [1, 2, 7]. In theabove derivation, however, the particle orbits remain fixed, and only the worldsheetsof the Dirac strings are moved around by monopole gauge transformation and thequantization follows from the requirement of invariance under these transformations.

Observe that after the quantization of the charge, the total action (8.32) isdouble-gauge invariant—it is invariant under the ordinary electromagnetic gaugetransformations (8.21) and the monopole gauge transformations (8.23).

8.3 Electric and Magnetic Current-Current Interactions

If we integrate out the Aa-field in the partition function associated with the action(8.63) we obtain the interaction

Aint =∫

d4x

1

4c

[(

FMab )2 + 2∂aF

Mab (−∂2)−1∂cF

Mcb

)]

+1

2c3ja(−∂2)−1ja +

i

2c2∂aF

Mab (−∂2)−1jb

. (8.37)

The second term

Ajj =1

2c3

d4xja(−∂2)−1ja (8.38)

is the usual electric current-current interaction, where (−∂2)−1 denotes the euclideanversion of retarded Green function of the vector potential Aa(x)

(−∂2)−1(x, x′) = G(x− x′) ≡∫

d4k

(2π)4

eik(x−x′)

k2. (8.39)

Indeed, inserting the components of the four-component current density ja = (cρ, j)[recall (1.191)], the interaction (8.38) reads

Ajj =1

2c

d4xd4x′ ρ(x)G(x, x′)ρ(x′) +1

2c3

d4xd4x′ j(x)G(x, x′)j(x′). (8.40)

1In many textbooks, the action (8.18) has a prefactor 1/4π, leading to Dirac’s charge quantiza-tion condition in the form 2eg/hc =integer. In these conventions, e2 = hc α, so that the condition(8.36) is the same.

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214 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

For the static charges and currents in Minkowski spacetime, this becomes [compare(4.97)]

Ajj =1

2

dtd3xd3x′ ρ(t,x)1

|x−x′| ρ(t,x′) − 1

2c2

dtd3xd3x′ j(t,x)1

|x−x′| j(t,x′).

(8.41)The first term is the Coulomb interaction, the second the Biot-Savart interaction ofan arbitrary current distribution.

The first two terms in the interaction (8.37) reduce to the magnetic current-current interaction [compare (4.99)]

A =1

2c3

d4x a(−∂2)−1a, (8.42)

This follows from (8.1) and the simple calculation with the help of the tensor identity(1A.23):

j2 =(

c

2εabcd∂bF

Mcd

)2

= c2[

∂2(FMcd )2 − 2(∂aF

Mab )2

]

. (8.43)

The magnetic interaction (8.42) can be decomposed into time- and space-like com-ponents in the same way as in Eq. (8.40), but with magnetic and electric charge andcurrent densities.

The last term in (8.37)

Aj =∫

d4xi

2c2∂aF

Mab (−∂2)−1jb (8.44)

specifies the interaction between electric and magnetic currents. It is the relativisticversion of the interaction (4.101).

All three interactions are invariant under monopole gauge transformations (8.23).For the electric and magnetic current-current interactions (8.38) and (8.42) this isimmediately obvious since they depend only on the world lines of electric and mag-netic charges. Only for the mixed interaction (8.44) is the invariance not obvious. Infact, performing a monopole gauge transformation (8.23), and using the world linerepresentation (8.10) of the electric four-vector current, this interaction is changedby

∆Aj = i∫

d4xg

c2∂2δb(x;V )(−∂2)−1jb = i

ge

c

d4xδb(x;V )δ(x;L) =ge

cI. (8.45)

This is nonzero, but the theory is still invariant since (ge/c)I is equal to 2πh timesa number which is integer due to Dirac’s charge quantization condition (8.35). Thuse−∆Aj/h is equal to one and the theory invariant. The reader will recognize theanalogy with the three-dimensional situation in the mixed interaction Eq. (4.101).

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8.4 Dual Gauge Field Representation 215

8.4 Dual Gauge Field Representation

It is instructive to subject the total action (8.32) to a similar duality transformationas to the Hamiltonian (4.85) by which we derived the dual gauge formulation (4.90).Thus we introduce an independent fluctuating field fab and replace the action (8.18)by the equivalent one [the four-dimensional analog of (4.86)]

A0,mg =∫

d4x[

1

4cf 2ab +

i

2cfab

(

Fab − FMab

)

]

, (8.46)

with the two independent fields Aa and fab. Inserting Fab ≡ ∂aAb − ∂aAa, thepartition function (8.28) becomes

Z =∫

DAaT∫

Dfab∫

DFMab e

−A0,mg . (8.47)

Here we may integrate out the vector potential Aa to obtain the constraint

∂bfab = 0. (8.48)

This can be satisfied identically (as a Bianchi identity) by introducing a dual mag-netoelectric vector potential Aa and writing [compare (4.88)]

fab ≡ εabcd∂cAd. (8.49)

If we also introduce a dual field tensor

Fab ≡ ∂aAb − ∂bAa, (8.50)

the action (8.46) takes the dual form

A0,mg ≡ A0 + Amg =∫

d4x(

1

4cF 2ab +

i

c2Aaa

)

, (8.51)

with the magnetoelectric source

a ≡c

2εabcd∂bF

Mcd . (8.52)

By inserting (8.11) and (8.15), we see that a is the magnetic current density (8.6).Since (8.52) satisfied trivially the current conservation law ∂aa = 0 [recall (8.3)],the action (8.51) allows for an additional set of gauge transformations which are themagnetoelectric ones

Aa → Aa + ∂aΛ (8.53)

with arbitrary integrable functions Λ,

(∂a∂b − ∂b∂a)Λ = 0. (8.54)

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216 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

If we include the electric current (8.30) into the dual form of the action (8.51) itbecomes

Atot =∫

d4x[

1

4cf 2ab +

i

2cfab

(

Fab − FMab

)

+i

c2jaAa

]

. (8.55)

Extremizing this with respect to the field Aa gives now

∂afab = −1

cjb, (8.56)

rather than (8.48). The solution of this requires the introduction of a gauge fieldanalog to (8.11), the charge gauge field

FEab = e δab(x;S

′). (8.57)

Then (8.56) is solved by

fab ≡1

2εabcd(Fab − FE

ab). (8.58)

The identity (8.15) ensures (8.56).When inserting (8.58) into (8.55), we find

Atot =∫

d4x[

1

4c(Fab − FE

ab)2 − i

4cFab εabcdF

Mcd +

i

4cFEab εabcdF

Mcd

]

. (8.59)

Integrating the second term by parts and using Eq. (8.16) we obtain to

d4x[

1

4c(Fab − FE

ab)2 +

i

c2Aaa

]

+ ∆A, (8.60)

where

∆A =i

4c

d4x FEab εabcdF

Mcd . (8.61)

Remembering Eqs. (8.11) and (8.57), this can be shown to be an integer multiple ofeg/c:

∆A = egi

4c

d4x δab(x;S) εabcd δcd(x;S′) = i

eg

cn, n = integer. (8.62)

To see this we simply choose the surface S to be the 12-plane and S ′ to be the34-plane. Then δ12(x;S) = −δ21(x;S) = δ(x1)δ(x2) and δ34(x;S

′) = −δ43(x;S ′) =δ(x3)δ(x4), and all other components vanish, so that

d4x εabcdδab(x;S)δab(x;S′) =

4∫

d4x δ(x1)δ(x2)δ(x3)δ(x4) = 4. This proof can easily be generalized to arbitraryS, S ′ configurations.

We now impose Dirac’s quantization condition (8.35), which was required in(8.35) to guarantee the invariance under monopole gauge transformations (8.23)ensuring the invariance under the string deformations (8.17). This makes the phasefactor e−∆A/h equal to unity, so that it has no influence upon any quantum process.

The dual version of the total action (8.32) of monopoles and charges is therefore

Atot ≡ A0 + Ael + Amg =∫

d4x[

1

4c(Fab − FE

ab)2 +

i

c2Aaa

]

. (8.63)

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8.5 Monopole Gauge Fixing 217

It describes the same physics as the action (8.32). Here the magnetic monopoleis coupled locally whereas the world line of the charged particle is represented bythe charge gauge field (8.57). With the predominance of electric charges in nature,however, this dual action is only of academic interest.

Note that just as before the electromagnetic action with monopoles (8.32), alsothe dual magnetoelectric action (8.63) is double-gauge invariant after the Diracquantization of the charge—it is invariant under the magnetoelectric gauge trans-formations (8.53) and the deformations of the surface S monopole gauge transfor-mations (8.23).

8.5 Monopole Gauge Fixing

First we should eliminate the superfluous monopole gauge transformation (8.27) withthe special gauge functions ΛM

a = g∂a∑

V4δ(x;V4) which do not give any change in

FMab . They may be removed from ΛM

a by a gauge-fixing condition such as

naΛMa ≡ 0, (8.64)

where na is an arbitrary fixed unit vector.The remaining monopole gauge freedom can be used to bring all Dirac strings

to a standard shape so that FMab (x) becomes a function of only the boundary lines

L. In fact, for any choice of the above unit vector na, we may always go to the axialmonopole gauge defined by

naFMab = 0. (8.65)

To see this we take na along the 4-axis and consider the gauge fixing equations

F4i + ∂4ΛMi − ∂iΛ

M4 = 0, i = 1, 2, 3. (8.66)

With (8.64) we have ΛM4 ≡ 0 and ΛM

i could certainly all be determined if theywere ordinary real functions. It is nontrivial to show that the gauge (8.65) can bereached using only the restricted class of gauge functions of the form (8.24). Thisis seen most easily by approximating the four-space by a fine-grained hypercubiclattice of spacing ε and imagining FM

ab to be functions defined on the plaquettes.Then the above defined δ-functions (8.26), (8.24), (8.13), and (8.5) correspond tointeger-valued functions on sites δ(x;V4)=N(x), on links δa(x;V )=Na/ε, on plaque-ttes δab(x;S)=Nab/ε

2, and on links δa(x;L)=Na/ε3, respectively, and the derivatives

∂a correspond to 1/ε times lattice differences ∇a across links. Thus FMab can be writ-

ten as gNab(x)/ε2 with integer Nab(x). The gauge fixing in (8.66) with the restricted

gauge functions amounts then to solving a set of integer-valued equations of thetype

N4i + ∇4NMi −∇iN4 = 0, i = 1, 2, 3, (8.67)

with N4 ≡ 0. This is always possible as has been shown with similar equations inRef. [9].

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218 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

Having fixed the gauge in this way we can solve Eq. (8.1) uniquely by themonopole gauge field

FMab = −2εabcdnc(n∂)

−1d. (8.68)

With this, the interaction between electric and magnetic currents in the last termof (8.37) becomes

Aj = εabcd

d4x ja(n∂ ∂2)−1nb∂cd. (8.69)

This interaction can be found in textbooks [10].

8.6 Quantum Field Theory of Spinless Electric Charges

The full Euclidean quantum field theory of electrically and magnetically charged par-ticles is obtained from the functional integral over the Boltzmann factors e−Atot/h

with the action (8.32). The functional integral has to be performed over all elec-tromagnetic fields Aa and over all electric and magnetic world line configurations Land L′. These, in turn, can be replaced by fluctuating disorder fields which accountfor grand-canonical ensembles of world lines [11]. This replacement, the Euclideananalog of second quantization, in many-body quantum mechanics, was explained inthe last chapter.

Let us assume that only a few fixed worldlines L of monopoles are present. Theelectric world lines, on the other hand will be assumed to consist of a few fixedworld lines L′ plus a fluctuating grand-canonical ensembles of closed world lines L′′.The latter are converted into a single complex field ψe whose Feynman diagrams arepictures of the lines L′′ [12]. The technique was explained in Subsection 5.1.10 andcorresponds to the second quantization of many-body quantum mechanics. In otherwords, we shall study the following partition function

Z =∫

DATa e−Atot

Dψe∫

Dψ∗e e−Aψe , (8.70)

with the field action of the fluctuating electric orbits

Aψe =∫

d4x1

2

[

|Dψe|2 +m2|ψe|2 + λ|ψe|4]

, (8.71)

where Da denotes the covariant derivative involving the gauge field Aa:

Da ≡ ∂a − ie

cAa (8.72)

When performing a perturbation expansion of this functional integral in powers ofthe coupling constant e, the Feynman loop diagrams of the ψe field provide directpictures of the different ways in which the fluctuating closed charged worldlinesinteract in the ensemble.

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8.7 Theory of Magnetic Charge Confinement 219

8.7 Theory of Magnetic Charge Confinement

The field action of fluctuating electric charges is the four-dimensional version of theGinzburg-Landau Hamiltonian (5.140). We have learned in the previous chapter thatthis Hamiltonian allows for a phase transition as a function of the mass parameterm2

in (8.71). There exists a critical value of m2 where the system changes from orderedto disordered. At the mean-field level, the critical value is zero. For m2 > 0, onlya few vortex loops are excited. In this phase, the field has a vanishing expectationvalue 〈ψe〉. For e < ec, on the other hand, the configurational entropy wins over theBoltzmann suppression and infinitely long vortex loops too small the m2 is negativeand the disorder field ψe develops a nonzero expectation 〈ψe〉 whose absolute value

is equal to√

|m2|/2λ. This is a condensed phase where the charge worldlines areinfinitely long and prolific. The passage of e through ec is a phase transition. Fromthe derivative term |Dψe|2, the vector field Aa receives a mass term (m2

A/2c)A2a with

m2A equal to q2c|m2|/λ. For very small e ec, the penetration depth 1/mA of the

vector potential is much larger than the coherence length 1/m of the disorder fieldand the system behaves like a superconductor of type II.

Between magnetic monopoles of opposite sign, the magnetic field lines aresqueezed into the four-dimensional analogs of the Abrikosov flux tubes. Withinthe present functional integral, the initially irrelevant surfaces S enclosed by thecharge worldlines L acquire, via the phase transition, an energy proportional totheir area which removes the charge gauge invariance of the action. They becomephysical fluctuating objects and generate the linearly rising static potential betweenthe charges, thus causing magnetic charge confinement.

The confinement mechanism is particularly simple to describe in the hydrody-namic or London limit. In this limit, the magnitude |ψe| of the field is frozen so itcan be replaced by a constant |ψe| multiplied by a spacetime-dependent phase factoreiθ(x), and the functional integral over ψe and ψ∗e in (8.70) reduces to

V

∫ ∞

−∞Dθ exp

−m2Ac

2q2

d4x[

∂aθ − θva(x) −

e

cAa

]2

, (8.73)

where θva(x) is the four-dimensional vortex gauge field

θva(x) ≡ 2πδa(x;V ). (8.74)

This may be chosen in a specific gauge, for instance in the axial gauge with δ4(x;V ) =0, so that V is uniquely fixed by its surface S, the worldsheet of a vortex line in theφe-field. Thus the action in (8.70) reads, in the hydrodynamic limit,

Ahy =∫

d4x

1

4c(Fab − FM

ab )2 +i

c2

d4x ja(x)Aa(x) +m2Ac

2q2

[

∂aθ − θva(x) −

e

cAa

]2

.

(8.75)

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220 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

If we ignore the vortices and eliminate the θ-fluctuations from the functional integral,we generate a transverse mass term

m2A

2cATa

2 (8.76)

where ATa ≡ (gab − ∂a∂b/∂2)Ab. This causes the celebrated Meissner effect in the

superconductor. In the hydrodynamic limit the action becomes, therefore,

Ahy =∫

d4x

[

1

4c(Fab − FM

ab )2 +m2A

2cATa

2

]

. (8.77)

If we now integrate out the Aa-fields in the partition function (8.70), we obtain theinteraction between the worldlines of electric charges L and the surfaces S whoseboundaries are the monopole worldlines:

Ahyint =

d4x

1

4c

[

(FMab )2 − 2∂aF

Mab (−∂2 +m2

A)−1∂cFMcb

]

+1

2c3∂aF

Mab (−∂2 +m2

A)−1jb +1

2c3ja(−∂2 +m2

A)−1ja

. (8.78)

This is a generalization of the previous current-current interaction (8.37), to whichit reduces for mA = 0. Using the relation (8.43), this becomes

Ahyint =

d4x∫

d4x′[

mA2

16πFMab (x)GmA(x− x′)FM

ab (x′)

+1

2∂aF

Mab GmA(x− x′)ja(x− x′) +

2ja(x)GmA(x− x′)ja(x

′)]

(8.79)

with the massive correlation function

(−∂2)−1(x, x′) = GmA(x− x′) ≡∫

d4k

(2π)4e−ik(x−x

′) 1

k2 +m2A

. (8.80)

Due to the mass mA, the interactions have changed. The last term in (8.79) is nowa short-range Yukawa-type interaction between the electric charges.

The second term is a short-range interaction between the surfaces and the bound-ary lines.

The first term is most interesting. It gives an energy to the previously irrelevantsurfaces S enclosed by the magnetic worldlines L. The energy covers S and aneighborhood of it up to a distance 1/mA. It converts S into a thick worldsheet.This is the world surface of a thick flux tube of thickness 1/mA between connectingthe magnetic charges. To lowest order in the thickness, this causes a surface tension,giving rise to a linearly rising potential between magnetic charges, and thus toconfinement. To next order, it causes a curvature stiffness [14].

The fact that the energy of the surface S enclosed by a monopole world linecauses confinement can be phased as a criterion for confinement due to Wilson. In

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8.8 Second Quantization of the Monopole Field 221

the duality transformation of the monopole part of the action (8.32) to (8.51) wehave observed that a surface S in the monopole gauge field FM

ab corresponds to alocal coupling (i/c)

d4x Aaa in the dual action. This implies that the expectationvalue of the exponential

exp(

i

c

Ld4x Aaa

)⟩

(8.81)

falls off like exp (−area enclosed by L) in the confined phase where the interactionis given by (8.79), but only like exp (−length of L) in the unconfined phase wherethe interaction is given by (8.37).

If the charged particles are electrons, the field ψe(x) must consist of four anti-commuting Grassmann components and the action must be of the Dirac type whichhas the form in Minkowski spacetime:

ADirace =

d4x

ψe(x)[

γa(

ih∂a −e

cAa

)

ψe(x) −mec2ψe(x)ψe(x)

]

, (8.82)

where me is the mass of the electron and ψe(x) are the standard Dirac fields of theelectron. Fermi fields cannot form a condensate, so that there is no confinement ofmonopoles, and the second quantization leads to the standard quantum field theoryof electromagnetism (QED) with the minimal electromagnetic interaction:

Ael =i

c2

d4xAaja, ja = e cψγaψ. (8.83)

8.8 Second Quantization of the Monopole Field

For monopoles described by the action (8.71), second quantization seems at firstimpossible since the partition function contains sum over a grand-canonical ensembleof surfaces S rather than lines. Up to now, there exists no satisfactory second-quantized field theory which could replace such a sum. According to present belief,the vacuum fluctuations of some non-abelian gauge theory is able to do this, but aconvincing theoretical formulation is still missing.

Fortunately, the monopole gauge invariance of the action (8.63) under (8.23)makes most configurations of the surfaces S ′ physically irrelevant and allows us toreturn to a worldline description of the monopoles after all. We simply fix the gaugeas described above, which makes the monopole gauge field unique. It is given byEq. (8.68) and thus completely specified by the worldlines L′ of the monopoles. Withthis we can rewrite the action as [13]

A = A′1 + Ael + Ac 1 + Ac 2

=∫

d4x[

1

4c(Fab − fMab )2 +

i

c2Aaja

]

+i

c2λab

(

nσ∂σfMab + 2εabcdncd

)

, (8.84)

where fMab and λab are now two arbitrary fluctuating fields, i.e., fMab is no longer ofthe restricted δ-function type implied by (8.11). This form restriction is enforced by

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222 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

the fluctuating λab-field. The two terms in the action in which λab appears have beendenoted by Aλab 1 and Aλab 2. The monopole worldline appears only in the magneticcurrent coupling

Amg ≡ Aλab 2 =i

c2

d4x And d, (8.85)

where Ana is short for the vector field

And ≡ 2λabεabcdnc. (8.86)

In the partition function associated with this action we may now sum over a grand-canonical ensemble of monopole worldlines L by converting it into a functionalintegral over a single fluctuating monopole field φg as in the derivation of (8.70).If monopoles carry no spin, this obviously replaces the sum over all fluctuatingmonopole world lines with the magnetic interaction (8.85) by a functional integral

DφgDφ∗g e−Ang , (8.87)

where Ag is the action of the complex monopole field

Ang =

d4x1

2

[

|Dnaφg|2 +m2

g|φg|2 + λ|φg|4]

, (8.88)

and Dna the covariant derivative

Dna ≡ ∂a −

g

cAna . (8.89)

By allowing all fields ψe, φg, Aa, fMab , and λab, to fluctuate with a Euclidean amplitude

e−A/h we obtain the desired quantum field theory of electric charges and Diracmonopoles [15]. The total field action of charged spin-1/2 particles and spin-zeromonopoles is therefore

A =∫

d4x[

1

4c(Fab − fMab )2 +

i

c2λabnσ∂σf

Mab

]

+ ADirace + An

g . (8.90)

Note that the effect of monopole gauge invariance is much more dramatic thanthat of the ordinary gauge invariance in pure QED. The electromagnetic gaugetransformation Aa → Aa + ∂aΛ eliminated only the longitudinal polarization ofthe photons. The monopole gauge transformations, in contrast, (8.23) reduce thedimensionality of the fluctuations from surfaces S to lines L, which is crucial forsetting up the disorder field theory (8.87).

It is obvious that there exists a dual formulation of this theory with the action

A =∫

d4x[

1

4c(Fab − fEab)

2 +i

c2λabnσ∂σ f

Eab

]

+ Ag + ADirace

n, (8.91)

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8.9 Quantum Field Theory of Electric Charge Confinement 223

where Ag is the action (8.87) with the covariant derivative

Da ≡ ∂a −g

cAa, (8.92)

and ADirace

n is the Dirac action coupled minimally to the vector potential (8.86):

ADirace

n =∫

d4x

ψe(x)[

γa(

ih∂a −e

cAna

)

ψe(x) −mec2ψe(x)ψe(x)

]

. (8.93)

8.9 Quantum Field Theory of Electric Charge Confinement

It has long been known that quantum electrodynamics on a lattice with a cyclicvector potential (called compact QED) shows quark confinement for a sufficientlystrong electric charge e. The system contains a grand-canonical ensemble of mag-netic monopoles which condense at some critical value ec. The condensate squeezesthe electric field lines emerging from any charge into a thin tube giving rise to aconfining potential [16, 17, 18]. It is possible to transform the partition function tothe dual version of a standard Higgs model coupled minimally to the dual vectorpotential Aa [19]. The Higgs field is the disorder field [9] of the magnetic monopoles,i.e., its Feynman graphs are the direct pictures of the monopole worldlines in theensemble. Two electric charges in this model are connected by Abrikosov vorticesproducing the linearly rising potential between the charges and thus confinement.The system is a perfect dielectric. While there is no problem in taking the dualHiggs field description of quark confinement to the continuum limit [19], the samething has apparently never been done in the original formulation in terms of thegauge field Aa. The reason was a lack of an adequate continuum description of theinteger-valued jumps in the electromagnetic gauge field Aa across the worldsheetsspanned by the worldlines of the magnetic monopoles. After the development ofthe previous sections we can easily construct a simple quantum field theory whichexhibits electric charge confinement. It is based on a slight modification of the dualmagnetoelectric action (8.63). The modification will lead to the formation of thinelectric flux tubes between opposite electric charges.

For a fixed set of electric and magnetic charges, the euclidean action reads

A =1

16π

d4x[

Fab(x) − FMab (x)

]2+ i

d4xja(x)Aa(x), (8.94)

where Fab = ∂aAb − ∂bAa is the usual field tensor,

ja(x) ≡ e δa(x;L) (8.95)

is the charge distribution along closed worldlines L of the electric charges withδa(x;L) being δ-functions singular on the lines L

δa(x;L) =∫

dτdxadτ

δ(4)(x− x(τ)), (8.96)

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224 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

while FMab (x) is the gauge field of monopoles . It is defined as follows: Let L be the

worldline of a monopole and S an arbitrary surface enclosed by L, then we take theδ-function on this surface

δab(x; S) =∫

dσdτ

[

∂xa∂σ

∂xb∂τ

− (a↔ b)

]

δ(4)(x− q(σ, τ)) (8.97)

and define FMab in terms of the dual of this

FMab (x) ≡ 4πg

1

2εabcdδcd(x; S). (8.98)

This field has the property that its curl is singular on the boundary line L

1

2εabcd∂bF

Mcd (x) = 4πg δa(x; L) = 4πja(x), (8.99)

this being a reformulation of Stokes’ integral theorem in terms of distributions.The constant g is the magnetic charge of the monopoles which is assumed to sat-isfy Dirac’s charge quantization condition (8.35). The euclidean quantum partitionfunction of the system is found by summing, in a functional integral, the Boltzmannfactor e−A over all field configurations Aa, all line configurations L in ja, and allsurface configurations S in FM

ab . It was pointed out in [4, 5, 6] that the action (8.94)is invariant under two types of gauge transformations, the ordinary electromagneticgauge transformations

Aa → Aa + ∂aΛ (8.100)

and the completely independent monopole gauge transformations

Aa → Aa + ΛMa (8.101)

FMab → FM

ab + ∂aΛMa − ∂bΛ

Ma (8.102)

which involve an arbitrary superposition of δ-functions on three-volumes V

ΛMa (x) = 4πg

V

δa(x; V ), (8.103)

with

δa(x, V ) ≡ εabcd

dσdτdλ∂xb∂σ

∂xc∂τ

∂xd∂λ

δ(4)(x− x(σ, τ, c)). (8.104)

The invariance of the gradient term in the action (8.94) is obvious. The current term,on the other hand, changes under the two gauge transformations by i

d4xja∂aΛ andby i

d4xjaΛMa , respectively. The first change vanishes after a partial integration

since for closed worldlines ∂aδa(x;L) = 0 ensuring the electric current conserva-tion law ∂aja = 0. The second change is irrelevant since

d4xδa(x;L)δa(x; V ) is aninteger, counting the number of times by which the line L pierces the volume V .

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8.9 Quantum Field Theory of Electric Charge Confinement 225

The exponential e−A governing the fluctuations in the functional integral changesby exp(−i4πegn) which is a trivial unit factor due to (8.35). Certainly, the func-tional integrals over Aa and the surfaces S require gauge fixing to remove infinitedegeneracies. The options for gauge fixing Aa are well known; for S one may fix thesurface shapes in such a way that they are uniquely determined by their boundarylines L. This was the key for constructing a field theory of magnetic monopoles in[6]. It was also shown in [4, 5, 6] that a duality transformation brings A to thecompletely equivalent form.

A =1

16π

d4x[

Fab(x) − FEab(x)

]2+ i

d4xja(x)Aa(x) (8.105)

where Fab = ∂aAb − ∂bAa is the dual field tensor Fab ≡ (1/2)εabcdFab and ja(x) =g δa(x; L) the dual current density singular on the magnetic monopole worldlines L.Now the electric charges are described by a charge gauge field FE

ab which is singularon arbitrary worldsheets S enclosed by the electric worldlines L:

FEab(x) ≡ 4πe

1

2εabcdδcd(x;S). (8.106)

This action is, of course, invariant under the magnetoelectric gauge transformations

Aa → Aa + ∂aΛ (8.107)

and under the discrete-valued charge gauge transformations

Aa → Aa + ΛEa ,

FEab → FE

ab + ∂aΛEb − ∂bΛ

Ea . (8.108)

We proceed as in the derivation of the disorder theory (8.70), but apply thetransformation to the worldlines of the magnetic monopoles in the dual action (8.63).The resulting second-quantized action is

Atot =∫

d4x1

4c(Fab − FE

ab)2+∫

d4x[

|Dφg|2+m2|φg|2+ λ|φg|4]

, (8.109)

whereDa ≡ ∂a −

g

cAa. (8.110)

If we choose the mass parameter m2 of the monopole field φg to be negative, then

φg acquires a nonzero expectation value√

−m2/2λ, which generates a Meissner mass

m2A

for the dual vector potential Aa. The energy has again the form (8.79), but withelectric and magnetic sources exchanged:

Ahyint =

d4x∫

d4x′[

mA2

16πFEab(x)Gm

A(x− x′)FE

ab(x′)

+1

2∂aF

EabGm

A(x− x′)a(x− x′) +

2a(x)Gm

A(x− x′)a(x

′)]

.(8.111)

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226 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

This gives the surfaces S ′ enclosed by the electric world lines an energy with theproperties discussed above, causing now the confinement of electric charges. Thethick energetic surface has tension and curvature stiffness as proposed independentlyby the author [20] and Polyakov [21] for world sheets of hadronic strings.

In the case of electric charge confinement, the expectation value of the dual ofthe Wilson loop (8.81)

exp(

i

c

Ld4xAaja

)⟩

(8.112)

behaves like exp (−area enclosed by L).It goes without saying that in order to apply the model to quarks, the action

(8.82) has to be replaced by a Dirac action with three colors and six flavors in agauge-invariant coupling

AD =∫

d4x ψ(D/−M)ψ, (8.113)

where Dµ+iGµ is a covariant derivative in color space an Gµ a traceless 3×3-matrixcolor-electric gauge field with the field action

AGµ= −1

4

d4x tr(

∂µGν − ∂νGµ − [Gµ, Gν ])2. (8.114)

The symbol M denotes a mass matrix in the six-dimensional flavor space ofu, d, c, s, t, b.

If one applies the above model interaction (8.79) to quarks, one may study low-energy phenomena by approximating it roughly by a four-Fermi interaction. Thiscan be converted into a chirally invariant effective action for pseudoscalar, scalar,vector, and axial-vector mesons by functional integral technique (hadronization) [22].The effective action reproduces qualitatively many of the low-energy properties ofthese particles, in particular their chiral symmetry, its spontaneous breakdown, andthe difference between the observed quark masses and the masses in the action(8.113) (current quark masses). It also explains why the quarks u, d in a nucleonhave approximately a third of a nucleon mass while their masses Mu, Md in theaction (8.113) are very small.2

The technique of hadronization developed in [22] has been generalized in variousways, in particular by including the color degree of freedom [23]. It has also beenused to describe the low-lying baryons and the restoration of chiral symmetry bythermal effects [24].

An interesting aspect of (8.79) is that the local part of the four-Fermi interaction,which is proportional to 1/m2

A, arises by the same mechanism as the confining poten-tial, whose tension is proportional to m2

A log(Λ2/m2A), with Λ being some ultraviolet

cutoff parameter. One would therefore predict that at an increased temperatureof the order of mA the spontaneous symmetry breakdown, which is caused by the

2The quark masses in the action (8.113) are Mu ≈ 4MeV,Md ≈ 8 MeV, Mc ≈ 1.5GeV,Ms ≈0.150GeV,Mt ≈ 176GeV,Mb ≈ 4.7GeV).

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Notes and References 227

four-Fermi interaction, takes place at the same temperature at which the potentiallooses its deconfinement properties. This initially surprising coincidence has longbeen observed in Monte Carlo simulations of lattice gauge theories.

It is an important open problem to generalize the above hydrodynamic discussionto the case of colored gluons. In particular, the existence of three- and four-stringvertices must be accounted for in a simple way. A promising intermediate solutionis suggested by ’t Hooft’s [25] hypothesis of dominance of abelian monopoles [26].

Notes and References

[1] Magnetic monopoles were first introduced inP.A.M. Dirac, Proc. Roy. Soc. A 133, 60 (1931); Phys. Rev. 74, 817 (1948).817 (1948).

[2] M.N. Saha, Ind. J. Phys. 10, 145 (1936);J. Schwinger, Particles, Sources and Fields , Vols. 1 and 2, Addison Wesley,Reading, Mass., 1970 and 1973;G. Wentzel, Progr. Theor. Phys. Suppl. 37, 163 (1966);E. Amaldi, in Old and New Problems in Elementary Particles, ed. by G. Puppi,Academic Press, New York (1968);D. Villaroel, Phys. Rev. D 14, 3350 (1972);Y.D. Usachev, Sov. J. Particles Nuclei 4, 92 (1973);A.O. Barut, J. Phys. A 11, 2037 (1978);J.D. Jackson, Classical Electrodynamics , John Wiley and Sons, New York,1975, Sects. 6.12-6.13.

[3] H. Kleinert, Defect Mediated Phase Transitions in Superfluids, Solids, andRelation to Lattice Gauge Theories , Lecture presented at the 1983 CargeseSummer School on Progress in Gauge Theories, publ. in Gauge Theories, ed. byG. ’t Hooft et al., Plenum Press, New York, 1984, pp 373-401 (kl/118), wherekl is short for the www address http://www.physik.fu-berlin.de/~klei-

nert.;

[4] H. Kleinert, The Extra Gauge Symmetry of String Deformations in Electro-magnetism with Charges and Dirac Monopoles, Int. J. Mod. Phys. A 7, 4693(1992) (kl/203).

[5] H. Kleinert, Double-Gauge Invariance and Local Quantum Field Theory ofCharges and Dirac Magnetic Monopoles, Phys. Lett. B 246, 127 (1990)(kl/205).

[6] H. Kleinert, Abelian Double-Gauge Invariant Continuous Quantum Field The-ory of Electric Charge Confinement , Phys. Lett. B 293, 168 (1992) (kl/211).

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228 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

[7] See, most notably, the textbook by Jackson in Ref. [1], p. 258, where it is statedthat “a choice of different string positions is equivalent to different choices of(electromagnetic) gauge”; also his Eq. (6.162) and the lines below it. In theunnumbered equation on p. 258 Jackson observes that the physical monopolefield is Fmonop

ab = Fab − FMab but the independent gauge properties of FM

ab andthe need to use the action (8.18) rather than (8.19) are not noticed.

[8] Compare the review article byP. Goddard and D. Olive, Progress in Physics 41, 1357 (1978) who use intheir review article “general gauge transformations” [their Eq. (2.46)]. Theyobserve that the field tensor F obs

ab = Fab+(1/a3a)Φ·(∂aΦ×∂bΦ) introduced by’t Hooft into his SU(2) gauge theory with Higgs fields Φ to describe magneticmonopoles can be brought to the form F obs

ab ≡ Fab − FMab by a gauge transfor-

mation within the SU(2) gauge group which moves magnetic fields from Fab toFMab without changing F obs

ab [see their Eq. (4.30) and the last two equations intheir Section 4.5]. Note that these are not permissible gauge transformation ofthe electromagnetic type. They are similar to our monopole gauge transforma-tions of the form (8.25), although with more general transformation functionsthan (8.24) due to the more general SU(2) symmetry.

[9] H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, Superflow and VortexLines, World Scientific, Singapore, 1989 (kl/b1).

[10] J. Schwinger, Particles, Sources and Fields , Vol 1, Addison Wesley, Reading,Mass., 1970, p. 235.

[11] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets, 4th ed., World Scientific, Singapore, 2006(kl/b5).

[12] K. Symanzik, Varenna Lectures 1986, in Euclidean Quantum Field Theory ,ed. R. Jost, Academic Press, New York, 1969.

[13] See pp. 570–578 in the textbook [9] (kl/b1/gifs/v1-570s.html).

[14] Surface tension and stiffness coming from interactions like the first term in(8.79) have been calculated for biomembranes inH. Kleinert, Dynamical Generation of String Tension and Stiffness in Stringsand Membranes, Phys. Lett. B 211, 151 (1988) (kl/177).

[15] The construction of a quantum field theory of monopoles is impossible in thetheory of “exorcized” Dirac strings byT.T. Wu and C.N. Yang, Phys. Rev. D 14, 437 (1976); Phys. Rev. D 12, 3845,(1075); D 16, 1018 (1977); Nuclear Physics B 107, 365 (1976);C.N. Yang, Lectures presented at the 1976 Erice summer school, in GaugeInteractions, Plenum Press, New York 1978, ed. by A. Zichichi and at the

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 229

1982 Erice summer school, in Gauge Interactions, Plenum Press, New York1984, ed. by A. Zichichi.

[16] Y. Nambu, Phys. Rev. D 10, 4262 (1974).

[17] S. Mandelstam, Phys. Rep. C 23, 245 (1976); Phys. Rev. D 19, 2391 (1979).

[18] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974); and in High Energy Physics , ed.by A. Zichichi, Editrice Compositori, Bologna, 1976.

[19] H. Kleinert, Higgs Particles from Pure Gauge Fields, Lecture presented at theErice Summer School, 1982, in Gauge Interactions, ed. by A. Zichichi, PlenumPress, New York, 1984 (kl/117).

[20] H. Kleinert, Phys. Lett. B 174, 335 (1986) (kl/149).

[21] A.M. Polyakov, Nucl. Phys. B 268, 406 (1986); see also his monograph GaugeFields and Strings , Harwood Academic, Chur, 1987.

[22] H. Kleinert, On the Hadronization of Quark Theories, Lectures presentedat the Erice Summer Institute 1976, Understanding the Fundamental Con-stituents of Matter, Plenum Press, New York, 1978, A. Zichichi (ed.), pp. 289-390 (kl/53). See also Hadronization of Quark Theories and a Bilocal form ofQED , Phys. Lett. B 62, 429 (1976).

[23] See for exampleK. Rajagopal and F. Wilczek, The Condensed Matter Physics of QCD , (hep-ph/0011333);S.B. Ruster, V. Werth, M. Buballa, I.A. Shovkovy, D.H. Rischke, Phase dia-gram of neutral quark matter at moderate densities, Phys. Rev. D 73, 034025(2006) (nucl-th/0602018),and references therein.

[24] R. Cahill, C.D. Roberts, and J. Praschifka, Aust. J. Phys. 42, 129 (1989);R. Cahill, J. Praschifka, and C.J. Burden, Aust. J. Phys. 42, 161 (1989);R. Cahill, Aust. J. Phys. 42, 171 (1989);H. Reinhardt, Phys. Lett. B 244, 316 (1990);V. Christov, E. Ruiz-Arriola, K. Goeke, Phys. Lett. B 243, 191 (1990);R. Cahill, Nucl. Phys. A 543, 63c (1992).

[25] G. ’t Hooft, Nucl. Phys. B 190, 455 (1981).

[26] A.S. Kronfeld, G. Schierholz, and U.-J. Wiese, Nucl. Phys. B 293, 461 (1987);A.S. Kronfeld, M.L. Laursen, G. Schierholz, and U.-J. Wiese, Phys. Lett. B198, 516 (1987);F. Brandstaeter, G. Schierholz, and U.-J. Wiese, DESY preprint 91-040 (1991);T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42, 4257 (1991);

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230 8 Relativistic Magnetic Monopoles and Electric Charge Confinement

S. Hioki, S. Kitahara, S. Kiura, Y. Matsubara, O. Miyamura, S. Ohuo, and T.Suzuki, Phys. Lett. B 272, 326 (1991);J. Smit and A. Van der Sijs, Nucl. Phys. B 355, 603 (1991);V.G. Bornyakov, E.M. Ilgenfritz, M.L. Laursen, V.K. Mitrijushkin, M. Muller-Preussker, A.J. Van der Sijs, and A.M. Zadorozhyn, Phys. Lett. B 261, 116(1991);J. Greensite and J. Winchester, Phys. Rev. D 40, 4167 (1989);J. Greensite and J. Iwasaki, Phys. Lett. B 255, 415 (1991);A. Di Giacomo, M. Maggiore, and S. Olejn´ik, Nucl. Phys. B 347, 441 (1990);M. Campostrini, A. Di Giacomo, H. Panagopoulos, and E. Vicari, Nucl. Phys.B 329, 683 (1990);M. Campostrini, A. Di Giacomo, M. Maggiore, S. Olejn´ik, H. Panagopoulos,and E. Vicari, Nucl. Phys. B (Proc. Suppl.) 17, 563 (1990);L. Del Debbio, A. Di Giacomo, M. Maggiore, and S. Olejn´ik, Phys. Lett. B267, 254 (1991);L. Polley and U.-J. Wiese, Nucl. Phys. B 356, 629 (1991);H.G. Evertz, K. Jansen, J. Jersak, C.B. Lang, and T. Neuhaus, Nucl. Phys. B285, 590 (1987);T.L. Ivanenko, A.V. Pochinsky, and M.I. Polikarpov, Phys. Lett. B 302, 458(1993);P. Cea and L. Cosmai, Nuovo Cim. A 106, 1361 (1993).

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Women love us for our defects. If we have enough of them,

they will forgive us everything, even our intellects.

Oscar Wilde (1854 - 1900)

9

Multivalued Mapping from Ideal Crystals

to Crystals with Line-Like Defects

In the last chapter we have learned how multivalued gauge transformations allow usto transform theories in field-free space into theories coupled to electromagnetism.By analogy, we expect that multivalued coordinate transformations can be used tomap theories in flat space into theories in spaces with curvature and torsion. Thisis indeed possible. The mathematical methods have been developed in the theory ofline-like defects in crystals [1, 2, 3]. Let us briefly review those parts of the theorywhich will be needed for our purposes.

9.1 Defects in Crystals

No crystal produced in the laboratory is perfect. It always contains a great numberof defects. These may be chemical, electrical, or structural in character, involvingforeign atoms. They may be classified according to their space dimensionality. Thesimplest type of defect is the point defect. It is characterized by the fact that withina certain finite neighborhood only one cell shows a drastic deviation from the perfectcrystal symmetry. The most frequent origin of such point defects is irradiation or anisotropic mechanical deformation under strong shear stresses. There are two typesof intrinsic point defects. Either an atom may be missing from its regular lattice site(vacancy) or there may be an excess atom (interstitial) (see Fig. 9.1). Vacancies andinterstitials are mobile defects. A vacancy can move if a neighboring atom movesinto its place, leaving a vacancy at its own former position. An interstitial atom canmove in two ways. It may hop directly from one interstitial site to another. Thishappens in strongly anisotropic materials such as graphite but also in some cubicmaterials like Si or Ge. Or it may move in a way more similar to the vacancies byreplacing atoms, i.e., by pushing a regular atom out of its place into an interstitialposition which, in turn affects the same change on its neighbor, etc.

Intrinsic point defects have the property that if a number of them move closetogether, the total energy becomes smaller than the sum of the individual energies.The reason for this is easily seen. If two vacancies in a simple cubic lattice come

231

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232 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

to lie side by side, there are only 10 broken valencies compared to 12 when theyare separated. If a larger set of vacancies comes to lie side by side forming anentire disc of missing atoms, the crystal planes can move together and make thedisc disappear (see Fig. 9.2). In this way, the crystal structure is repaired. Onlyclose to the boundary line, such a repair is impossible. The boundary line forms aline-like defect.

Certainly, line-like defects can arise also in an opposite process of clusteringof interstitial atoms. If they accumulate side by side forming an interstitial disc,the crystal planes move apart and accommodate the additional atoms in a regularatomic array, again with the exception of the boundary line. Line-like defects of thistype are called dislocation lines .

It is obvious that a dislocation line need not only consist of a single disc of missingor excessive atoms. There can be several discs stacked on top of each other. Theirboundary forms a dislocation line of higher strength. The energy of such a higherdislocation line increases roughly with the square of the strength. Dislocations are

Figure 9.1 Intrinsic point defects in a crystal. An atom may become interstitial, leaving

behind a vacancy. It may perform random motion via interstitial places until it reaches

another vacancy where it recombines. The exterior of the crystal may be seen as a reservoir

of vacancies.

Figure 9.2 Formation of a dislocation line (of the edge type) from a disc of missing

atoms. The atoms above and below the missing ones have moved together and repaired

the defect, except at the boundary.

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9.1 Defects in Crystals 233

created and set into motion if stresses exceed certain critical values. This is whythey were first seen in plastic deformation experiments of the nineteenth century inthe form of slip bands. The grounds for their theoretical understanding were laidmuch later by Frenkel who postulated the existence of crystalline defects in order tounderstand why materials yield to plastic shear about a thousand times more easilythan one might expect on the basis of a naive estimate (see Fig. 9.3).

Figure 9.3 Naive estimate of maximal stress supported by a crystal under shear stress

as indicated by the arrows. The two halves tend to slip against each other.

Assuming a periodic behavior σ = σmax sin(2πx/a), this reduces to σ ∼σmax2π(x/a) ∼ µ(x/a). Hence σmax = µ/2π. Experimentally, however, σmax ∼10−3µ to 10−4µ.

The large discrepancy was explained by Frenkel who noted that the plastic slipwould not proceed by the two halves moving against each other as a whole butstepwise, by means of defects. In 1934, Orowan, Polanyi and Taylor identified thesedefects as dislocation lines. The presence of a single moving edge dislocation allowsfor a plastic shear movement of the one crystal half against the other. The movementproceeds in the same way as that of a caterpillar. This is pictured in Fig. 9.4. One legis always in the air breaking translational invariance and this is exchanged againstthe one in front of it, etc. In the crystal shown in the lower part of Fig. 9.4, thesingle leg corresponds to the lattice plane of excess atoms. Under stress along thearrows, this moves to the right. After a complete sweep across the crystal, the upperhalf is shifted against the lower by precisely one lattice spacing.

If many discs of missing or excess atoms come to lie close together there existsa further cooperative phenomenon. This is illustrated in Fig. 9.5. On the left-hand side, an infinite number of atomic half planes (discs of semi-infinite size) hasbeen removed from an ideal crystal. If the half planes themselves form a regularcrystalline array, they can fit smoothly into the original crystal. Only at the origin isthere a breakdown of crystal symmetry. Everywhere else, the crystal is only slightlydistorted. What has been formed is again a line-like defect called a disclination.Dislocations and disclinations will play a central role in our further discussion.

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234 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Figure 9.4 Dislocation line permits the two crystal pieces to move across each other in

the same way as a caterpillar moves through the ground. The bonds can flip direction

successively which is a rather easy process.

Before coming to this let us complete the dimensional classification of two-dimensional defects. They are of three types. There are grain boundaries wheretwo regular lattice parts meet, with the lattice orientations being different on bothsides of the interface (see Fig. 9.6).

They may be considered as arrays of dislocation lines in which half planes ofpoint defects are stacked on top of each other with some spacing, having completelyregular lattice planes between them. The second type of planar defects are stackingfaults . They contain again completely regular crystal pieces on both sides of theplane, but instead of being oriented differently they are shifted one with respect tothe other (see Fig. 9.7). The third unavoidable type is the surface of the crystal.

From now on we shall focus attention upon the line-like defects.

9.2 Dislocation Lines and Burgers Vector

Let us first see how a dislocation line can be characterized mathematically. Forthis we look at Fig. 9.8 in which a closed circuit in the ideal crystal is mapped

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9.2 Dislocation Lines and Burgers Vector 235

Figure 9.5 Formation of a disclination from a stack of layers of missing atoms (cf.

Fig. 9.2). Equivalently, one may cut out an entire section of the crystal. In a real crystal,

the section has to conform with the symmetry angles. In the continuum approximation,

the angel Ω is meant to be very small.

Figure 9.6 Grain boundary where two crystal pieces meet with different orientations in

such a way not every atomic layer matches (here only every other one does).

into the disturbed crystal. The orientation is chosen arbitrarily to be anticlockwise.The prescription for the mapping is that for each step along a lattice direction,a corresponding step is made in the disturbed crystal. If the original lattice sitesare denoted by xn, the image points are given by xn + u(xn), where u(xn) is thedisplacement amount field: At each step, the image point moves in a slightly differentoriginal point. After the original point has completed a closed circuit, call it B0,the image point will not have arrived at the point of departure. The image of theclosed contour B0 is no longer closed. This failure to close is given precisely by alattice vector b(x) called a local Burgers vector , which points from the beginning to

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236 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Figure 9.7 Two typical stacking faults. The first is called growth-stacking fault or twin

boundary, the second deformation-stacking fault.

the end of the circuit.1 Thus, the dislocation line is characterized by the followingequation,

B0

∆ui(x) = bi, (9.1)

where ∆ui(xn) are the increments of the displacement vector from step to step.Equivalently, we can consider a closed circuit in the disturbed crystal, call it B, andfind that its counter image in the ideal crystal does not close by a vector b called thetrue Burgers vector which now points from the end to the beginning of the circuit.

If we consider the same process in the continuum limit, we can write

B0

dui(xa) = bi,∫

Bdui(x

′a) = bi. (9.2)

The closed circuit B is called Burgers circuit . The two Burgers vectors are the sameif both circuits are so large that they lie deep in the ideal crystal. Otherwise theydiffer by an elastic distortion.

A few remarks are necessary concerning the convention employed in defining theBurgers vector. The singular line L is in principle without orientation. We mayarbitrarily assign a direction to it. The Burgers circuit is then taken to encirclethis chosen direction in the right-handed way. If we choose the opposite direction,Burgers vector changes sign. However, the products bi, dxj, where dxj is the in-finitesimal tangent vector to L, are invariant under this change. Note that this issimilar to the magnetic case discussed in Part II. There one defined the direction ofthe current by the flow of positive charge. The Burgers circuit gives

du = I. Onecould, however, also reverse this convention referring to the negative charge. Then

1Our sign convention is the opposite of Bilby et al. and the same as Read’s (see Notes andReferences). Note that in contrast to the local Burgers vector, the true Burgers vector is definedon a perfect lattice.

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9.2 Dislocation Lines and Burgers Vector 237

Figure 9.8 Definition of Burgers vector. In the presence of a dislocation line the image

of a circuit which is closed in the ideal crystal fails to close in the defected crystal. The

opposite is also true. The failure to close is measured by a lattice vector, called Burgers

vector. The dislocation line in the figure is of the edge type and the Burgers vector points

orthogonally to the line.

du would give −I. Again, I · dxi is an invariant. Only these products can appearin physical observables such as the biot-Savart law.

The invariance of bidxj under reversal of the orientation has a simple physicalmeaning. In order to see this, consider once more the above dislocation line whichwas created by removing a layer of atoms. We can see in Fig. 9.8 that in this caseb × dx points inwards, namely, towards the vacancies. Consider now the oppositecase in which a layer of new atoms is inserted between the crystal planes forcingthe planes apart to relax the local stress. If we now calculate

B dui(x) = bi, wefind that x × dx points outwards, i.e. away from the inserted atoms. This is againthe direction in which there are fewer atoms. Both statements are independentof the choice of the orientation of the Burgers circuit. Since the second case hasextra atoms inside the circle, where the previous one had vacancies, the two can beconsidered as antidefects of one another. If the boundary lines happen to fall on topof each other, they can annihilate each other and re-establish a perfect crystal. Thiscan happen only piece-wise in which case the parts where the lines differ remaina dislocation lines. In both of the examples, the Burgers vectors are everywhere

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238 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Figure 9.9 Screw disclination which arises when tearing a crystal. The Burgers vector

is parallel to the vertical line.

orthogonal to the dislocation line and one speaks of a pure edge dislocation (seeFig. 9.2).

There is no difficulty in constructing another type of dislocation by cutting acrystal along a lattice half-plane up to some straight line L, and translating oneof the lips against the other along the direction of L. In this way one arrives atthe so-called screw dislocation shown in Fig. 9.9 in which the Burgers vector pointsparallel to the line L.

When drawing crystals out of a melt, it always contains a certain fraction ofdislocations. Even in clean samples, at least one in 106 atoms is dislocated. Theirboundaries run in all directions through space. We shall see very soon that theirBurgers vector is a topological invariant for any closed dislocation loop. Therefore,the character “edge” versus “screw” of a dislocation line is not an invariant. Itchanges according to the direction of the line with respect to th invariant Burgersvector bi. It is obvious from the Figs. 9.2 to 9.9 that a dislocation line destroys thetranslational invariance of the crystal by multiples of the lattice vectors. If thereare only a few lines this destruction is not very drastic. Locally, i.e., in any smallsubspecimen which does not lie too close to the dislocation line, the crystal can stillbe described by a periodic array of atoms whose order is disturbed only slightly bya smooth displacement field ui(x).

9.3 Disclination Lines and Frank Vector

Since the crystal is not only invariant under discrete translations but also undercertain discrete rotations we expect the existence of another type of defect whichis capable of destroying the global rotational order, while maintaining it locally.These are the disclination lines of which one example was given in Fig. 9.5. It aroseas a superposition of stacks of layers of missing atoms. In the present context, itis useful to construct it by means of the following Gedanken experiment. Take a

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9.3 Disclination Lines and Frank Vector 239

regular crystal in the form of cheese and remove a section subtending an angle Ω(see Fig. 9.10). The free surfaces can be forced together. For large Ω this requiresconsiderable energy. Still, if the atomic layers on the free surfaces match togetherperfectly, the crystal can re-establish locally its periodic structure. This happens forall symmetries of the crystal. In a simple cubic crystal, Ω can be 900, 1800, 2700.The 900 case is displayed in Fig. 9.11.

Figure 9.10 Volterra cutting and welding process leading to a wedge disclination.

Figure 9.11 Lattice structure at a wedge disclination in a simple cubic lattice. The

Frank angle Ω is equal to the symmetry angles 900 or -900. The crystal is locally perfect

except close to the disclination line.

In Fig. 9.11 we can imagine also the opposite procedure going from the right inFig. 9.10 to the left. We may cut the crystal, force the lips open by Ω and insert newundistorted crystalline matter to match the atoms in the free surfaces. These arethe disclinations of negative angles. The case for Ω = −900 is shown in Fig. 9.11.

The local crystal structure is destroyed only along the singular line along theaxis of the cheese. The rotation which has to be imposed upon the free surfacesin order to force them together may be represented by a rotation vector Ω which,in the present example, points parallel to L and to the cut. This is called a wedgedisclination. It is not difficult to construct other rotational defects. The threepossibilities are shown in Figs. 9.12. Each case is characterized by a vector. Inthe first case, Ω pointed parallel to the line L and the cut. Now, in the secondcase, it is orthogonal to the line L and Ω points parallel to the cut. This is a splaydisclination. In the third case, Ω points orthogonal to the line and cut. This is atwist disclination .

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240 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

The vector Ω is referred to as the Frank vector of the disclination. Just as in theconstruction of dislocations, the interface at which the material is joined togetherdoes not have any physical reality. For example, in Fig. 9.12a we could have cut outthe piece along any other direction which is merely rotated with respect to the firstaround L by a discrete symmetry irregular piece as long as the faces fit togethersmoothly (recall Fig. 9.10). Only the singular line is a physical object.

The Gedanken experiments of cutting a crystal, removing or inserting slicesor sections, and joining the free faces smoothly together were first performed byVolterra in 1907. For this reason one speaks of the creation of a defect line asa Volterra process and calls the cutting surfaces, where the free faces are joinedtogether, Volterra surfaces.

Figure 9.12 Three different possibilities of constructing disclinations: (a) wedge, (b)

splay, and (c) twist disclinations.

9.4 Interdependence of Dislocation and Disclinations

It must be pointed out that dislocation and disclination lines are note completelyindependent. We have seen before in Fig. 9.5 that a disclination line was created byremoving stacks of atomic layers from a crystal. But each layer can be considered as

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9.4 Interdependence of Dislocation and Disclinations 241

a dislocation line running along the boundary. Thus a disclination line is apparentlyindistinguishable from a stack of dislocation lines, placed with equal spacing on top ofeach other. Conversely, a dislocation line is very similar to a pair of disclination linesrunning in opposite directions close to each other. This is illustrated in Fig. 9.13.What we have here is a pair of opposite Volterra processes of disclination lines. Wehave cut out a section of angle Ω, but instead of removing it completely we havedisplaced it merely by one lattice spacing a. This is equivalent to generating adisclination of the Frank vector Ω and another one with the opposite Frank vector−Ω whose rotation axis is displaced by a. It is obvious from the figure that theresult is a dislocation line with Burgers vector b.

Because of this interdependence between dislocations and disclinations, the de-fect lines occurring in a real crystal will, in general, be of a mixed nature. It must bepointed out that disclinations were first observed and classified by F.C. Frank in 1958in the context of liquid crystals. Liquid crystals are mesophases. They are liquidsconsisting of rod-like molecules. Thus, they cannot be described by a displacementfield ui(x) alone but require an additional orientational field ni(x) for their descrip-tion. This orientation is independent of the rotational field ωi(x) = 1

2εijk∂juk(x).

The disclination lines defined by Frank are the rotational defect lines with respect tothis independent orientational degree of freedom. Thus, they are a priori unrelatedto the disclination lines in the rotation field ωi(x) = 1

2εijk∂juk(x). In fact, the liquid

is filled with dislocations and ω-disclinations even if the orientation field nj(x) iscompletely ordered.

Friedel in his book on dislocations (see the references at the end) calls th nj-disclinations, rotation dislocations. But later the name disclinations became cus-tomary (see Kleman’s article cited in the Notes and References). In general, there islittle danger of confusion, if one knows what system and phase one is talking about.

Figure 9.13 Generation of dislocation line from a pair of disclination lines running in

opposite directions at a fixed distance b. The Volterra process amounts to cutting out a

section and reinserting it, but shifted by the amount b.

9.5 Defect Lines with Infinitesimal Discontinuities

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242 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

in Continuous Media

The question arises as to how one can properly describe the wide variety of line-likedefects which can exist in a crystal. In general, this is a rather difficult task dueto the many possible different crystal symmetries. For the sake of gathering someinsight it is useful to restrict oneself to continuous isotropic media. Then defectsmay be created with arbitrarily small Burgers and Frank vectors. Such infinitesimaldefects have the great advantage of being accessible to differential analysis. This isessential for a simple treatment of rotational defects. It permits a characterizationof disclinations in a way which is very similar to that of dislocations via a Burgerscircuit integral. Consider, for example, the wedge disclination along the line L(shown in Figs. 9.5, 9.10, 9.11 or 9.12a), and form an integral over a closed circuitB enclosing L.

Just as in the case of dislocations this measures the thickness of the materialsection removed in the Volterra process. Unlike the situation for dislocations, thisthickness increases with distance from the line. If Ω is very small, the displacementfield across the cut has a discontinuity which can be calculated from an infinitesimalrotation

∆ui = (Ω × x)i , (9.3)

where x is the vector pointing to the place where the integral starts and ends. Inorder to turn this statement into a circuit integral it is useful to remove the explicitdependence on x and consider not the displacement field ui(x) but the local rotationfield accompanying the displacement instead. This is given by the antisymmetrictensor field

ωij(x) ≡ 1

2

[

∂iuj(x) − ∂jui(x)]

. (9.4)

The rotational character of this tensor field is obvious by looking at the change ofan infinitesimal distance vector under a distortion

dx′i − dxi = (∂jui) dxj = uij dxj − ωij dxj. (9.5)

The tensor field ωij is associated with a vector field ωi as follows:

ωij(x) = εijkωk(x) (9.6)

i.e.,

ωij(x) ≡ 1

2εijkωjk(x) =

1

2(∇× u)i . (9.7)

The right-hand side of (9.6) separates the local distortion into a sum of a localchange of shape and a local rotation. Now, when looking at the wedge disclination

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9.6 Multivaluedness of Displacement Field 243

in Fig. 9.12a, we see that due to (9.3), the field ωi(x) has a constant discontinuityΩ across the cut. This can be formulated as a circuit integral

∆ωi =∮

Bdωi = Ωi. (9.8)

The value of this integral is the same for any choice of the circuit B as long as itencloses the disclination line L.

This simple characterization depends essentially on the infinitesimal size of thedefect. If Ω were finite, the differential expression (9.3) would not be a rotationand the discontinuity across the cut could not be given in the form (9.8) withoutspecifying the circuit B. The difficulties for finite angles are a consequence of thenon-Abelian nature of the rotation group. Only infinitesimal local rotations haveadditive rotation angles, since the quadratic and higher-order corrections can beneglected.

9.6 Multivaluedness of Displacement Field

As soon as a crystal contains a few dislocations, the definition of displacementfield is intrinsically non-unique. The displacement field is intrinsically non-unique.The displacement fields is really multivalued . In a perfect crystal, in which theatoms deviate little from their equilibrium positions x, it is natural to draw thedisplacement vector from the lattice places x to the nearest atom. In principle,however, the identity of the atoms makes such a specific assignment impossible.Due to thermal fluctuations, the atoms exchange positions from time to time bya process called self-diffusion. After a very long time, the displacement vector,even in a regular crystal, will run through the entire lattice. Thus, if we describea regular crystal initially by very small displacement vectors ui(x), then, after avery long time, these will have changed to a permutation of lattice vectors, each ofthem occurring precisely once, plus some small fluctuations around them. Hence thedisplacement vectors are intrinsically multivalued, with ui(x) being indistinguishablefrom ui(x) + aNi(x), where Ni(x) are integer numbers and a is the lattice spacing.

It is interesting to realize that this property puts the displacement fields on thesame footing with the phase variable γ(x) of superfluid 4He. There the indistin-guishability of γ(x) and γ(x) + 2πN(x) has an entirely different reason: it followsdirectly from the fact that the physical field is the complex field ψ(x) = |ψ(x)|eiγ(x),which is invariant under the exchange γ(x) → γ(x) + 2πN(x).

Thus, in spite of the different physics described by the variables γ(x), ui(x),they both share the characteristic multivaluedness. It is just as if the rescaled ui(x)variables γi(x) = (2π/a)ui(x) were phases of three complex fields

ψi(x) = |ψi(x)|eiγi(x),

which serve to describe the positions of the atoms in a crystal.In a regular crystal, the multivaluedness of ui(x) has no important physical

consequences. The atoms are strongly localized and the exchange of positions occurs

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244 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

very rarely. The exchange is made irrelevant by the identity of the atoms andsymmetry of the many-body wave function. This is why the natural assignment ofui(x) to the nearest equilibrium position x presents no problems. As soon as defectsare present, however, the full ambiguity of the assignment comes up: When removinga layer of atoms, the result is a dislocation line along the boundary of the layer.Across the layer, the positions ui(x) jump by a lattice spacing. This means that theatoms on both sides are interpreted as having moved towards each other. Figure9.14 shows that the same dislocation line could have been constructed by removing acompletely different layer of atoms, say S ′, just as long as it has the same boundaryline. The jump of the displacement field across the shifted layer S ′ correspondsto the neighboring atoms of this layer having moved together and closed the gap.Physically, there is no difference. There is only a difference in the descriptions whichamounts to a difference in the assignment of the equilibrium positions from where tocount the displacement field ui(x). In contrast to regular crystals there now existsno choice of the nearest equilibrium point. It is this multivaluedness which will formthe basis for the geometric description of the defects in solids.

Figure 9.14 This is due to the fact that the surface S on which the atoms have been

removed is arbitrary as long as the boundary line stays fixed. Shifting S implies shifting

of the reference positions, from which to count the displacements ui(x).

9.7 Smoothness Properties of Displacement Fieldand Weingarten’s Theorem

In order to be able to classify a general defect line we must first give a characteriza-tion of the smoothness properties of the displacement field away from the singularity.In physical terms, we have to make sure that the crystal matches properly togetherwhen cutting and rejoining the free faces.

In the gradient representation of magnetism in Subsection 4.2, the presence ofa magnetic field was signalized by a violation of the integrability condition [recall([email protected])]

(

∂i∂j − ∂j∂i)

Ω(x) = 0. (9.9)

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9.7 Smoothness Properties of Displacement Field and Weingarten’s Theorem 245

In the crystal, this property will hold away from the cutting surface S, whereui(x) is perfectly smooth and satisfies the corresponding integrability condition

(

∂i∂j − ∂j∂i)

uk(x) = 0. (9.10)

Across the surface, ui(x) is discontinuous. However, the open faces of the crystallinematerial must fit properly to each other. This implies that the strain as well as itsfirst derivatives should have the same values on both sides of the cutting surface S:

∆uij = 0, (9.11)

∆∂kuij = 0. (9.12)

This severely restricts the discontinuities of ui(x) across S. In order to see this letx(1),x(2) be two different crystal points slightly above and below S and C+, C−

be two curves connecting the two points. (See Fig. 9.15) We can then calculate thedifference of the discontinuities as follows:

∆ui(1) − ∆ui(2) =[

ui(1−)ui(1

+)]

−[

ui(2−) − ui(2

+)]

=∫ 2+

1+

C+

dxj∂jui −∫ 2−

1−

C−

dxj∂jui. (9.13)

Using the local rotation field ωij(x) we can rewrite this as

∆ui(1) − ∆ui(2) =∫ 2+

1+

C+

dxjdxj(uij − ωij −∫ 2−

1−

C−

dxj(uij − ωij). (9.14)

The ωij pieces may be integrated by parts:

Figure 9.15 Geometry used in the derivation of Weingarten’s theorem [Eqs. (9.13)–

(9.22)].

−(

xj − xj(1+)

ωij

2+

1+

+∫ 2+

1+dxk

(

xj − xj(1+))

∂kωij

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246 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

+(

xj − xj(1−))

ωij

2−

1−dxk

(

xj − xj(1−))

∂kωij (9.15)

=

[

−(

xj(2+) − xj(1

+))

ωij(2+) +

∫ 2+

1+dxk

(

xj − xj(1+))

∂kωij

]

− [+ → −].

Since

xj(1+) = xj(1

−), xj(2+) = xj(2

−),

we arrive at the relation

∆ui(1) − ∆ui(2) = −(

xj(1) − xj(2)) (

ωij(2−) − ωij(2

+))

+∮

C+−

dxk

uik + (xj − xj(1))∂kωij

, (9.16)

where C+− is the closed contour consisting of C+ followed by −C−. Since C+ and−C− are running back and forth on top of each other, the closed contour integralcan be rewritten as a single integral along −C− with uik and ∂kωij replaced bytheir discontinuities across the sheet S. Moreover, the discontinuity of ∂kωij can bedecomposed in the following manner:

∆(

∂kωij)

=1

2∂k(

∂iuj − ∂jui)

(x−) −(

x− → x+)

= ∂iukj(x−) − ∂juki(x

−) +1

2(∂k∂i − ∂i∂k) uj(x

−) (9.17)

− 1

2

(

∂k∂j − ∂j∂k)

ui(x−) +

1

2

(

∂j∂i − ∂i∂j)

uk(x−) −

(

x− → x−)

.

Since above and below the sheet, the displacement field is smooth, the two derivativesin front of u(x±) commute. Hence the integral in (9.16) becomes

−∫

C−

dxk

∆uij +(

xj − xj)(1)∆(∂iukj − ∂juki)

(9.18)

This expression vanishes due to the physical requirements (9.11) and (9.12). As aresult we find that the discontinuities between two arbitrary points 1 and 2 on thesheet have the simple relation

∆ui(2) = ∆ui(1) − Ωij

(

xj(2) − xj(1))

, (9.19)

where Ωij is a fixed infinitesimal rotation matrix given by

Ωij = ∆ωij(2) = ωij(2−) − ωij(2

+). (9.20)

We now define the rotation vector

with components

Ωk =1

2εijkΩij (9.21)

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9.8 Integrability Properties of Displacement Field 247

in terms of which (9.19) takes the form

∆u(2) = ∆u(1) + × (x(2) − x(1)) . (9.22)

This forms the content of Weingarten’s theorem: The discontinuity of the displace-ment field across the cutting surface can only be a constant vector plus a fixedrotation.

Note that these are precisely the symmetry elements of a solid continuum. Whenlooking back at the particular dislocation and disclination lines in Figs. 9.2–9.12 wesee that all the discontinuities obey this theorem, as they should. The vector Ω isthe Frank vector of the disclination lines. For a pure disclination line, Ω = 0 and∆u(1) = ∆u(2) = b is the Burgers vector.

9.8 Integrability Properties of Displacement Field

The rotation field ωij(x) has also nontrivial integrability properties. Taking Wein-garten’s theorem (9.19) and forming derivatives, we see that the jump of the ωij(x)field is necessarily a constant, namely Ωij . Hence ωkl also satisfies the integrabilitycondition

(

∂i∂j − ∂j∂i)

ωkl = 0, (9.23)

everywhere except on the defect line. The argument is the same as that for thevortex lines. We simply observe that the contour integral over a Burgers circuit

∆ωij =∮

Bdωij =

Bdxk∂kωij (9.24)

can be cast, by Stokes’ theorem, in the form

∆ωij =∫

SBdSmεmkl∂k∂lωij, (9.25)

where SB is some surface enclosed by the Burgers circuit. Since the result is inde-pendent of the size, shape, and position of the Burgers circuit as long as it enclosesthe defect line L, this implies

εmkl∂k∂lωij(x) = 0 (9.26)

everywhere away from L, which is what we wanted to show.In fact, the constancy of the jump in ωij could have been derived somewhat more

directly, without going through (9.23)–(9.26), by taking again the curves C+, C− onFig. 9.15 and calculating

∆ωij(1) − ∆ωij(2) =∫ 2+

1+

C+

dxk∂kωij −∫ 2−

1−

C−

dxk∂kωij = −∫ 2−

1−

C−

dxk∆(

∂kωij)

. (9.27)

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248 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

From the assumptions (9.11) and (9.12), together with (9.18), we see that ωij(x)does not jump across the Volterra surface S. But then (9.27) shows us that ∆ωij isa constant.

Let us now consider the displacement field itself. As a result of Weingarten’stheorem, the integral over the Burgers circuit B2 in Fig. 9.15 gives

∆ui(2)∮

B2

dui = ∆u(1) − Ωij

[

xj(2) − xj(1)]

. (9.28)

∆ui(1) − Ωij

[

xj(2) − xj(1)]

=∮

B2

dxk

uik +[

xj − xj(2)]

∂kωij

. (9.29)

Here we observe that the factors of xi(2) can be dropped on both sides by (9.24) and∆ωij = Ωij . By Stokes’ theorem, the remaining equation then becomes an equation

for the surface integral over SB2 ,

∆ui(1) + Ωijxj(1) =∫

SB2

dSl εlmk∂m(

uik + xj∂kωij)

=∫

SB2

dSl εlmk[

(∂muik + ∂kωm) + xj∂m∂kωij]

. (9.30)

This must hold for any size, shape, and position of the circuit B2 as long as itencircles the defect line L. For all these different configurations, the left-hand sideof (9.30) is a constant. We can therefore conclude that

SdSl

[

εlmk (∂muik + ∂kωim) + xjεlmk∂kωij]

= 0 (9.31)

for any surfaces S which does not enclose L. Moreover, from (9.23) we see thatthe last term cannot contribute. The first two terms, on the other hand, can berewritten, using the same decomposition of ∂kωim as in (9.18), in the form

−∫

SdSl εlmk (Skmi − Smik + Sikm) =

SdSl εlmkSmki, (9.32)

where we have abbreviated

Skmi(x) ≡ 1

2(∂k∂m − ∂m∂k)ui(x). (9.33)

Since this has to vanish for any S, we conclude that away from the defect line, thedisplacement field ui(x) also satisfies the integrability condition

(∂k∂m − ∂m∂k)ui(x) = 0. (9.34)

On the line L, the integrability conditions for ui and ωij are, in general, both violated.Let us first consider ωij. In order to give the constant result ∆ij(x) ≡ Ωij in (9.25)the integrability condition must be violated by a singularity in the form of a δ-function along the line L (4.10), namely:

εlmk∂kωij = δl(x;L). (9.35)

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9.9 Dislocation and Disclination Densities 249

Then (9.25) gives ∆ωij = Ωij via the formula

SBdSl δl(x;L) = 1. (9.36)

In order to see how the integrability condition is violated for ui(x), consider nowthe integral (9.30) and insert the result (9.32). This gives

∆ui(1) + Ωijxj(1) =∫

SB2

dSl εlmk(

Smki + xj∂m∂kωij)

. (9.37)

The right hand side is a constant independent of the position of the surface SB2 .This implies that the singularity along L is of the form

εlmk(

∂m∂kui + xj∂m∂kωij)

= biδl(x;L), (9.38)

where we have introduced the quantity

bi ≡ ∆ui(1) + Ωijxj(1). (9.39)

Inserting (9.35) into (9.38) leads to the following violation of the integrability con-dition for ui(x) along L,

εlmk∂m∂kui =(

bi − Ωijxj)

δl(x;L). (9.40)

In terms of the tensor (9.33), this reads

εlmkSmki =(

bi − Ωijxj)

δl(x;L). (9.41)

9.9 Dislocation and Disclination Densities

The violation of the integrability condition for displacement and rotation fields pro-portional to δ-functions along lines L is analogous to the situation in the multivalueddescription of the magnetic field in the last chapter. The analogy can be carriedfurther. Consider, for example, the current density of magnetism in Eq. (4.7), whichby Eqs. (4.30) and Eqs. (4.36) can be rewritten in the multivalued description as

ji(x) = εijk∂jBk =I

4πεijk∂j∂kΩ = Iδi(x;L), (9.42)

where Ω is the solid angle under which the loop L is seen from the point x. Byanalogy, we introduce densities for dislocations and disclinations, respectively, asfollows:

αij(x) ≡ εikl∂k∂luj(x), (9.43)

θij(x) ≡ εikl∂k∂lωj(x), (9.44)

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250 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

where we have used the vector form of the rotation field ωi = (1/2)εijkωjk, in orderto save one index. For the general defect line along L, these densities have the form

αij(x) = δi(x;L)(

bi − Ωjkxk)

, (9.45)

θij(x) = δi(x;L)Ωj , (9.46)

where Ωi = (1/2)εijkΩjk is the Frank vector.Note that in terms of the tensor Sijk of Eq. (9.33), the dislocation density (9.43)

reads

αij(x) ≡ εiklSlkj. (9.47)

In (9.45) and (9.46) the rotation by Ω is performed around the origin. Obviously, theposition of the rotation axis can be changed to any other point x0 by a simple shift

in the constant bj → b′j + ( × x0)j. Then αij(x) = δi(x;L)

b′j + ( × (x − x0)j

.Note that due to the identity

∂iδi(x;L) = 0 (9.48)

for closed lines L, the disclination density satisfies the conservation law

∂iθij = 0, (9.49)

which implies that disclination lines are always closed. This is not true for mediawith a directional field, e.g., nematic liquid crystal. Such media are not consideredhere since they cannot be described by a displacement field alone. Differentiating(9.45) we find the conservation law for disclination lines ∂iαij = −Ωijδi(x;L) which,in turn, can be expressed in the form

∂iαij = −εjklθkl. (9.50)

In terms of the tensor Sijk, this becomes

εjkl (∂iSkli + ∂kSlnm − ∂lSknn) = −εjklθkl. (9.51)

Indeed, inserting Sklj = (1/2)εkliαij from (9.47), this reduces to the conservationlaw (9.50) for the dislocation density.

From the linearity of the relations (9.43) and (9.44) in uj and ωj, respectively, itis obvious that these conservation laws remain true for any ensemble of infinitesimaldefect lines. The conservation law (9.50) may, in fact, be derived by purely differen-tial techniques from the first smoothness assumption (9.11). Using Stokes’ theorem,∆uij can be expressed in the same way as ∆ωij in (9.25), and by the same argumentas the one used for ωij we conclude that the strain is an integrable function in allspace and satisfies

(∂i∂k − ∂k∂i) ulj(x) = 0. (9.52)

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9.9 Dislocation and Disclination Densities 251

If we then look at αij in the general definition (9.43), rewrite it as

αij = εikl∂k∂luj = εikl∂k(

ulj + ωlj)

= εikl∂kulj + δij∂kωk − ∂jωi, (9.53)

and apply the derivative ∂i, this gives directly (9.50).In a similar way, the conservation law (9.49) can be derived by combining both

smoothness assumptions (9.11) and (9.12). The first can be stated, via Stokes’theorem, as an integrability condition for the derivative of strain, i.e.,

(∂l∂n − ∂n∂l) ∂kuij(x) = 0. (9.54)

Let us recall that from the first assumption (9.11) we have concluded in (9.18) that∂kωij(x) is also a completely smooth function across the surface S. Hence, ∂kωijmust also satisfy the integrability condition

(∂l∂n − ∂n∂l) ∂kωij(x) = 0.

Together with (9.54) this implies that ∂k∂iuj(x) is integrable:

(∂l∂n − ∂n∂l) ∂k∂iuj(x) = 0. (9.55)

If we write down this relation three times, each time with l, n, k exchanged cyclically,we find

∂lRnkij + ∂nRklij + ∂kRlnij = 0, (9.56)

where Rnkij is an abbreviation for the expression,

Rnkij = (∂n∂k − ∂k∂n) ∂iuj(x). (9.57)

Contracting k with i and l with j gives us

∂jRniij + ∂nRijij + ∂iRjnij = 0. (9.58)

Now we observe that because of (9.52), Rnkij is anti-symmetric not only in n and kbut also in i and j so that

2∂jRinji − ∂nRijji = 0.

This, however, is the same as

2∂j

(

1

4εjpqεnklRpqkl

)

= 0, (9.59)

as can be verified using the identity

εjpqεnkl = δjnδpkδql + δjkδplδqn + δjlδpnδqk − δjnδplδqk − δjkδpnδql − δjlδpkδqn. (9.60)

Recalling now the definition (9.57) and ωn = (1/2)εnkl∂kul, Eq. (9.59) becomes

2εipq∂i∂p∂qωn = 0, (9.61)

and this is precisely the conservation law ∂iθik = 0 for disclinations (9.49), whichwe wanted to prove. The educated reader will have noted the appearance of torsionand curvature in Eqs. (9.33) and (9.57), and Eqs. (9.51) and (9.56) as the linearizedfundamental identity and the linearized Bianchi identity, to be discussed in detailin Sections 12.1 and 12.5 [see Eqs. (12.108) and (12.120), respectively].

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252 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Figure 9.16 Illustration of Volterra process in which an entire volume piece is moved

with the vector bi.

9.10 Mnemonic Procedure for ConstructingDefect Densities

There exists a simple mnemonic procedure for constructing the defect densities andtheir conservation laws. This we now explain.

Suppose we perform the Volterra cutting procedure on a closed surface S, di-viding it mentally in two parts, joined along some line L (see Fig. 9.16). On onepart of S, say S+, we remove material of thickness bi and on the other we add thesame material. This corresponds to a simple translational movement of crystallinematerial by bi, i.e., to a displacement field

ul(x) = −δ(x;V )bl, (9.62)

where the δ-function on a volume V was defined in Eq. (4.29). By this transformationthe elastic properties of the material are unchanged.

Consider now the distortion field ∂kul(x). Under (9.62), it changes by

∂kul(x) → ∂kul(x) − ∂kδ(x;V )bl. (9.63)

The derivative of the δ-function is singular only on the surface of the volume V . Infact, in Eq. (4.34) we already derived the formula ∂kδ(x;V ) = −δk(x;S), so that(9.63) reads

∂kul(x) → ∂kul(x) + δk(x;S)bl. (9.64)

From this trivial transformation we can now construct a proper dislocation line byassuming S to be no longer a closed surface but an open one, i.e. we may restrict Sto the shell S+ with a boundary L. Then we can form the dislocation density

αil(x) = εijk∂j∂kul(x) = εijk∂jδk(x;S)bl. (9.65)

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9.10 Mnemonic Procedure for Constructing Defect Densities 253

The superscript + was dropped. Using Stokes’ theorem for the δk(x;S)-function inthe form (4.23), this becomes simply

αil(x) = δi(x;L)bl. (9.66)

For a closed surface, this vanishes.For a general defect line, the starting point is the trivial Volterra operation

of translating and rotating a piece of crystalline volume. This corresponds to adisplacement field

ul(x) = −δ(x;V )(

bl + εlqrΩqxr)

. (9.67)

If we now form the distortion, we find

∂kul(x) = δk(x;S)(

bl + εlqrΩqxr)

− δ(x;V )εlqkΩq. (9.68)

In the expression it is still impossible to assume S to be an open surface. If we,however, form the symmetric combination, the strain tensor

ukl =1

2(∂kul + ∂luk) =

1

2

[

δk(x;S)(

bl + εlqrΩqxr)

+ (kl)]

(9.69)

is well defined for an open surface, in which case we shall refer to ukl as the plasticstrain and denote it by upkl. The field

βpkl ≡ δk(x;S)(

bl + εlqrΩqxr)

(9.70)

plays the role of a dipole density of the defect line across the surface S. It is usuallycalled a plastic distortion. It is a single valued field (i.e., derivatives in front of itcommute). In terms of βpkl, the plastic strain is simply

upkl =1

2(βpkl + βpkl) . (9.71)

The full displacement field (9.67) is not defined for an open surface due to theδ(x;V ) term. It is multiple valued. The dislocation density, however, is singlevalued. Indeed, we can easily calculate

αil = εijk∂j∂kul(x) = εijk∂j[

δk(x;S)(

bl + εlqrΩqxr)

− δ(x;V )εlqkΩq

]

= δi(x;L)(

bl + εlqrΩqxr)

, (9.72)

and see that this is the same as (9.45).Let us now turn to the disclination density θpj = εpmn∂m∂nωj . From (9.67) we

find the gradient of the rotation field

∂nωj =1

2εjkl∂n∂kul

=1

2εjkl∂n

[

δk(x;S)(

bl + εlqrΩqxr)

− δ(x;V )εlqkΩq

]

=1

2εjkl∂nβ

pkl + δn(x;S)Ωj. (9.73)

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254 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

This is defined for an open surface S in which case it is called the plastic bend-twistand denoted by κpnj ≡ ∂nω

pj . It is useful to define the plastic rotation

φpnj ≡ δn(x;S)Ωj, (9.74)

which plays the role of a dipole density for disclinations. With this, the plasticgradient of ωj is given by

κpnj = ∂nωpj =

1

2εjkl∂nβ

pkl + φpnj. (9.75)

We can now easily calculate the disclination density:

θpj = εpmn∂m∂nωj = εpmn∂mκpnj =

1

2εjklεpmn∂m∂nβ

pkl + εpmn∂mφ

pnj.

In front of βpkl, the derivatives commute [see (9.70)] so that the first term vanishes.Use of Stokes’ theorem on the second term gives

θpj = εpmn∂mφpnj = δp(x;L)Ωj , (9.76)

in agreement with (9.48).

Note that according to the second line of (9.72), the dislocation density can alsobe expressed in terms of βpkl and φpli as

αil = εijk∂jβpkl + δilφ

ppp − φpli, (9.77)

In fact, this is a direct consequence of the decomposition (9.53), which can be writtenin terms of plastic strain and bend-twist as

αij = εikl∂kuplj + δijκ

pqq − κpji (9.78)

Expressing upli in terms of βpli and κpij in terms of φpij [see (9.71) and (9.75)] we find

αij =1

2εikl∂kβ

plj + δijφ

pqq − φpqq − φpji +

1

2

(

εijk∂kβpjl + δijεqkl∂qβ

pkl − εikl∂iβ

pkl

)

.

But the quantity inside the parentheses is equal to 12εikl∂kβ

plj, as can be seen from

applying the identity

δijεqkl = δjqεikl + δjkεilq + δjl + δiqk

to ∂qβkl. Thus αij takes again the form (9.77).

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9.11 Defect Gauge Invariance 255

9.11 Defect Gauge Invariance

A given defect distributions can be derived from many plastic strains and rotations.This is an obvious consequence of the freedom in choosing the Volterra surfacesfor the construction of the defect lines. They run along the boundary lines of thesesurfaces. The ambiguity can be formulated mathematically as a gauge freedom. Thishas already been seen in Subsection 4.6 in the context of a gradient representationof the magnetic field of a current loop. The current density was given in Eq. (4.91)by a curl of a δ-function on a surface:

j(x) = I∇ × (x;S), (9.79)

and this representation was invariant under a gauge transformation, which ariseswhen shifting the surface S to a new position S ′ with the same boundary line:

(x;S) → (x;S ′) = (x;S) − ∇δ(x;V ). (9.80)

This gauge invariance can be found in all expressions for defect densities whichinvolve δ-functions over Volterra surfaces S. The relevant gauge transformationscan be derived from the basic translational movement (9.63) which does not changethe defect configurations but does change the plastic fields. Thus we perform whatwe may call the basic trivial Volterra transformation

ul(x) → ul(x) − δ(x;V )bl. (9.81)

According to Eq. (9.64), the plastic distortion field changes by

βpkl → βpkl − ∂kδ(x;V )bl. (9.82)

This has the typical form of a gauge transformation, where the gradient of an ar-bitrary field is added to a gauge field. In the context of defects there is, however,an important difference: the function is not completely arbitrary but contains aδ-function over an arbitrary volume. The Burgers vector accounts for the latticeproperties.

Comparison with Eqs. (9.84) and (9.72) shows that the transformation (9.82)corresponds to a shift of the Volterra surface S → S ′ if the defect is a pure dislocationline. If it contains also a Frank vector Ωk, we must supplement the translation (9.62)by a rotation and transform

ul(x) → ul(x) − δ(x;V )bl − εlmnδ(x, V )Ωmxn. (9.83)

The we find from (9.84) that

∂kul(x) → ∂kul(x) − ∂kδ(x;V )(

bl + εlqrΩqxr)

− δ(x;V )εlqkΩq, (9.84)

so that the plastic strain transforms like

upkl → upkl −1

2

[

∂kδ(x;V )(

bl + εlqrΩqxr)

+ (kl)]

. (9.85)

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256 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Comparison with (9.69) shows that this corresponds, via Eq. (9.80), precisely to ashift of the Volterra surface S zo S ′. The change of the plastic distortion βpkl inEq. (9.70) is, however, not yet obtained, due to the last term in (9.84). This isremoved if we accompany (9.83) by a trivial Volterra transformation of the rotationfield

ωj =1

2εjkl∂kul → ωj − δ(x;V )Ωj . (9.86)

Adding this to (9.84), the last term is removed and we obtain the combined trans-formation

∂kul(x) → ∂kul(x) − ∂kδ(x;V )(

bl + εlqrΩqxr)

. (9.87)

This is precisely the gauge transformation of the plastic distortion (9.70) if theVolterra surface is shifted from S to S ′. Simultaneously, the gradient of the rotationfield ω in Eq. (9.73) undergoes the defect gauge transformation

∂nωj → ∂nωj − ∂nδ(x;V )Ωj , (9.88)

corresponding to a gauge transformation of the plastic rotation (9.74) transformslike

φpnj → φpnj − ∂nδ(x;V )Ωj, (9.89)

The vortex gauge transformations (9.82), (9.85), and (9.89) summarize the invari-ance of a defect line under the shift of the Volterra surface.

9.12 Branching Defect Lines

We recall that from the geometric point of view, the conservation laws state thatdisclination lines never end and dislocations end at most at a disclination line.Consider, for example, a branching configuration where a line L splits into two linesL and L′. Assign an orientation to each line and suppose that their disclinationdensity is

θij(x) = Ωiδj(x;L) + Ω′iδj(x;L′) + Ω′′i δj(x;L′′), (9.90)

with their dislocation density being

αij(x) = δi(x;L)

bj + [ × (x − x0)]j

+ δi(x;L′)

b′j +[

Ω′ ×(

x − x′0

)]

j

δi(x;L′)

b′′j +[

Ω′′ ×(

x − x′′0

)]

j

. (9.91)

The conservation law ∂iθij = 0 then implies that the Frank vectors satisfy theequivalent of Kirchhoff’s law for currents

Ωj + Ω′′i = Ω′i. (9.92)

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9.13 Defect Density and Incompatibility 257

This follows directly from the identity for lines

∂iδi(x;L) =∫

dsdxidsδ(3) (x − x(3)) = δ(3)(x − xi) − δ(3)(x − xf ),

where xi and xf are the initial and final points of the curve L. The conservationlaw ∂iαij = εiklθkl, on the other hand, gives

bi − [ × (x − x0)]i + b′′i −

[

Ω′′ ×(

x − x′′0

)]

i= b′i −

[ ×(

x′ − x′0

)]

i. (9.93)

If the same position is chosen for all rotation axes, the Burgers vectors bi satisfyagain a Kirchhoff-like law:

bi + b′′i = b′i. (9.94)

But Burgers vectors can be compensated for by different rotation axes, for example,L′ and L′′ could be pure disclination lines with different axes through x′0,x

′′0 and L′

a pure dislocation line with x′ = −Ω′ × (x′0 − x′′0) which ends on L′, L′′. Equation(9.92) renders different choices equivalent.

9.13 Defect Density and Incompatibility

As far as classical linear elasticity is concerned, the information contained in αij andθij can be combined into a single symmetric tensor, called the defect density ηij(x)[In higher gradient elasticity this is no longer true; see chapter 18.]. It is defined asthe double curl of the strain tensor

ηij(x) ≡ εiklεjmn∂k∂muln(x). (9.95)

In order to see its relation with αij and θij , we take (9.43) and contract the indicesi and j, obtaining

αii = 2∂iωi. (9.96)

Using this, (9.53) can be written in the form

εikl∂kuln = ∂nωi −(

−αin +1

2δinαkk

)

. (9.97)

The expression in parentheses was first introduced by Nye and called contortion2

Kni ≡ −αin +1

2δinαkk. (9.98)

2In terms of the plastic quantities introduced in the last section the plastic part of Kij reads

Kpij = −εikl∂kβp

li +1

2δijεnkl∂kβp

ln + φpij .

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258 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

The inverse relation is

αij = −Kji + δijKkk. (9.99)

Multiplying (9.95) by εjmn∂m, we find with (9.44)

ηij = εjmnεikl∂m∂kuln = εjmn∂m∂nωi − εjmn∂mKni

= θij − εjmn∂mKni. (9.100)

The final expression is not manifestly symmetric. Let us verify that it is, neverthe-less. Contracting it with the antisymmetric tensor εlij, we find εlijθij∂lKii−∂iKli =εlijθijθij + ∂iαil. But this vanishes due to the conservation law (9.50) for the dislo-cation density. Thus ηij is symmetric.

There is yet another version of the decomposition (9.100) which is obtainedafter applying the identity εijnδmq + εjmnδiq + εminδjq = εijmδnq to ∂mαqn giving

εnjm∂m(

αin − 12δinαkk

)

= −12∂m

(

εmjnαin + (ij) + εijnαmn)

. Hence

ηij = θij −1

2∂m

(

εminαjn + (ij) − εijnαmn)

. (9.101)

This type of decomposition will be encountered in the context of general relativitylater in Part IV.

The double curl operation is a useful generalization of the curl operation onvector fields to symmetric tensor fields. Recall that the vanishing of a curl of avector field E implies that E can be written as the gradient of a scalar potentialφ(x) which satisfies the integrability condition (∂i∂j − ∂j∂i)φ(x) = 0:

∇ × E = 0 ⇒ Ei = ∂iφ(x). (9.102)

The double curl operation implies a similar property for the symmetric tensor, aswas shown a century ago by Riemann and by Christoffel. If the double curl of asymmetric tensor field vanishes everywhere in space, his field can be written as thestrain of some displacement field ui(x) which is integrable in all space [i.e., it satisfies(9.34)]. We may state this conclusion briefly as follows:

εiklεjmn∂k∂muln(x) = 0 ⇒ uij =1

2

(

∂iuj + ∂jui)

. (9.103)

If the double curl of uln(x) is zero one says that uln(x is compatible with a displace-ment field and calls the double curl the incompatibility , i.e.,

(inc u)ij ≡ εiklεjmn∂k∂muln. (9.104)

The proof of statement (9.103) follows from (9.102) for a vector field: we simplyobserve that every vector field Vk(x) vanishing at infinity and satisfying the inte-

grability condition(

∂i∂j − ∂j∂i)

Vk(x) = 0 can be decomposed into transverse and

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9.13 Defect Density and Incompatibility 259

longitudinal pieces, namely, a gradient whose curl vanishes and a curl whose gradientvanishes,

Vi = ∂iϕ+ εijk∂jAk, (9.105)

both fields ϕ and Ak being integrable. Explicitly these are given by

ϕ =1

∂2∂iVi, (9.106)

Ak = − 1

∂2 εklm∂lVm + ∂kC, (9.107)

where 1/∂2 is a short notation for the Coulomb Green function (1/∂2)(x,x′) whichacts on an arbitrary function in the usual way:

− 1

∂2 f(x) ≡∫

d3x1

4π|x − x′|f(x′) (9.108)

is the Coulomb Green function. Note that the field Ak is determined by (9.107) onlyup to an arbitrary pure gradient ∂kC.

By repeated application of this formula, we find the decompositions of an arbi-trary, not necessarily symmetric, tensor uil:

uil = ∂iϕ′l + εijk∂jA

′kl = ∂iϕ

′l + εijk∂j (∂lϕk + εlmn∂mAkn) . (9.109)

Setting

ϕ′′i ≡ εijk∂jϕk, (9.110)

this may be cast as

uil = ∂iϕ′l + ∂lϕ

′′i + εijkεlmn∂j∂mAkn. (9.111)

For the special case that uil is symmetric we can symmetrize this result and decom-pose it as

uil = ∂iuj + ∂jui + εijkεlmn∂j∂mASkn, (9.112)

where

ui =1

2

(

ϕ′i + ϕ′′i)

, (9.113)

and ASkn is the symmetric part of Akn, both being integrable fields. The first termin (9.112) has zero incompatibility, the second has zero divergence when applied toeither index.

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260 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

In the general case, i.e., when there is no symmetry, we can use the formulas(9.106), (9.107) twice and determine the fields ϕ′l, ϕ

′′i , Akn as follows:

ϕ′l =1

∂2∂kukl, (9.114)

A′kl = − 1

∂2 εkpq∂puql + ∂kCl, (9.115)

ϕk = − 1

∂4 εkpq∂p∂luql +1

∂2∂k∂lCl, (9.116)

Akn = − 1

∂4 εklmεnpqumq + ∂k

(

− 1

∂2 εnjl∂jCl

)

+ ∂nDk, (9.117)

so that from (9.109)

ϕ′′i = − 1

∂4∂i∂p∂qupq +1

∂2∂luil. (9.118)

Reinserting this into decomposition (9.111) we find the identity

uil =1

∂2 (∂i∂kukl + ∂l∂kuik) −1

∂4∂i∂l(

∂p∂qupq)

+1

∂4 εijkεlmn∂j∂m(

εkprεnqs∂p∂qurs)

, (9.119)

which is valid for any tensor of rank two. This may be verified by working out thecontractions of the ε tensors.

While the statements (9.102) and (9.103) for vector and tensor fields are com-pletely analogous to each other, it is important to realize that there exists an im-portant difference between the two. For a vector field with no curl, the potentialcan be calculated uniquely (up to boundary conditions) from

ϕ =1

∇2∂iEi. (9.120)

This is no longer true, however, for the compatible tensor field uil. The point ofdeparture lies in the non-uniqueness of functions ϕ′l and ϕ′′i in the decomposition(9.111). They are determined only modulo a common arbitrary local rotation fieldωi(x). In order to see this we perform the replacements

∂iϕ′l(x) → ∂iϕ

′l(x) + εilqωq(x), (9.121)

∂lϕ′′i (x) → ∂lϕ

′′i (x) + εliqωq(x), (9.122)

and see that (9.111) is still true. The field (9.113) is only a particular example of adisplacement field which has the strain tensor equal to the given ukl

u0kl =

1

2

(

∂ku0l + ∂lu

0k

)

= ukl. (9.123)

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9.13 Defect Density and Incompatibility 261

This displacement field may not, however, be the true displacement field ul(x) inthe crystal, which also satisfies

1

2(∂kul + ∂luk) = ukl (9.124)

In order to find the latter, we need additional information on the rotation field

ωkl =1

2(∂kul − ∂luk) . (9.125)

We must know both ukl(x) and ωkl(x) to calculate

∂kul(x) = ukl(x) + ωkl(x) (9.126)

and solve this equation for ul(x).In order to make use of this observation we have to be sure that ωi = 1

2εijkωjk

can be written as the curl of a displacement field ui(x). This is possible if

∂iωi = εijk∂i∂juk = 0, (9.127)

which implies that [see (9.96)]

αii(x) = 0. (9.128)

In later discussions we shall be confronted with the situation in which ukl and ∂iωjare both given. In order to obtain ωi from the latter we have to make sure that ωiis an integrable field, which is assured by the constraint

θij = εikl∂k∂lωj = 0. (9.129)

Thus we can state the following important result: Suppose a crystal is subject to astrain ukl(x) and a rotational distortion ωkl(x). There exists an associated single-valued displacement field ul(x), if and only if the crystal possesses a vanishing defectdensity ηij(x), a vanishing disclination density θij(x), and a vanishing αii = 0, i.e.,

ηij(x) = 0, θij(x) = 0, αii(x) = 0. (9.130)

Relation (9.101) implies that this is true if only two of these densities vanish:

ηij(x) = 0, αij(x) = 0, (9.131)

or

θij(x) = 0, αij(x) = 0. (9.132)

Note that it is possible to introduce, into a given elastically distorted crystal, nonzerorotational and translational defects in such a way that θij and αij in (9.101) canceleach other. Then the elastic distortions do in fact remain unchanged. The localrotation field, however, can be changed drastically. In particular it may no longerbe integrable.

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262 9 Multivalued Mapping from Ideal Crystals to Crystals with Line-Like Defects

Notes and References

[1] The first observations and theoretical studies of dislocations are found inO. Mugge, Neues Jahrb. Min. 13 (1883);A. Ewing and W. Rosenhain, Phil. Trans. Roy. Soc. A 193, 353 (1899);J. Frenkel, Z. Phys. 37, 572 (1926);E. Orowan, Z. Phys. 89, 605, 634 (1934);M. Polanyi, Z. Phys. 89, 660 (1934);G.I. Taylor, Proc. Roy. Soc. A 145, 362 (1934);J.M. Burgers, Proc. Roy. Soc. A 145, 362 (1934);F. Kroupa and P.B. Price, Phil. Mag. 6, 234 (1961);W.T. Read, Jr. , Disclinations in Crystals , McGraw-Hill, New York, 1953;F.C. Frank and W.T. Read, Jr. , Phys. Rev. 79, 722 (1950);B.A. Bilby, R. Bullough, and E. Smith, Proc. Roy. Soc. A 231, 263 (1955);Disclination lines were described byF.C. Frank, Disc. Farad. Soc. 25, 1 (1958).More details can be found in the books quoted below.Weingarten’s theorem is derived inF.R.N. Nabarro, Theory of Dislocations , Clarendon Press, Oxford, 1967;C. Truesdell and R. Toupin, in Handbook of Physics , Vol III(1), e.d S. Flugge,Springer, Berlin 1960.The plastic strain described in Section 2.9 was used efficiently byJ.D. Eshelby, Brit. J. Appl. Phys. 17, 1131 (1966),to calculate elastic field configurations. The plastic distortions and rotationswere introduced byT. Mura, Phil. Mag. 8, 843 (1963), Arch. Mech. 24, 449 (1972),Phys. Stat. Sol. 10, 447 (1965), 11, 683 (1965).See also his book Micromechanics of Defects in Solids, Noordhoof, Amster-dam, 1987.For an excellent review seeE. Kroner, in The Physics of Defects , eds. R. Balian et al. , North-Holland,Amsterdam, 1981, p. 264.Further useful literature can be found inJ.P. Friedel, Disclinations , Pergamon Press, Oxford, 1964.See also the 1980 lectures at les Houches in The Physics of Defects ,eds. R. Balian et al. , North-Holland, Amsterdam, 1981, p. 264;M. Kleman, The General Theory of Disclinations, in Dislocations in Solids ,ed. F.R.N. Nabarro, North-Holland, Amsterdam, 1980;W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin,1970.B. Henderson, Defects in Crystalline Solids , Edward Arnold, London, 1972.

[2] We follow closely the theory presented inH. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore,

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Notes and References 263

1989, Vol. II, Part IV, Differential Geometry of Defects and Gravity with Tor-sion, p. 1432 (kl/b2).

[3] For details on vortex lines see inH. Kleinert, Gauge Fields in Condensed Matter , World Scientific, Singapore,1989, Vol. I, Superflow and Vortex Lines, pp. 1–742 (kl/b2).

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Women love us for our defects. If we have enough of them,

they will forgive us everything, even our intellects.

Oscar Wilde (1854 - 1900)

10Defect Melting

In Chapter 5 we have seen that the phase transitions in superfluid helium and insuperconductors can be explained by a the proliferation of vortex lines at the criticaltemperature. A similar proliferation mechanism of dislocation and disclination linescan be shown to lead to the melting of crystals.

10.1 Specific Heat

The specific heat of solids has several parallels with the specific heat of the λ-transition. For low temperature it starts out like T 3, as a signal for massless exci-tations [see Fig. 10.1]. These are the longitudinal and transverse phonons, whichare the Goldstone modes caused by the fact that the crystalline ground state breaksspontaneously the translational symmetry of the Hamiltonian. For higher tempera-tures the specific heat saturates at a value 6 × 1/2kBN as required by the Dulong-Petit rule. which requires the value 1/2 for each particle and harmonic degree offreedom (3 potential and three kinetic degrees).

The transition between the two regimes lies at the Debye temperature ΘD whichis determined by the longitudinal and transversal sound velocities cLs , c

Ts and the

particle density n ≡ N/V . For one atom per lattice cell end three equal soundvelocities, it is given by

ΘD = 2πhcskB

(

2n

)1/3

. (10.1)

The internal energy is given by the universal Debye function

D(z) ≡ 3

z3

∫ z

0

x3

ex − 1(10.2)

as

U = 3NkBTD(ΘD/T ) (10.3)

The specific heat follows from this:

C =∂U

∂T= 3NkBT

[

D(ΘD/T ) − (ΘD/T )D′(ΘD/T )]

. (10.4)

264

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10.2 Elastic and Plastic Energies 265

Using the limiting behavior

D(z) =

π2

5z3 − 3e−z + . . . for z 1,

1 − 3

8z2 + . . . for z 1.

(10.5)

we find

C = 3NkB

4π4

5

(

T

ΘD

)3

, for T ΘD,

1 for T ΘD.(10.6)

The result agrees well with experiments as shown in Fig. 10.1.

T/ΘD

Figure 10.1 Specific heat of various solids. By plotting the data against the ratio T/ΘD,

where ΘD is the Debye temperature, the data fall on a universal curve. The insert lists

ΘD-values and melting temperature Tm.

10.2 Elastic and Plastic Energies

In crystals, the elastic energy is usually expressed in terms of a material displacementfield ui(x) as

E =∫

d3x

[

µu2ij(x) +

λ

2u2ii(x)

]

, (10.7)

where µ is the shear module, λ the Lame constant, and

uij(x) =1

2[∂iuj(x) + ∂juj(x)] (10.8)

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266 10 Defect Melting

is the strain tensor (9.69). The elastic energy goes to zero for infinite wave lengthsince in this limit ui(x) reduces to a pure translation and the energy of the systemis translationally invariant. The crystallization process causes a spontaneous break-down of the translational symmetry of the system. The elastic distortions describethe Nambu-Goldstone-modes resulting from this symmetry breakdown.

As discussed in the previous chapter, a crystalline material always contains de-fects. In their presence, the elastic energy is

E =∫

d3x

[

µ(uij − upij)

2 +λ

2(uii − up

ii)2

]

. (10.9)

where upij is so-called plastic strain tensor (9.71) describing the defects.

The above energy is the continuum limit of a lattice energy which we shallassume, for mathematical simplicity, to be simple cubic of spacing 2π. Then theenergy of ui(x) and ui(x)+2πNi(x) must be indistinguishable for any integer-valuedfield Ni(x), which correspond to permutations of the lattice sites.

The plastic strain tensor allows for this ambiguity. It guarantees the defect gaugeinvariance of the energy (10.9). By analogy with the superfluid in Eq. (5.175) , wemay define the expectation value

Oi ≡ 〈Oi(x)〉 = 〈eui(x)〉 (10.10)

as an order parameter of the system. It will be nonzero in the crystalline phase sinceui(x) fluctuates around zero, and zero in the molten state in which ui(x) fluctuatethrough the entire crystal.

The plastic distortions contain three types of surfaces where the displacementfield ui(x) jumps by 2π, one for each lattice direction, characterized by the threeBurgers vectors b(1),(2),(3). They are describes by the plastic distortion of Eq. (9.70)

βpil(x) = δi(x;S)bl, (10.11)

where bl are the components of any of the three Burgers vectors b(i). The irrelevantsurfaces S are the Volterra surfaces of the dislocation lines.

We can now calculate the partition function of lattice fluctuations governed bythe energy (10.9) from the functional integral and the sum over all Volterra surfaces.

Z =∫

Du∑

S

e−H/kBT = e−βF , (10.12)

where β ≡ 1/kBT . This is most easily done by Monte-Carlo simulations. Theresulting specific heat near the melting transition is shown in Fig. 10.2.

It is possible to write down an elastic energy which disentangles dislocations anddisclinations by including higher gradients of the displacements field. This energyreads [1, 2]

E = µ∫

d3x[

(

uij − upij

)2+ `2

(

∂iωj − κpij

)2]

. (10.13)

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10.2 Elastic and Plastic Energies 267

β

CV3NkB

Figure 10.2 Specific heat of various solids. By plotting the data against the ratio T/ΘD,

where ΘD is the Debye temperature, the data fall on a universal curve. The insert lists

ΘD-values and melting temperature Tm.

The parameter ` is the length scale over which the crystal is rotationally stiff.

The partition function contains integrals over ui and sums over the jumping sur-faces of dislocations and disclinations. By integrating out the ui-fields, one obtains aBiot-Savart type of interaction energy between the defect lines in which dislocationline elements interact with each other via a Coulomb potential, whereas disclinationline elements interact via a linearly rising potential.

It is again possible to eliminate the jumping surfaces from the partition functionby introducing conjugate variables and associated stress gauge fields. For this werewrite the elastic action of defect lines as [1, 3]

E =∫

d3x

[

1

(

σij + σji)2

+1

8`2τ 2ij

+iσij(

∂iuj − εijkωk − βpij

)

+ iτij(

∂iωj − φpij

)]

, (10.14)

and form the partition function by integrating over σij , τij, ui, ωj and summing overall jumping surfaces S in the plastic fields. A functional integral over the antisym-metric part of σij has been introduce the obtain an independent integral over ωi [ifwe were to integrate out the antisymmetric part of σij, we would enforce the relation

ωi = 12εijk(∂juk + βp

ij)]. By integrating out ωj and ui, we find the conservation laws

∂iσij = 0, ∂iτij = −εjklσkl. (10.15)

These are dual to the conservation laws for dislocation and disclination densities(9.50) and (9.49).

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268 10 Defect Melting

Figure 10.3 Separation of first-order melting transition into two successive Kosterlitz-

Thouless transitions in two dimension when increasing the length scale ` of rotational

stiffness of the defect model (after Ref. [1]).

They can be enforced as Bianchi identities by introducing the stress gauge fieldsAij and hij and writing

σij = εikl∂kAljτij = εikl∂khlj + δijAll − Aji. (10.16)

This allows us to reexpress the energy as

E =∫

d3x[

1

4

(

σij + σji)2

+1

8l2τij

2 + Aijαij + hijθij

]

. (10.17)

The stress gauge fields couple locally to the defect densities which are singular onthe boundary lines of the Volterra surfaces. In the limit of a vanishing length scale`, τij is forced to be identically zero and (10.16) allows us to express Aij in terms ofhij . Then the energy becomes

E =∫

d3x[

1

4

(

σij + σji)2

+ hijηij

]

, (10.18)

where the defect density ηij contains dislocation and disclination lines.

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10.2 Elastic and Plastic Energies 269

Figure 10.4 Phase diagram in the T -`-plane in two-dimensional melting (after Ref. [1])

.

Depending on the length parameter ` of rotational stiffness, the defect systemwas predicted to have either a single first-order transition (for small `), of twosuccessive continuous melting transitions. In the first transitions, dislocation linesproliferate and destroy the translational order, in the second transition, disclinationlines proliferate and destroy the rotational order [2] (see Figs. 10.3 and 10.4).

The existence of two successive continuous transitions was conjectured a longtime ago [4, 5, 6, 7] for two dimensional melting, where these transitions would beof the Kosterlitz-Thouless type. However, the simplest lattice defect models con-structed to illustrate this behavior displayed only a single first-order transition [8].Only after introducing the angular stiffness ` was it possible to separate the first-order melting transition into two successive Kosterlitz-Thouless transitions [9], thusconfirming theoretical predictions made by the author in 1983 [3]. The dependenceon ` is shown in Figs. 10.3 and 10.4.

[1] H. Kleinert, Gauge fields in Condensed Matter , Vol. II: Stresses and De-fects, Differential Geometry, Crystal Defects, World Scientific, Singapore, 1989(kl/b2).

[2] H. Kleinert, Lett. Nuovo Cimento 37, 425 (1983). (kl/97).

[3] H. Kleinert, Phys. Lett. A 130, 443 (1988) (kl/174).

[4] D. Nelson, Phys. Rev. B 18, 2318 (1978).

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270 10 Defect Melting

[5] D. Nelson and B.I. Halperin, Phys. Rev. B 19, 2457 (1979).

[6] A.P. Young, ibid., 1855 (1979).

[7] D. Nelson, Phys. Rev. B 26, 269 (1982).

[8] W. Janke and H. Kleinert, Phys. Lett. A 105, 134 (1984); (kl/120);Phys. Lett. A 114, 255 (1986) (kl/135).

[9] W. Janke and H. Kleinert, Phys. Rev. Lett. 61, 2344 (1988) (kl/179).

H. Kleinert, MULTIVALUED FIELDS

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Better die than live mechanically a life

that is a repetition of repetitions

D. H. Lawrence (1885 - 1930)

11Relativistic Mechanics in Curvilinear Coordinates

The basic idea which led Einstein to his formulation of the theory of gravitationin terms of curved spacetime was the observation by Galileo Galilei (1564-1642) allbodies would fall with equal velocity if in the absense of air friction. This observationwas confirmed with a higher accuracy by C. Huygens (1629-1695). In 1889 R. Eotvos[1]. found a simple trick to remove the air friction completely enabling him to limitthe relative difference between the falling speeds of wood and platinum to one partin 109. This implies that the inertial mass m which governs the acceleration of abody if subjected to a force f(t) in Newton’s equation of motion

m x(t) = f(t), (11.1)

which appears on the left-hand side of the equations of motion (1.2), and the grav-itational mass on the right-hand side of Eq. (1.2) cannot differ by more than thisextremely small amount.

11.1 Equivalence Principle

Einstein considered the result of the Eotvos experiment as evidence that inertialand gravitational masses are exactly equal. From this he concluded that the motionof all point particles under the influence of a gravitational field can be describedcompletely in geometric terms. The basic thought experiment which led him to thisconclusion consisted in imagining an elevator in a large sky scraper to fall freely.Since all bodies in it would fall with the same speed, they would appear weightless.Thus, for an observer inside the cabin, the gravitational attraction to the earth wouldhave disappeared. Einstein concluded that gravitational forces can be removed byacceleration. This is the content of the equivalence principle.

Mathematically, the cabin is just an accelerating coordinate frame. If the originalspacetime coordinates with gravity are denoted by xµ, the coordinates of the smallcabin are given by a function xµ(xa). Hence the equivalence principle states thatthe behavior of particles under the influence of gravitational forces can be foundby going to a new coordinate frame xa(xµ) in which the motion within the cabinproceeds without gravity.

271

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272 11 Relativistic Mechanics in Curvilinear Coordinates

There is a converse way of stating this principle. Given an inertial frame xa,we can simulate a gravitational field at a point by going to a small cabin with xµ,which is accelerated with respect to the inertial frame xa. In the coordinates xµ,the motion of the particle looks the same as if a gravitational field were present.

This suggests a simple way of finding the equations of motion of a point particlein a gravitational field: one must simply transform the known equations of motionin an inertial frame to arbitrary curvilinear coordinates xµ. When written in generalcoordinates xµ, the flat-spacetime equations must be valid also in the presence ofgravitational fields.

In formulating the equivalence principle it must be realized that by a coordinatetransformation the gravitational field can only be removed at a single point. In afalling cabin, a point particle will remain at the same place only if it resides at thecenter of mass of the cabin. Particles in the neighborhood of this point will moveslowly away from this point. The force causing this are called tidal forces. They arethe same forces which give rise to the tidal waves of the oceans. Earth and mooncircle around each other and their center of mass circles around the sun. The centerof mass is in “free fall”, the gravitational attraction proportional to the gravitationalmass being canceled by the centrifugal force proportional to the inertial mass. Anypoint on the earth which lies farther from the sun than the center of mass is pulledoutwards by the centripetal force, those which lie closer are pulled inwards by thegravitational force.

It is important to realize that the existence of tidal forces makes it impossible tosimulate gravitational forces by coordinate transformation in quantum mechanics.Due to Heisenberg’s uncertainty relation, quantum particles can never be localizedto a point but always occur in the form of wave packets. These flow apart andare therefore increasingly sensitive to the tidal forces. If one wants to remove thegravitational forces for a wave packet, multivalued coordinate transformations willbe necessary of the type used in the last chapter to create defects. These will supplyus with a quantum equivalence principle to be derived in Chapter 12.

11.2 Free Particle in General Coordinates Frame

As an first application of Einstein’s equivalence principle, consider the action (2.19)of a free massive point particle in Minkowski spacetime:

m

A = −mc∫ σb

σads (11.2)

where

ds = dσ√

gab xa(σ)xb(σ). (11.3)

is the increment of the invariant length along the path x(σ). The quantity

τ ≡ s/c (11.4)

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11.2 Free Particle in General Coordinates Frame 273

is called the proper time.A free particle moves along a straight line

x a(τ) = 0, (11.5)

which is the shortest spacetime path between initial and final points. This pathextremizes the action:

δm

A = 0 (11.6)

When going to an arbitrary curvilinear description of the same Minkowski spacein terms of coordinates xµ carrying latin indices

xµ = xµ(xa), (11.7)

the invariant length ds is given by

ds = dσ√

gµν(x(σ))xµ(σ)xν(σ). (11.8)

The 4 × 4 spacetime-dependent matrix

gµν(q) = gab∂xa

∂xµ∂xb

∂xν(11.9)

plays the role of a metric in the spacetime. Note that the inverse metric is given by

gµν(x) = gab∂xµ

∂xa∂xν

∂xb(11.10)

Since spacetime has not really changed, only its parametrization, the path is stillstraight. The equation of motion in the new curvilinear coordinates xµ can be foundin two ways. One is to simply transform the free equation of motion in Minkowskispace (11.5) to curvilinear coordinates. This will be done at the end of this section.

We begin with the more complicated but instructive way by extremizing thetransformed action

m

A =∫ σb

σadσ

m

L (xµ(σ)) , (11.11)

with the transformed Lagrangian [compare (2.19)]

m

L (xµ(σ)) = −mcdsdσ

= −mc[

gµν(x(σ))xµ(σ)xν(σ)] 1

2 (11.12)

As observed in Subsection (2.2), the action (11.11) is invariant under arbitraryreparametrizations

σ → σ′ = f(σ). (11.13)

Variation of the action yields

δm

A =∫ σb

σadσ δL (xµ(σ))

= −m2c21

2

∫ σb

σa

L (xµ(σ))

[

(

∂λgµν)

δxλ xµ(σ)xν(σ) + 2gλνdδxλ

dσxν(σ)

]

.(11.14)

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274 11 Relativistic Mechanics in Curvilinear Coordinates

On the right-hand side we have used the property (2.7) that the variation of thederivative is equal to the derivative of the variation.

The factor before the bracket is equal to dσ/(−mcds/dσ) = −(dσ/ds)2ds/mc.Thus, if we choose σ to be the proper time τ for which dσ/ds = dτ/ds = 1/c, wemay rewrite the variations as

δm

A = −m 1

2

∫ τb

τadτ

[

(

∂λgµν)

δxλ xµ(τ)xν(τ) + 2gλνdδxλ

dτxν(τ)

]

. (11.15)

This shows that if we use the proper time τ to parameterize the paths, the equationsof motion can alternatively be derived from the simpler action

m

A =∫ τb

τadτ

m

L (xµ(τ)) , (11.16)

wherem

L (xµ(τ)) ≡ −m2gµν(x(τ))x

µ(τ)xν(τ). (11.17)

This has the same form as the the action of a nonrelativistic point particle in four-dimensional spacetime parameterized by a pseudotime τ . Note that although the

actionm

A has the same same extrema, it has only half the size ofm

A .The second integral in (11.15) can be performed by parts to yield

2gλν(x(τ))δxλ(τ)xν(τ)

τb

τa− 2

∫ τb

τadτ δxλ(τ)

d

dτ[gλν(x(τ))x

ν(τ)] . (11.18)

According to the extremal principle of classical mechanics, we derive the equationsof motion by varying the action with vanishing variations of the paths δqµ at theend points [recall (2.3), which leads to the equation

1

2

∫ τb

τadτ[(

∂λgµν − 2∂µgλν)

xµ(τ)xν(τ) − 2gλν xν(τ)

]

δxλ(τ) = 0. (11.19)

This is valid for all δxµ(τ) vanishing at the end points, in particular for the infinites-imal local spikes:

δxµ(τ) = εδ(τ − τ0). (11.20)

Inserting these into (11.19) we obtain the equations of motion

gλν xν(τ) +

(

∂µgλν −1

2∂λgµν

)

xµ(τ)xν(τ) = 0. (11.21)

It is convenient to introduce a quantity called the Riemann connection of Christoffelsymbol :

Γµνλ ≡ µν, λ =1

2

(

∂µgνλ + ∂νgµλ − ∂λgµν)

, (11.22)

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11.3 Minkowski Geometry formulated in General Coordinates 275

Then the equation of motion can be written as

gλν xν(τ) + Γµνλ x

µ(τ)xν(τ) = 0. (11.23)

By further introducing the modified Christoffel symbol

Γµνκ ≡

κµν

= gκλΓµνλ = gκλ µνλ , (11.24)

we can bring Eq. (11.21) to the form

xλ(τ) + Γµνλxµ(τ)xν(τ) = 0. (11.25)

A path xλ(τ) satisfying this differential equation of shortest length is called a geodesictrajectory . It is Einstein’s postulate, that this equation describes correctly the mo-tion of a point particle in the presence of a gravitational field.

Now we turn to the simpler direct derivation of the equation of motion applyingthe coordinate transformation xa(xµ) to the straight-line equation of motion (11.5)in Minkowski space:

x a(τ) =d

dt

[

∂xa

∂xµxµ(τ)

]

=∂xa

∂xµxµ(τ) +

(

d

dt

∂xa

∂xµ

)

xµ(τ) = 0, (11.26)

where we have written ∂xa/∂xµ for the coordinate transformation matrix evaluatedon the trajectory x(τ). Multiplying this by ∂xλ/∂xa and summing over repeatedindices a yields

xλ(τ) +∂xλ

∂xa

(

d

dt

∂xa

∂xµ

)

xµ(τ) = xλ(τ) +∂xλ

∂xa(∂µ∂νx

a) xµ(τ)xν(τ) = 0. (11.27)

The second term can be processed using (11.10) as follows:

∂xλ

∂xa(∂µ∂νx

a) = gλσ(∂σxa)(∂µ∂νxa). (11.28)

It takes a little algebra to verify that this is equal to Γλκµ, so that the tranformed

equation of motion (11.27) coincides, indeed, with the geodesic equation (11.25).

11.3 Minkowski Geometry formulated in GeneralCoordinates

In Einstein’s theory, all gravitational effects can be completely described by a non-trivial geometry of spacetime. As a first step towards developing this theory it isimportant to learn to distinguish between inessential properties of the geometrywhich are merely due to the formulation in terms of general coordinates, as in thelast section, and those which are caused by the presence of gravitational forces. Forthis purpose we study in more detail the mathematics of coordinate transformationin Minkowski spacetime.

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276 11 Relativistic Mechanics in Curvilinear Coordinates

11.3.1 Local Basis tetrads

As in Eq. (1.25) we use coordinates xa (a = 0, 1, 2, 3) to specify the points inMinkowski spacetime. From now on it will be convenient to use fat latin lettersto denote four-vectors in spacetime. Thus we shall denote the four-dimensionalbasis vectors by ea, and an arbitrary four-dimensional vector with coordinates xa byx = eax

a. The basis vectors are orthonormal with respect to the Minkowski metricgab of Eq. (1.29):

eaeb = gab. (11.29)

The basis vectors ea define an inertial frame of reference.Let us now reparametrize this Minkowski spacetime by a new set of coordinates

xµ whose values are given by a mapping

xa → xµ = xµ(xa). (11.30)

Since xµ still describe the same spacetime we shall assume the function xµ(xa) topossess an inverse xa = xa(xµ) and to be sufficiently smooth so that xµ(xa) andxa(xµ) have at least two smooth derivatives. These will always commute with eachother. In other words, the general coordinate transformation (11.30) and theirinverse xa(xµ) will satisfy the integrability conditions of Schwartz:

(

∂µ∂ν − ∂ν∂µ)

xa(xκ) = 0, (11.31)(

∂µ∂ν − ∂ν∂µ)

∂λxa(xκ) = 0. (11.32)

The conditions xµ(xa) = const. define a network of new coordinate hypersurfaceswhose normal vectors are given by (see Fig. 11.1)

xµ(x′a) ↔ x′a(xµ)(a) (b)

outside oberverinside oberver

Figure 11.1 Illustration of crystal planes (xµ = const.) before and after elastic distortion,

once seen from within the crystal (a) and once from outside (b).

eµ(x) ≡ eaeaµ(x) = ea

∂xa

∂xµ. (11.33)

These are called local basis vectors. Their components eaµ(x) are called local basistetrads. The difference vector between two points x′ and x has, in the inertial frame

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11.3 Minkowski Geometry formulated in General Coordinates 277

of reference, the description ∆x = ea (x′a − xa). When going to coordinates x′µ, xµ,this becomes

∆x = ea

∫ x′

xeaµ(x)dx

µ. (11.34)

The length of an infinitesimal vector dx is given by

ds =√

dx2 =

(

eµdxµ)2

=√

eµeνdxµdxν , (11.35)

The right-hand side shows that the metric in the curvilinear coordinates can beexpressed as a scalar product of the local basis vectors:

gµν(x) = eµ(x)eν(x) = gabeaµ(x)e

bν(x). (11.36)

Indeed, inserting here (11.33) we verify:

gµν(x) = eµeν = eaeb∂xa

∂xµ∂xb

∂xν= gab

∂xa

∂xµ∂xb

∂xν. (11.37)

as in (11.8).In the sequel, we shall freely raise and lower the latin index using the Minkowski

metric gab = gab, and define

eaµ ≡ gabebµ, eaµ ≡ gabe

bµ. (11.38)

Then we can rewrite (11.36) as

gµν(x) = eµ(x)eν(x) = eaµ(x)eaν(x). (11.39)

Since the general coordinate formulation (11.29) was assumed to have an in-verse, we can also calculate the derivatives ∂xµ/∂xa. These are orthonormal to thederivatives ∂xa/∂xµ in two ways:

∂xa

∂xµ∂xµ

∂xb= δab,

∂xµ

∂xa∂xa

∂xν= δµν . (11.40)

It is useful to denote the inverse derivatives ∂xµ/∂xa by eaµ and introduce the vectors

eµ = eagabeb

µ, called the reciprocal multivalued basis tetrads (see also Fig. 11.2).With this notation, (11.40) become orthonormality and completeness relations ofthe tetrads:

eaµ(x)ebµ(x) = δab, ea

µ(x)eaν(x) = δµν . (11.41)

The scalar product

gµν(x) = eµeν = eaµ(x)eaν(x) (11.42)

is obviously the inverse metric, satisfying

gµν(x)gνλ(x) = δνλ. (11.43)

The metric gµν(x) and its inverse gµν(x) can be used to freely lower and raise greekidices on any tensor, and to form invariants under coordinate transformations bycontraction of all indices.

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278 11 Relativistic Mechanics in Curvilinear Coordinates

11.3.2 Vector- and Tensor Fields, and Lorentz Invariance

When formulating the laws of physics in Minkowski spacetime it is important to an-alyze physical quantities according to their transformation properties under Lorentztransformations, which change the coordinates of one inertial frame of reference tothose of another

xa → x′a ≡ (Λx)a = Λabxb. (11.44)

The 4 × 4 matrices Λab preserve the metric gab. The length elements ds =

gabdxadxb =

gabdx′adx′b are the same in both coordinate frames xa and x′a.

This implies that the transformation matrices Λab satisfy

gabΛaa′Λ

ab′ =

(

ΛTgΛ)

a′b′= ga′b′. (11.45)

Infinitesimally, we parametrize Λab and (Λ−1)

ab as

Λab = δab + ωab,

(

Λ−1)a

b = δab − ωab, (11.46)

and the relation (gΛ)T = gΛ−1 implies that

ωab ≡ gaa′ωa′

b (11.47)

is an antisymmetric matrix, i.e., ωab = −ωba. It has six independent matrix elements.The three components ωk = 1

2εkjiωij parametrize infinitesimal rotations, |ω| being

the angle and ≡ /‖‖ the axis of rotation. The three components ωi0 = −ω0i = ζspecify the infinitesimal relative rapidity of the two coordinate frames

).

Since the physical points are the same before and after a Lorentz transformation,the basis vectors ea change according to the law

ea → e′a ≡(

eΛ−1)

a= eb

(

Λ−1)b

a =(

egΛTg)

a. (11.48)

This gives

x ≡ eaxa ≡ egx −→ x′ = eΛTgΛx = egx = x (11.49)

showing that the vectors in the external frame are the same before and after thetransformation.

Consider now a vector field va(x). It assigns to every point P a vector

v(P ) = eava(x). (11.50)

Under a Lorentz transformations of the coordinates xa, the basis vectors ea change.The observable vector v(P ), however, must remain the same at the same point inspace, i.e.,

v′(P) = v(P). (11.51)

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11.3 Minkowski Geometry formulated in General Coordinates 279

Writing this as

v′(P ) = e′av′a(x′) = v(P ) = eav

a(x) (11.52)

we see that the components of the vector in the two frames have to be related inthe same way as the coordinate x′a and xa, i.e.,

v′a(x′) = Λabv′b(x), (11.53)

or, written differently,

v′a(x) = Λabvb(

Λ−1x)

. (11.54)

For infinitesimal transformations,

Λabxb = (δab + ωab) x

b (11.55)

with(

Λ−1x)a

= xa − ωabxb, (11.56)

the vector va(x) goes over into

v′a(x) = va(x) + ωabvb(x) − ωb

bxb∂b′v

a(x). (11.57)

The infinitesimal local change of a function f(x) when evaluated at the same nu-merical values of the coordinates x (which correspond to two differential points Pin space, namely x and (Λ−1x), is substantial change of f(x) the denoted by δsf(x),

δsf(x) ≡ f(x) − f ′(x). (11.58)

Using the symbol δs, the infinitesimal transformation law (11.57) reads

δsva(x) = v′a(x) − va(x)

= ωabvb(x) − ωbb′x

b′∂bva(x). (11.59)

For reasons of constructing Lorentz-invariant equations of motion we introduce forevery vector field va(x) a contravariant vector field va(x) = gabv

b(x). Its transforma-tion properties correspond to those of the derivative with respect to the coordinatesin Eq. (1.80). Infinitesimally, a substantial change of va is equal to

δsva(x) = ωabvs(x) − ωb

bxb∂b′va(x)

= ωabvb(x) + ωb

b′xb∂b′va(x) (11.60)

where we have introduced the matrix elements

ωbb′ = gabg

a′b′ωab′ = ga′b′ωba. (11.61)

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280 11 Relativistic Mechanics in Curvilinear Coordinates

The derivatives of contravariant vector field with respect to changes of xa are highertensor fields. Infinitesimally, derivatives transform via the sum of operations (11.60),one applied to each index. This follows directly from (11.60) and the commutationrule [∂a, xb] = gab:

δs∂bva = ∂bδsva

= ∂b(

ωaa′va′ + ωa

c′xc∂c′va)

(11.62)

= ωaa′∂bva′ + ωb

b ∂b′va + ωcc′xc∂c′∂bva.

This simple rule can easily be extended to arbitrary higher derivatives thereby form-ing higher tensor fields. Note that since the arguments in f and f ′ in (11.58) arethe same, the operation “substantial change” commutes with the derivative.

Consider now the same physical objects but described in terms of curvilinearcoordinates xµ(xa). Then the components of v are measured not with respect tothe basis ea but with respect to the local basis eµ(x) = eae

aµ(x). It is then natural

to specify v(x) in terms of its local components vµ(x) = va(x)eaµ(x). On the fields

vµ(x) one cannot only perform Lorentz transformations but any general coordinatetransformation xµ → x′µ(xµ) which, in the following, will shortly be referred to asEinstein transformations.

Under Einstein transformations, the vectors eaµ(x) being derivatives of the co-

ordinate transformation functions xµ(xa), undergo the following changes

eaµ → e′a

µ(x′) ≡ ∂x′µ

∂xa=

∂x′µ

∂xν∂xν

∂xa

= αµν(x)eaν(x) (11.63)

eaµ → e′aµ(x′) =

∂xa

∂x′µ=

∂xν

∂x′µ∂xa

∂x′µ

= αµν(x)eaν(x).

The matrices

αµν(x) ≡ ∂x′µ

∂xν≡ (α)µν

αµν(x) ≡ ∂xν

∂x′µ(11.64)

are orthogonal to each other

ανλανµ = δλ

µ,

ανµαλµ = δν

λ, (11.65)

i.e.,(

α−1)ν

λ = αλν (11.66)

is a right- as well as a left-inverse of the matrix ανµ. Infinitesimally, we may set

x′µ ≡ xµ − ξµ(x), (11.67)

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11.3 Minkowski Geometry formulated in General Coordinates 281

and see that Einstein transformations can be interpreted as local translations. Theinfinitesimal transformation matrices are

αλν ≈ δνλ − ∂νξ

λ(x)

αµν ≈ δµ

ν + ∂µξν(x) (11.68)

and the substantial changes of eµa(x) are given by

δseaµ ≡ e′a

µ(x) − eaµ(x) = e′a

µ(x′) − eaµ(x′)

= eaµ(x) − ea

µ(x′) + e′aµ(x′) − ea

µ(x)

= ξλ∂λeaµ(x) − ∂λξ

µeaµ(x), (11.69)

δseaµ = ξλ∂λe

aµ(x) + ∂µξ

λeaλ(x). (11.70)

Analogous transformation laws can be derived for the components of the vectorfields vµ(x) and vµ(x). They follow from the fact that the components va(xb), va(x

b)

do not change under a change of the general coordinates from xµ to xµ′

. Thus wehave the obvious relation

v′a(xb) = va(xb). (11.71)

When reparametrizing the point xb in the two different coordinates x′µ and xµ, thisrelation takes the form

v′a(x′) = va(x) (11.72)

where we have omitted the greek superscripts of x′ and x. Thus the substantialchanges, i.e., the changes at the same values of the general coordinates xµ, are

δsva(x) = v′a(x) − va(x) = ξλ∂λv

a(x). (11.73)

Using this and (11.70), we derive from (11.72)

v′µ(x′) = αµνvν(x),

v′µ(x′) = αµ

νvν(x), (11.74)

with the substantial changes

δsvµ(x) = v′µ(x) − vµ(x) = ξλ∂λv

µ − ∂λξµvλ (11.75)

δsvµ(x) = v′µ(x) − vµ(x) = ξλ∂λvµ∂µξλvλ. (11.76)

Any four-component field with these transformation properties is called Einsteinvector or world vector .

This definition can trivially be extended to higher Einstein- or world tensors.We merely apply separately the transformation matrices (11.68) to each index.

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282 11 Relativistic Mechanics in Curvilinear Coordinates

In particular, the metric gµν(x) transform as

g′λκ(x′) = αλ

µακνgµν(x),

g′λκ(x′) = αλµακνg

µν(x), (11.77)

or, infinitesimally, as

δsgµν = ξλ∂λgµν + ∂µξλgλν + ∂νξ

λgµλ, (11.78)

δsgµν = ξλ∂λg

µν − ∂λξµgλν − ∂λξ

νgµλ. (11.79)

This can be rewritten in a manifestly covariant form as follows:

δsgµν = Dµξν + Dνξµ, (11.80)

δsgµν = Dµξν + Dνξµ. (11.81)

It is now obvious from (11.65) that one can multiply any set of world tensors witheach other by a simple contraction of upper and lower indices and always obtainnew world tensors. In particular, one obtains an Einstein- or world invariant if sucha contraction is complete, i.e., if no index is left.

11.3.3 Affine Connections and Covariant Derivatives

The multiplication rules for world tensors are completely analogous to those forLorentz tensors. There is, however, one important difference. Contrary to theLorentz case, derivatives of world tensors are no longer tensors. In curvilinear coor-dinates, certain modifications of the derivatives are required in order to make themproper tensors. It is quite easy to find these modifications and construct objectsanalogous to the derivative tensors in the Lorentz frames. For this we rewrite thederivative tensors in terms of the general curvilinear components. Take, for exam-ple, the tensor ∂bva(x). Going over to curvilinear components xµ we can write thisas

∂bva = ∂b(

eaµvµ

)

. (11.82)

But if we take the derivative ∂b past the basis tetrad eaµ we find

∂bva = eaµ∂bvµ + ∂bea

µvµ. (11.83)

Using the relation

∂b = ebλ∂λ (11.84)

we see that

∂bva = eaµeb

ν∂νvµ +(

ebν∂νea

λ)

vλ. (11.85)

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11.3 Minkowski Geometry formulated in General Coordinates 283

The right-hand side can be rewritten as

∂bva ≡ eaµeb

νDνvµ (11.86)

where the symbol Dν stands for the modified derivative

Dνvµ = ∂νvµ − ecλ∂νeµ

cvλ ≡ ∂νvµ − Γνµλvλ. (11.87)

The explicit form on the right-hand side follows from the simple relation

∂νeaλ = −eaµ

(

ecλ∂νe

)

(11.88)

∂νeaλ = −eaµ (ecλ∂νec

µ) (11.89)

which, in turn, is a consequence of differentiating the orthogonality relation eaλebλ =

δab. Similarly, we can find the Einstein version of the derivative of a contravariant

vector field ∂bva(x), which can be rewritten as

∂bva = ∂b

(

eaµvµ)

= eaµebν∂νv

µ + (ebν∂νe

aλ) v

λ (11.90)

and brought to the form

eaµebνDνv

µ, (11.91)

with a covariant derivative

Dνvµ = ∂νv

µ − ecλ∂νecµvλ = ∂νv

µ + ecµ∂νecλv

λ ≡ ∂νvµ + Γνλ

µvλ. (11.92)

The extra term appearing in (11.87) and (11.92):

Γµνλ ≡ ea

λ∂µeaν ≡ −eaν∂µeaλ (11.93)

is called the affine connection of the space. In general, a spacetime with a metricgµν and an affine connection to define covariant derivatives, is called a affine space,

and the geometry carried by gµν ,Γµνλ, is referred to a metric-affine geometry . Note

that by definition, the covariant derivatives of eaν and eaν vanishes:

Dµeaν = ∂µe

aν − Γµν

λeλa = 0, (11.94)

Dµeaν = ∂µea

ν + Γµλνea

λ = 0. (11.95)

Since gµν = eaµeaν , the same property holds for the metric tensor1

Dλgµν = ∂λgµν − Γλµσgσν − Γλν

σgµσ = 0, (11.96)

Dλgµν = ∂λg

µν + Γλσµgσν + Γλσ

νgµσ = 0. (11.97)

1In the present context where the spacetime is still flat and only reparametrized with curvilinearcoordinates, this a rather trivial statement. For the situation in general geometries see the remark[2] in Notes and References.

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284 11 Relativistic Mechanics in Curvilinear Coordinates

It is worth noting that the metric satisfies once more relations like (11.97), in whichthe connections are replaced by Christoffel symbols. In fact, from the definition(11.22) we can verify directly that

Dλgµν = ∂λgµν − Γλµσgσν − Γλν

σgµσ = 0, (11.98)

Dλgµν = ∂λg

µν + Γλσµgσν + Γλσ

νgµσ = 0. (11.99)

Since the left-hand sides of (11.86) and (11.90) are tensors with respect to Lorentztransformations, the covariant derivatives Dνvµ andDνv

µ in Eqs. (11.87) and (11.92)must be tensors with respect to general coordinate transformation, i.e., world ten-sors. In fact, one can easily verify that they transform covariantly:

∂′µ′vν′(x′) = αµ′

µαν′ν∂µvν(x). (11.100)

Working out the derivative on the left-hand side we obtain

∂′µ′vν′(x′) = αµ′

µ∂µ [αν′νvν(x)]

= αµ′µαν′

ν∂µvν(x) + αµ′µ∂µαν′

ν (11.101)

The last term is an obstacle to covariance. It is compensated by a similar term innon-tensorial behavior of Γµν

λ:

Γ′µ′ν′λ′(x′) = e′a

λ′∂′µ′e′αν′ = αλ

λαµ′µea

λ∂µ (xν′νeaν)

= αµ′µ[

αν′ναλ

λΓµνλ(x) + αλ

ν∂µαν′ν]

, (11.102)

Γ′µ′ν′λ′(x′) = −e′aν′∂µ′eaλ

= −αν′ναµ′µeaν∂µ(

αλ′

λeaλ)

= αµ′µ[

αν′ναλ

λΓλµν(x) − αν′

ν∂µαλ′

ν

]

. (11.103)

Infinitesimally, the transformation matrices are αµν = δµ

ν + ∂µξν and αµν = δµν −

∂νξµ, and we easily verify that the covariant derivatives Dµvν , Dµv

ν have the correctsubstantial transformation properties of world tensors:

δsDµvν = ξλ∂λDµvp + ∂µξλDλvν + ∂vνξ

λDµvλ,

δsDµvν = ξλ∂λDµv

ν + ∂µξλDλvν − ∂νξ

νDµvλ. (11.104)

The last non-covariant piece in

δs∂µvν = ∂µδsvµ = ∂µ(

ξλ∂λνν + ∂νξλvλ

)

= ξλ∂λ∂µvν + ∂µξλ∂λνν + ∂νξ

λ∂µvλ + ∂µ∂νξλvλ (11.105)

is canceled by the last non-tensorial piece in δsΓκµν :

δsΓµνκ = ξλ∂λΓµν

κ + ∂µξλΓµν

κ + ∂νξλΓµν

κ + ∂µ∂νξκ. (11.106)

It is easily checked that the same cancellation occurs in the covariant derivative ofan arbitrary tensor field, defined as

Dµtν1...νn ≡ ∂µtν1...νn −∑

i

Γµνλtv1...λi...νn ,

Dµtν1...νn ≡ ∂µtν1...νn +∑

i

Γµνλtv1...λi...νn (11.107)

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11.4 Covariant Time Derivative and Acceleration 285

11.4 Covariant Time Derivative and Acceleration

In this context it is useful to introduce the concept of a covariant time derivativeof a vector field in spacetime. The four-velocity uµ(τ) = qµ(τ) transforms like afour-vector. By analogy with the covariant derivative of a vector field vµ(x) in alonga trajectory x = q(τ) in Eqs. (11.87) and (11.92), we define the covariant timederivative of the vector field along a trajectory x = q(τ) as

D

dτvµ(τ) ≡ d

dτvµ(τ) + Γλκ

µvλ(τ)uκ(τ),D

dτvµ(τ) ≡

d

dτvµ(τ) − Γλµ

κuλ(τ)vκ(τ),

(11.108)where we have abbreviated vµ(q(τ)) by vµ(τ).

If va(x) were a parallel vector field in a Minkowski spacetime, the equa-tion (11.108) would describe the change of the transformed components vµ(x) =eµa(x)v

a(x). By analogy, we shall define (11.108) as the change under a paralleltransport also for general metric-affine geometries. This will be discussed in detailin Subsection 12.3.1.

If the vector trajectory is the velocity trajectory of a point particle, the covariantderivative is the covariant acceleration.

We may also define a Riemann covariant derivative

D

dτvµ(τ) ≡ d

dτvµ(τ) + Γλκ

µvλ(τ)uκ(τ),D

dτvµ(τ) ≡

d

dτvµ(τ) − Γλµ

κuλ(τ)vκ(τ),

(11.109)which vanish along geodesic trajectories.

In Eq. (1.305) we have derived the time derivative of the spin four-vector ofa spinning point particle in Minkowski space. The multivalued or nonholonomicmapping principle transforms this to a general affine geometry:

DSµdτ

= SκDuκ

dτuµ. (11.110)

This equation shows that in the absence of external forces, the spin four-vector ofa point particle remains always parallel to its initial orientation along the entireautoparallel trajectory:

DSµdτ

= 0. (11.111)

11.5 Torsion tensor

Since the coordinate transformations xµ(xa) and xa(xµ) were assumed to be inte-grable, the derivatives of the infinitesimal local translation field ξµ(x) commute witheach other:

(

∂µ∂ν − ∂ν∂µ)

ξλ(x) = 0. (11.112)

This has the consequence that the antisymmetric part of the connection

Sµνλ ≡ 1

2

(

Γµνλ − Γνµ

λ)

= eaλ∂µe

aν − ea

λ∂νeaµ (11.113)

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286 11 Relativistic Mechanics in Curvilinear Coordinates

transforms like a proper tensor. This follows directly from the transformation law(11.106). The additional derivative term ∂µ∂νξ

κ arising in the transformation ofΓµν

κ disappears in the antisymmetrized expression (11.113). For this reason, Sµν iscalled the torsion tensor.

Minkowski spacetime has no torsion so that the tensor nature of Sµνλ implies that

it vanishes in all curvilinear coordinates. In order to see that there is no torsion wedescribe Minkowski space in terms of coordinates xµ which coincide with the inertialcoordinates xa

. Then the basis tetrads are eaµ = ∂µxa are unit matrices, so that

the connection vanishes, and so does its antisymmetric part, the torsion. If we nowperform a general coordinate transformation to curvilinear coordinates xµ(xa) theconnection will in general become nonzero. The torsion, however, being a tensor,remains zero for all coordinate transformations of Minkowski space.

It is useful to realize that with the help of the torsion tensor, the connectioncan be decomposed into a Christoffel part, given by (11.24), which depends onlyon the metric gµν(x), and a second part, called the contortion tensor, which is acombination of torsion tensors.

To derive this decomposition, which is valid in spaces with torsion, let us definethe modified connection

Γµνλ ≡ Γµνκgκλ = eaλ∂µe

aν ,

and decompose this trivially as follows:

Γµνλ =e

Γµνλ +e

Kµνλ, (11.114)

where

e

Γµνλ≡1

2

eaλ∂µeaν +∂µeaλe

aν + eaµ∂νe

aλ+ eaλ∂νe

aµ−eaµ∂λeaν −∂λeaµeaν

,(11.115)

e

Kµνλ≡1

2

eaλ∂µeaν−eaλ∂νeaµ−eaµ∂νeaλ+ eaµ∂λe

aν + eaν∂λe

aµ−eaν∂µeaλ

.(11.116)

The terms in the first expression can be combined to

e

Γµνλ=1

2

∂µ (eaλeaν) + ∂ν

(

eaµeaλ

)

− ∂λ(eaµeaν)

=1

2

(

∂µgλν + ∂νgµλ − ∂λgµν)

,

(11.117)

which shows thate

Γµνλ is equal to the Riemann connection Γµνλ in Eq. (11.22). Thesecond expression is a combination of three torsion tensors (11.113). Defining anassociated torsion tensor Sµνλ ≡ Sµν

κgκλ, we see that

e

Kµνλ ≡ Kµνλ ≡ Sµνλ − Sνλµ + Sλµν . (11.118)

The combination of the three torsion tensor Sµνλ is the so-called contortion tensorKµνλ. The order of the indices of the three torsion terms are easy to remember:The first starts out with the same indices as Kµνλ. The second and third terms are

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11.6 Curvature Tensor as Covariant Curl of Affine Connection 287

shifted cyclically to the left with alternating signs. Note that the antisymmetry ofSµνλ makes the contortion tensor Kµνλ antisymmetric in the last two indices.

Summarizing, we have found that the full affine connection Γµνλ can be decom-posed into a sum of a Riemann connection and a contortion tensor:

Γµνλ = Γµνλ +Kµνλ. (11.119)

11.6 Curvature Tensor as Covariant Curl of AffineConnection

In the last section we have seen that even though the connection Γµνλ is not a tensor,

its antisymmetric part, the torsion Sλµν , is a tensor. The question arises whether itis possible to form a covariant object which contains information on the contentof gravitational forces in the symmetric Christoffel part of the connection. Such atensor does indeed exist.

When looking back at the transformation properties (11.113) we see that thetensor character is destroyed by the last term which is additive in the derivative ofan arbitrary function ∂µ∂νξ

κ(x). Such additive derivative terms were encounteredbefore in Subsection 2.4.4 in gauge transformations of electromagnetism, Recall thatthe gauge field of magnetism transform with such an additive derivative term [recall(2.103)]

δAa(x) = ∂aΛ(x), (11.120)

where Λ(x) are arbitrary gauge functions with commuting derivatives [recall (2.104)].The experimentally measurable physical fields are given by the gauge invariant an-tisymmetric combination of derivatives (2.80):

Fab = ∂aAb − ∂bAa. (11.121)

The additional derivative terms (11.120) disappear in the antisymmetric combina-tion (11.121). This suggests that a similar antisymmetric construction exists alsofor the connection. The construction is slightly more complicated since the transfor-mation law (11.113) contains also contributions which are linear in the connection.

In a nonabelian gauge theory associated with an internal symmetry which isindependent of the spacetime coordinate x, the covariant field strength Fab is amatrix. If g are the elements of the gauge group and D(g) a representation of g inthis matrix space, the field strength transforms like a tensor

Fab → F ′ab = D(g)FabD−1(g). (11.122)

The gauge field Aa behaves under such transformations as

Aa(x) → A′a(x) = D(g)Aa(x)D−1(g) + [∂aD(g)]D−1(g), (11.123)

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288 11 Relativistic Mechanics in Curvilinear Coordinates

which is the generalization of the gauge transformations (2.103). The covariant fieldstrength with the transformation property (11.122) is obtained from this by formingthe nonabelian curl

Fab = ∂aAb − ∂bAa − [Aa, Ab]. (11.124)

This kind of gauge transformations and covariant field strengths appear in the non-abelian gauge theries used to describe the vector bosons W 0,± and Z0 of weakinteractions, where the gauge group is SU(2), and the octet of gluons G1,...,8 in thetheroy of strong interactions, where the gauge group is SU(3). In either case, therepresentation matrices D(g) belong to the adjoint representaion of the gauge group.

Now we observe that the transformation law (11.102) of the affine connectioncan be written in a way which is completely analogous to the transformation law(11.123) of a non-abelian gauge field. For this we consider Γλµν as the matrix elementsof a four 4 × 4 matrix Γµ:

Γµνλ =

(

Γµ)

ν

λ. (11.125)

Then we can rewrite (11.103) as the matrix equation

′µ′(x

′) = αµ′µ[

α

µ(x)α−1 + (∂µα)α−1

]

. (11.126)

This equation is a direct generalization of Eq. (11.123) to the case that the symmetrygroup acts also on the spacetime coordinates. To achieve covariance, the vector indexµ of the gauge field must be transformed accordingly.

Actually this is no surprise if we remember the original purpose of introducing theconnection Γµν

λ. It served to form covariant derivatives (11.87) and (11.92). Equa-tion (11.126) shows that the connection may be viewed as a non-abelian gauge fieldof the group of local general coordinate transformations αµν(x). Einstein vectorsand tensors in curvilinear coordinates are the associated gauge covariant quantities.

By analogy with the field strength (11.124), we can immediately write down acovariant curl of the matrix field

µ:

Rµν ≡ ∂µ

ν − ∂ν

µ −[

µ,

ν

]

, (11.127)

which should transform like a tensor under general coordinate transformations. Incomponent form, this tensor reads

Rµνλσ = ∂µΓνλ

σ − ∂νΓµλσ − Γµλ

δΓνδσ + Γνλ

δΓµδσ. (11.128)

The covariance properties of Rµνλκ follow most easily by realizing that in terms of

the basic tetrads eaµ, the covariant curl has the simple representation

Rµνλσ = ea

σ(

∂µ∂ν − ∂ν∂µ)

eaλ = −eaλ(

∂µ∂ν − ∂ν∂µ)

eaσ. (11.129)

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11.6 Curvature Tensor as Covariant Curl of Affine Connection 289

The first line is obtained directly by inserting Γµνλ = ea

λ∂µeaν into (11.127), and

executing the derivatives

[

∂µΓνλκ −

(

ΓµΓν)

λ

κ]

− [µ ↔ ν]

=(

∂µeaκ∂νe

aλ + ea

κ∂µ∂νeaλ + eb

ρ∂µebλeaρ∂νea

κ)

− (µ ↔ ν)

= eaκ(

∂µ∂ν − ∂ν∂µ)

eaλ. (11.130)

The second line in (11.130) is obtained from the first by inserting Γµνλ = −eaν∂µeaλ

or Γνρκ = ea

κ∂νeaρ.

We are now ready to realize another property of Minkowski space. Just as thisspacetime had a vanishing torsion tensor for any curvilinear parametrization, it alsohas a vanishing curvature tensor. The representation (11.129) shows that a spacexµ can have curvature only if the derivatives of the mapping functions xa → xµ arenot integrable in the Schwarz sense. Expressed differently, the vanishing of Rµνλ

κ

follows from the obvious fact that

Rµνλκ = ea

κ(

∂µ∂ν − ∂ν∂µ)

eaλ ≡ 0 (11.131)

for the trivial choice of the basis tetrad eaκ = δa

κ. Together with the tensor trans-formation law (11.132) we find that Rµνλ

κ remains identically zero in any curvilinearparametrization of Minkowski space.

From the tetrad expression for Rµνλκ the tensor transformation law is easily

found [using (11.64)]

Rµνλκ(x) → R′µ′ν′λ′

κ′(x′)

= eax′(x′)

(

∂′µ′∂′ν′ − ∂′ν′∂

′µ′

)

e′aλ′(x)

= ακ′

καµ′µea

κ(x)(

∂µ∂ν − ∂ν∂µ) (

αλ′λeaλ

)

= αµ′µαν′

ναλ′λακ

κRµνλκ(x)

+αµ′µαν′

νακ′

λ

[(

∂µ∂ν − ∂ν∂µ)

αλ′λ]

. (11.132)

Since general coordinate transformations are assumed to be smooth, the derivativesin front of αλ′

λ commute and Rµνλκ is a proper tensor. It is called the curvature

tensor .

By constructing, this curvature tensor is antisymmetric in the first index pair.What is not so easy to see is that it is also antisymmetric with respect to the secondindex pair, namely

Rµνλκ = −Rµνκλ (11.133)

where Rµνλκ ≡ Rµνλσgκσ.

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290 11 Relativistic Mechanics in Curvilinear Coordinates

Indeed, if we calculate this differences using the definition (11.129) we find

Rµνλκ +Rµνκλ = eaκ(

∂µ∂ν − ∂ν∂µ)

eaλ + eaλ(

∂µ∂ν − ∂ν∂µ)

eaκ

= ∂µ∂ν (eaκeaλ) − ∂ν∂µ (eaκe

aλ)

=(

∂µ∂ν − ∂ν∂µ)

gλκ. (11.134)

By assumption, the coordinate transformations xa(xµ) are smooth functions whichsatisfy the integrability condition (11.32). As a consequence, the metric

gλκ(x) =∂xa

∂xλ∂xa∂xκ

is a smooth function. In the following we shall always assume that it also admitsfor at least two derivatives. Since the metric is an observable quantity it must besingle valued. This implies the integrability condition

(

∂µ∂ν − ∂ν∂µ)

gλκ = 0. (11.135)

It is this property which makes the curvature tensor antisymmetric in the last twoindices.2 The curvature tensor gives a covariant characterization of the connectionwhich includes information on the Christoffel part.

Since Rµνλκ is a tensor it can be contracted with the metric tensor to form

covariant quantities of lower rank. There are two possibilities

Rµν ≡ Rκµνκ (11.136)

called the Ricci tensor and

R = Rµνgµν (11.137)

called the scalar curvature. A combination of both

Gµν ≡ Rµν −1

2gµνR (11.138)

was introduced by Einstein and is therefore called the Einstein curvature tensor . Itcan also be written as

Gνµ =1

4eµαβγeναδτRβγδτ . (11.139)

This follows directly from the curved-spacetime version of the identity (1A.24) (seeAppendix 11A).

2In more general geometries, where Dµgλκ = −Qµλκ 6= 0 (see the remark [2] in Notes andReferences), there would be a symmetric part

Rµνλκ + Rµνκλ = [DµQνλκ − (ν ↔ µ)] + 2SµνρQρλκ

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11.7 Riemann Curvature Tensor 291

11.7 Riemann Curvature Tensor

Actually, Einstein worked with a related tensor which deals exclusively with theRiemann part of the connection and the curvature tensor. Since the contortion Kµν

λ

is a tensor, the Riemann part Γµνλ of Γµν

λ has the same transformation properties

(11.106) as Γλµν , and we can form the Riemann curvature tensor

Rµνλσ = ∂µΓνλ

σ − ∂ν Γµλσ − Γµλ

ρΓνρσ + Γνλ

ρΓµρσ. (11.140)

Contrary to Rµνλκ in Eq. (11.128), this curvature tensor can be expressed completely

in terms of derivatives of the metric [recall (11.22), (11.24)]. The difference betweenthe two tensors is the following function of the contortion tensor

Rµνλκ − Rµνλ

κ = DµKνλκ − DνKµλ

κ −(

KµλρKνρ

κ −KνλρKµρ

κ)

, (11.141)

where Dµ denotes a covariant derivative which is formed with only the Christoffelpart of the connection. Note that the Riemann part of the curvature tensor has thesame antisymmetry in the first and second index pairs as Rµνκ

κ. For the first pairsthis follows directly from (11.140); for the second, it follows from (11.133) and theantisymmetry of the contortion tensor Kνλκ is the second index pair.

In three dimensions, the antisymmetry in both ij and kl suggests the introduc-tion of a tensor of second rank

Gji =1

4eiklejmnR

klmn. (11.142)

In addition the curvature tensor Rµνλκ is symmetric under the exchange of thefirst and the second index pair

Rµνλκ = Rλκµν . (11.143)

This can be shown by writing the first two terms in (11.140) explicitly as derivativesof the metric tensor

Rµνλκ=

[

gκδ∂µgδσ

2(∂νgλσ+∂λgνσ−∂σgνλ)

]

−[µ↔ ν]−gκδ(

ΓµλρΓνρ

δ−ΓνλρΓµρ

δ)

,

(11.144)

and using (11.99) to express ∂µgδσ in terms of Christoffel symbols,

gκδ∂µgδσ = −

(

∂µgκδ)

gδσ

= −(

Γµκτgτδ + Γµδ

τgκτ)

gδσ = −Γµκσ − Γµδκg

δσ (11.145)

we derive

Rµνλκ =1

2

[(

∂µ∂λgνκ − ∂µ∂κgνλ)

− (µ ↔ ν)]

−[(

Γµκσ + Γµκ′κg

λσ)

Γνλσ − (µ↔ ν)]

−(

ΓµλρΓνρκ − Γνλ

ρΓµρκ)

.(11.146)

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292 11 Relativistic Mechanics in Curvilinear Coordinates

A further use of relation (11.99) brings the second line to

−1

2

(

Γµκσ + gδσΓµδκ

) [(

Γνλσ + Γνσλ)

+ (λ↔ ν) − Γσνλ − Γσλν]

− µ↔ ν ,

and we find that almost all terms cancel, due to the symmetry of Γµνλ in µν. Only

−(

ΓµκσΓνλσ + ΓµδκΓνλ

δ)

+ (µ↔ ν)

survives, whose second term cancels the third line in (11.146), bringing Rµνλκ to theform

Rµνλκ =1

2

[(

∂µ∂λgνκ − ∂µ∂κgνλ)

− (µ↔ ν)]

−(

ΓµκσΓνλσ − Γνκ

σΓµλσ)

. (11.147)

This expression shows manifestly the symmetry µν → λκ as a consequence of theintegrability property

(

∂µ∂ν − ∂ν∂µ)

gλκ = 0. It is also antisymmetric under µ → ν

and λ → κ, the first trivially so, due to the definition (11.128), the second as aconsequence of the integrability property of the Christoffel symbol.

By contracting (11.147) with gνλgµκ, we can derive the following compact ex-pression for the curvature scalar√−gR = ∂λ

[

(gµν√−g)

(

Γµνλ−δµλΓνκκ

)]

+√−ggµν

(

ΓµλκΓνκ

λ−ΓµνλΓλκ

κ)

.(11.148)

It is instructive to check this equation as follows: We use the identity (11.145) inthe form

∂κgµν = −gµσgντ∂κgστ = Γκµν + Γκνµ, (11.149)

and another identity

∂λ√−g =

1

2

√−ggστ∂λgστ = Γλµµ, (11.150)

which follows directly from Eq. (11A.24), to derive

∂λ(gµν√−g) =

√−g[

−gµσΓλσν − gνσΓλσµ + gµνΓλσ

σ]

. (11.151)

This allows us to rewrite the first term in (11.148) as

∂λ(gµν√−g)

(

Γµνλ − δµ

λΓνκκ)

(11.152)

=√−g

[

−gµσΓσλν − gνσΓσλµ + gµνΓσλ

σ] (

Γλµν − δµλΓκν

κ)

= −2√−g

[

ΓµνσgµλgνκΓλκ

σ − ΓλσσgλκΓκµ

µ]

.

Using this, Eq. (11.148) becomes

√−gR =√−g

[

gµνgλκ(

∂λΓκµν − ∂µΓνλκ − gστ ΓλσµΓκτν + ΓσλσΓκµν

)]

, (11.153)

which is the contraction of the defining equation (11.140) for the Riemann curvaturetensor with δµσg

νλ.

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Appendix 11A Curvilinear Versions of Levi-Civita Tensor 293

Appendix 11A Curvilinear Versions of Levi-Civita Tensor

In Appendix 1A we have listed to the properties of the Levi-Civita tensor εa1...aD ineuclidean as well as Minkowski space. These properties acquire little change if thespaces are reparametrized with curvilinear coordinates. To be specific, we consideronly a four-dimensional Minkowski spacetime whose metric arises from a coordinatetransformation of (1A.21). The same formulas hold also if the spacetime is curved.The curvilinear Levi-Civita tensor is

eµ1... µD =1√−g ε

µ1... µD , (11A.1)

where

−g ≡ det(

−gµν)

. (11A.2)

is the determinant of the negative metric gµν . and√−g is the positive square root.

Just as εa1...aD was a pseudotensor under Lorentz transformations [recall (1A.11)],eµ1... µD is a pseudotensor under general coordinate transformations, which transform

εµ1... µD → αµ1

ν1· · ·αµDνDε

ν1... νD = det (α) εµ1... .µD . (11A.3)

Since gµν is transformed as

gµν → αµλαν

κgλκ (11A.4)

its determinant behaves like

g → det(

αµλ)2g = det (αµν)

−2 g. (11A.5)

Hence

eµ1... µD → det (αµν)

|det (αµν) |eµ1... µD , (11A.6)

showing the pseudotensor property.The same thing holds for the tensor

eµ1... µD=

√−g εµ1... µD. (11A.7)

It arises from eν1... µD by multiplication with gµ1ν1· · · gµDνD as it should.

The co- and contravariant antisymmetric tensors eµ1... µD, eµ1... µD share an im-

portant property with the symmetric tensors gµ1µ2, gµ1µ2 . Just as those, they are

invariant under covariant differentiation:

Dλeµ1... µD= 0, Dλe

µ1... µD = 0. (11A.8)

Indeed, since eµ1... µDis a tensor, we can write this equation explicitly as

∂λeµ1... µD= Γλµ1

ν1eν1µ2... µD+ Γλµ2

ν2eµ1ν2... µD+ . . .+ ΓλµD

νDeµ1µ2... νD. (11A.9)

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294 11 Relativistic Mechanics in Curvilinear Coordinates

Using eµ1... µD=

√−gεµ1... µD, the left-hand side can be rewritten as

1√−g(

∂λ√−g

)

eµ1... µD, (11A.10)

from which the equality follows by using the trivial identity

δστ εµ1... µD= δσµ1

ετµ2... µD+ δσµ2

εµ1τ ... µD+ . . .+ δσµDεµ1µ2... τ

, (11A.11)

after turning it into the covariant form

gστeµ1... µD= gσµ1

eτµ2... µD+ gσµ2

eµ1τ ... µD+ . . .+ gσµDeµ1µ2... τ

, (11A.12)

and multiplying it by gσδΓλδτ .

An important consequence of the vanishing covariant derivative of the antisym-metric tensors in Eq. (11A.8) is that antisymmetric products satisfy the covariantversion chain rule of differentiation without an extra term. For instance, the vectorproduct in three curved dimensions

(x ×w)µ = eµλκxλwκ (11A.13)

has the covariant derivative

Dσ(x ×w) = Dσx × w + x ×Dσw, (11A.14)

just as in flat space. The same rule applies, of course, to the scalar product

x · w = gµνvµωλ, (11A.15)

as a consequence of the vanishing covariant derivative of the metric in Eq. (11.96):

Dσ(x ·w) = Dσx · w + x ·Dσw. (11A.16)

The determinant of arbitrary tensor tµν is given by a formula similar to (1A.9)

det(

tµν)

=1

D!εµ1...µDεν1...νDtµ1ν1

. . . tµDνD = −g 1

D!eµ1...µDeν1...νDtµ1ν1

. . . tµDνD .(11A.17)

The determinant of tµν , on the other hand, is equal to

det(

tµν)

= − 1

D!εµ1...µDεν1...νDtµ1

ν1 = − 1

D!eµ1...µDeν1...νDtµ1

ν1 . . . tµDνD , (11A.18)

in agreement with the relation det(

tµν)

= det(

tµλgλν)

= det(

tµν)

g−1.The covariant tensors eν1...νD are useful for writing down explicitly the cofactors

Mνµ in the expansion of a determinant.

det(

tµν)

=1

DtµνMν

µ. (11A.19)

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 295

By comparison with Eq. (11A.18) we identify:

Mν1µ1 = − 1

(D − 1)!εµ1...µDεν1...νDtµ2

ν2 . . . tµDνD . (11A.20)

The inverse of the matrix tµν has then the explicit form

(

t−1)

ν

µ =1

det(

tµν)Mν

µ. (11A.21)

For a determinant det(

tµν)

we find, similarly,

det(

tµν)

=1

DtµνM

µν , (11A.22)

with

Mµ1ν1 =1

(D − 1)!eµ1...µDeν1...νDtµ2ν2

. . . tµDνD

= det(

tµν) (

t−1)µ1νν1

. (11A.23)

This equation is useful for calculating variations of the determinant g upon variationsof the metric gµν , which will be needed later in Eq. (15.24). Inserting gµν into(11A.22) and using the first line of (11A.23), we find immediately

δg =1

D!εµ1...µDεν1...νDδ

(

gµ1ν1gµ2ν2

. . . gµDνD

)

=1

(D − 1)!εµ1...µDεν1...νDδgµ1ν1

gµ2ν2. . . gµDνD = δgµνM

µν = det(

gµν)

gµνδgµν

= ggµνδgµν . (11A.24)

Notes and References

For more details see Chapters 10 and 11 of the textbookH. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics,anmd Financial Markets, World Scientific, Singapore, 4th edition, 2006 (kl/b5),where kl is short for the www address http://www.physik.fu-berlin.de/~klei-nert, and Part IV of the textbookH. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects, WorldScientific, Singapore, 1989 (kl/b2).

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Get your facts first,

and then you can distort them as much as you please

Mark Twain (1835 - 1910)

12Torsion and Curvature from Defects

In the last chapter we have seen that a Minkowski space has neither torsion norcurvature. The absence of torsion follows from its tensor property, which was a con-sequence of the commutativity of derivatives in front of the infinitesimal translationfield

(

∂µ∂ν − ∂ν∂µ)

ξκ(x) = 0. (12.1)

The absence of curvature, on the other hand, was a consequence of the integrabilitycondition (11.32) of the transformation matrices

(

∂µ∂ν − ∂ν∂µ)

ακλ(x) = 0. (12.2)

Infinitesimally, this implies that(

∂µ∂ν − ∂ν∂µ)

∂λξκ(x) = 0, (12.3)

i.e., that derivatives commute in front of derivatives of the infinitesimal translationfield.

The situation is similar to those in electromagnetism in Chapter 4. Arbitrarygauge transformations (2.103) whose gauge function Λ(x) has commuting derivatives[see (2.104)] does not change the electromagnetic fields in spacetime. In particular, afield-free spacetime remains field-free. In Subsection 4.3 we have seen however, thatit is possible to generate thin nonzero magnetic field tubes in a field-free space byperforming multivalued gauge transformations which violate Schwarz’ integrabilityconditions. It is useful to imagine these coordinate transformations as being plasticdistortions of a world crystal . The ordinary single-valued coordinate transforma-tions correspond to elastic distortions of the world crystal which do not change thegeometry represented by the defects.

In Chapter 9 we have shown that the theoretical description of crystals withdefects is very similar to that of electromagnetism in terms of a multivalued scalarfield. This suggests a simple way of constructing general affine spaces with torsion orcurvature or both from a Minkowski spacetime by performing multivalued coordinatetransformations which do not satisfy (12.1), (12.3).

296

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12.1 Multivalued Infinitesimal Coordinate Transformations 297

12.1 Multivalued Infinitesimal Coordinate Transformations

Let us study the properties of a spacetime at which we can arrive from basis tetradseaµ = δa

µ, eaµ = δaµ via such infinitesimal multivalued coordinate transformationsξκ(x). According to (11.70), the new basis tetrads are

eaµ = δa

µ − ∂aξµ

eaµ = δaµ + ∂µξa (12.4)

and the metric is

gµν = eaµeaν = ηµν +(

∂µξν + ∂νξµ)

, (12.5)

where ηµν denotes the Minkowski metric (1.29) which needs here a different notationdue to our convention that Greek subscripts refer to curvilinear coordinates.

The connection associated with the tetrads 12.4) is (see (11.93))

Γµνλ = ∂µ∂νξ

λ, (12.6)

and the curvature tensor becomes

Rµνλκ =

(

∂µ∂ν − ∂ν∂µ)

∂λξκ. (12.7)

Since ξλ are infinitesimal displacements, we can lower the index in both equations,with a mistake which is only of the order of ξ2 and thus negligible, such that

Γµνλ = ∂µ∂νξλ, (12.8)

Sµνλ =1

2

(

∂µ∂ν − ∂ν∂µ)

ξλ, (12.9)

Rµνλκ =(

∂µ∂ν − ∂ν∂µ)

∂λξκ. (12.10)

The curvature tensor is trivially antisymmetric in the first two indices [as in(11.133)].

For singular ξ(x), the metric and the connection are, in general, also singular.This could cause difficulties in performing consistent length measurements and par-allel displacements. In order to avoid such difficulties, Einstein postulated that themetric gµν and the connection Γµν

λ should be so smooth as to permit two differ-entiations which commute as stated earlier in (11.135). Due to (12.4), (12.5) thisimplies that we must consider only such singular coordinate transformations whichsatisfy the condition

(

∂µ∂ν − ∂ν∂µ)

(∂λξκ + ∂κξλ) = 0, (12.11)(

∂µ∂ν − ∂ν∂µ)

∂σ∂λξκ = 0. (12.12)

The integrability conditions (12.11) shows again, now for the linearized metric, thatthe curvature tensor (12.15) is antisymmetric in the last two indices [recall (11.134)].

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298 12 Torsion and Curvature from Defects

For completeness, let us also write down the pure Christoffel part of the connec-tion as it follows from inserting (12.5) into (11.24):

Γµνκ=1

2µν, κ=

1

2

[

∂µ (∂νξκ + ∂κξν) + ∂ν(

∂µξκ + ∂κξµ)

− ∂κ(

∂µξν + ∂νξµ)]

(12.13)thus negligible such that

Γµνλ = ∂µ∂νξλ

Sµνλ =1

2

(

∂µ∂ν − ∂ν∂µ)

∂λξκ (12.14)

Rµνλκ =(

∂µ∂ν − ∂ν∂µ)

∂λξκ.

For completeness, let us also write down the decomposition (11.119) of the con-nection into the Christoffel part and the contortion tensor (insert (12.5) into (11.24)

Γµνκ = µν, κ +Kµνκ (12.15)

with

µν, κ =1

2∂µ (∂νξκξν) +

1

2∂ν(

∂µξκ + ∂κξµ)

− 1

2∂κ(

∂µξν + ∂νξµ)

Kµνλ =1

2

(

∂µ∂ν − ∂ν∂µ)

ξλ −1

2(∂ν∂λ − ∂λ∂ν) ξµ +

1

2

(

∂λ∂µ − ∂µ∂λ)

ξν

=1

2∂µ (∂νξλ − ∂λξν) +

1

2∂λ(

∂ν + ∂µξν)

− 1

2∂ν(

∂λξµ + ∂µξλ)

. (12.16)

From the Christoffel symbol we find the Riemann curvature tensor

Rµνλκ =1

2∂µ [∂ν (∂λξκ + ∂κξλ) + ∂λ (∂νξκ + ∂κξν) − ∂κ (∂νξλ + ∂λξν)]

= −1

2∂ν[

∂µ (∂λξκ + ∂κξλ) + ∂λ (∂νξκ + ∂κξν) − ∂κ(

∂µξλ + ∂λξλ)]

. (12.17)

Due to the integrability condition (12.12) the first terms in each line cancel and thisbecomes

Rµνλκ =1

2

[

∂µ∂λ (∂νξκ + ∂κξν) − (µ ↔ ν)]

− (λκ)

. (12.18)

In order to understand the geometric properties of such a space generated by theinfinitesimal singular transformations

xa → xµ =[

xa − ξa(

xb)]

δaµ (12.19)

we like to point out that such transformations are encountered in the context ofcrystalline defects. There, one considers infinitesimal displacements of atoms givenby

xi → x′i = xi + ui(x) (12.20)

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12.1 Multivalued Infinitesimal Coordinate Transformations 299

where x′i are the shifted positions, as seen from an ideal reference crystal. If wechange the point of view to an intrinsic description, i.e., if we measure coordinatesby counting the number of atomic steps within the distorted crystal, then the atomsof the ideal reference crystal are displaced by

xi → x′i = xi − ui(x). (12.21)

This is the same as (12.19). Hence the non-commutativity of derivatives in front

of singular coordinate changes ξa(

xλ)

is completely analogous to that in front of

crystal displacements ui(x). In the crystals this was a signal for the presence ofdefects. For the purpose of a better visualization, let us restrict our considerationto the three-dimensional euclidian subspace of the Minkowski space. Then we haveto identify the physical coordinates of material points xa for a = 1, 2, 3 with theprevious spatial coordinates xi for i = 1, 2, 31 and ∂a = ∂/∂xa(a = i) with theprevious derivatives ∂i. The infinitesimal translations in (11.138), ξa=i(x) are equalto the displacements ui(x) such that the basis tetrads are

e′a = δia − ∂aui, eai = δai + ∂iua (12.22)

and the metric becomes, to linear approximation,

gij = eaieaj = δij + ∂iuj + ∂jui. (12.23)

Apart from the trivial unit matrix it coincides with twice the strain tensor uij =12

(

∂iuj + ∂jui)

. The connection is simply

Γijk = ∂i∂juk (12.24)

with torsion and curvature tensors

Sijk =1

2

(

∂i∂j − ∂j∂i)

uk

Rijkl =(

∂i∂j − ∂j∂i)

∂kul. (12.25)

The integrability conditions read(

∂i∂j − ∂j∂i)

(∂kul + ∂luk) = 0,(

∂i∂j − ∂j∂i)

∂n (∂kui + ∂1uk) = 0 (12.26)(

∂i∂j − ∂j∂i)

∂k (∂kui − ∂luk) = 0.

They state that the strain tensor, its derivative, and the derivative of the localrotation field are all twice-differentiable single-valued functions everywhere.

1When working with four-vectors it is conventional to consider the upper indices as physicalcomponents. In purely three dimensional calculations one usually employs the metric gab = δab

such that xa=i and xi are the same.

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300 12 Torsion and Curvature from Defects

It was argued that this is true in a crystal. We can take advantage of the firstcondition and write the curvature tensor alternatively as

Rijkl =(

∂i∂j − ∂j∂i) 1

2(∂kul − ∂luk) . (12.27)

The antisymmetry in ij and kl suggests, in three dimensions, the introduction of atensor of second rank

Gji ≡1

4eiklejmnR

klmn, (12.28)

where

eijk =√−gεijk = gii′gjj′gkk′e

i′j′k′ = gii′gjj′gkk′

(

1√−g εi′j′k′

)

(12.29)

is the ε-tensor in general metric spaces. The tensor Gji happens to coincide withthe Einstein tensor as defined in (11.138). Indeed, if we use the identity

eiklejmn = gijgkmgln+gimgknglj+gingkjglm−gijglmgkn−gimgkngkj−gingljgkm,(12.30)

and insert it into (12.28), we find

Gji = Rji −1

2gjiRkk (12.31)

which is the general definition [see (11.136), (11.138)] of the Einstein tensor in anydimension. To linear approximation, Gij becomes, sue to (12.25),

Gij = εikl∂k∂l

(

1

2εjmn∂mun

)

. (12.32)

The second factor is the local rotation ωj = 12εjmn∂mun, and we see that the Einstein

curvature tensor can be written as

Gji = εikl∂k∂lωj (12.33)

Let us also form the Einstein tensor Gij associated with the Riemann curvature

tensor Rijkl. Using (12.18) we find

Gji = εiklεjmn∂k∂m1

2(∂lun + ∂nul) . (12.34)

In the discussion of crystal defects we have introduced the following measures forthe non-commutativity of derivatives. The dislocation density

αij = εikl∂k∂luj (12.35)

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12.1 Multivalued Infinitesimal Coordinate Transformations 301

the disclination density

θij = εikl∂k∂lωj (12.36)

and the defect density

gij = εiklεjmn∂k∂mulm. (12.37)

Comparison with (12.18) shows that αij is directly related to the torsion tensor

Skli = 1

2(Γkl

i − Γlki):

αij ≡ εiklΓklj ≡ εiklSklj. (12.38)

Hence torsion is a measure of the translational defects contained in the multivaluedcoordinate transformations, which may be pictured as combinations of elastic plusplastic distortions of a world crystal.

We can also use the decomposition (11.119) and write, due to the symmetry ofthe Christoffel symbol kl, j in kl.

αij = εiklKklj. (12.39)

Where Kklj is the contortion tensor. In terms of the displacement field u(x),

Kijk =1

2∂j(

∂juk − ∂kuj)

− 1

2

[

∂j(

∂kuj + ∂iuk)

− (j ↔ k)]

= ∂iωjk −(

∂juki − ∂kuji)

. (12.40)

Since Kijk is antisymmetric in lj, it is useful to introduce the tensor of second rankcalled Nye’s contortion tensor

Kln =1

2Kkljεljn. (12.41)

Inserting this into (12.39) we see that

αij = −Kji + δijKll (12.42)

In terms of the displacement and rotation fields, one has

Kil = ∂iωl − εlkj∂jukj (12.43)

Consider now the disclination density θij . Comparing (12.37) with (12.33) we seethat it coincides exactly with the Einstein tensor Gjl formed from the full curvaturetensor

θij ≡ Gji. (12.44)

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302 12 Torsion and Curvature from Defects

The defect density (12.37), finally, coincides with the Einstein tensor formed fromthe Riemann curvature tensor.

gij = Gij (12.45)

Hence we can conclude: A spacetime with torsion and curvature can be generatedfrom a Minkowski spacetime via singular coordinate transformations and is com-pletely equivalent to a crystal which has undergone plastic deformation and is filledwith dislocations and disclinations.

In Minkowski space, the trajectories of free particles are straight lines. In thedefected space, they choose the shortest path which is no longer straight since defectsmay lie in its way. According to Einstein’s theory, the motion of mass points in agravitational field is governed by the principle of shortest path as defined by thedefected metric gµν . This defected metric contains all gravitational effects which area consequence of the defects presented in the world crystal. The natural length scaleof gravitation is the Planck length which is the following combination of Newton’sgravitational constant GN [≈ 6.673 × 10−8 cm3/g s2, recall (1.3)] with the lightvelocity c (≈ 3 × 1010 cm/s) and Planck’s constant h (≈ 1.05459 × 10−27 erg/s):

lP =

(

c3

GNh

)−1/2

≈ 1.616 × 10−33cm, (12.46)

The Planck length is an extremely small quantity. It is by a factor 10−25 smallerthan an atom, which is roughly the ratio between the radius of an atom (≈ 10−8 cm)and the radius of the solar system (≈ 1010 km). Such small distances are at presentbeyond any experimental resolution. The Planck length lP may easily be imag-ined as the lattice constant of a world crystal with defects, without running intoexperimental contradictions.

The mass whose Compton wavelength is lP,

mP =h

clP=

hc

GN

= 1.221 047(79)× 1019GeV

= 0.021 7671(14) mg = 1.30138(6)× 1019mproton, (12.47)

is extremely large.

12.2 Examples for Nonholonomic CoordinateTransformations

It may be useful to give a few explicit examples of multivalued mappings xµ(xa)leading from a flat spacetime to a spacetime with curvature and torsion. We shalldo so by appealing to actual physical situations. For simplicity, we consider twodimensions. Imagine an ideal crystal with atoms placed at xa = (n1, n2, n3) · b withinfinitesimal lattice constant b.

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12.2 Examples for Nonholonomic Coordinate Transformations 303

12.2.1 Dislocation

The simplest example for a crystalline defect is the edge dislocation and the edgedisclination shown in Fig. 12.2.1. The mapping transforms the lattice points to new

Figure 12.1 Edge dislocation in a crystal associated with a missing semi-infinite plane of

atoms. The multivalued mapping from the ideal crystal to the crystal with the dislocation

introduces a δ-function type torsion in the image space.

distorted positions of which xµ(xa) are the cartesian coordinates. There exists noone-to-one mapping between the two figures since the excessive atoms in the middlehorizontal layer xa < 0, x2 = 0 hae no correspondence in xa space. In the continuumlimit of an infinitesimally small Burgers vector, the mapping can be described bythe multivalued function

x1 = x1

x2 = x2 − b

2πφ (12.48)

where the

φ(x) = arctanx2

x1 (12.49)

with the multivalued definition of the arctg, which on the physical Riemann sheetis equal to ±π for x1 = 0, x2 = ±ε. Its differential version is

dx1 = dx1 (12.50)

dx2 = dx2 +b

1

(x1)2 + (x2)2

(

x2dx1 − x1dx2)

(12.51)

with the basis diads eaµ = ∂xa/∂xµ

eaµ =

1 0b

x2

(x1)2 +(

x2)2 − b

x1

(x1)2 + (x2)2

. (12.52)

We have used the notation xa ≡ xa in order to distinguish xa=1,2 from xµ=1,2.

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304 12 Torsion and Curvature from Defects

Let us now integrate dxµ over a Burgers circuit which consists of a closed circuitC (xµ) in xµ-space around the origin,

ba =∫

C(xµ)dxa =

C(xµ)dxµ

∂xa

∂xµ=∫

C(xµ)dxµeaµ (12.53)

Inserting (12.50) and (12.51) we see that

b1 =∮

C(xµ)dx1 =

C(xµ)dxµ

∂x1

∂xµ=∫

C(xµ)dxµe1µ = 0, (12.54)

b2 =∮

C(xµ)dx2 =

C(xµ)dxµ

∂x2

∂xµ=∫

C(xµ)dxµe2µ = −b. (12.55)

It is easy to calculate the torsion tensor Saµν associated with the multivalued mapping(12.50) and (12.51). Because of its antisymmetry, only S12

1 and S122 are indepen-

dent. These become

S122 = ∂1e

22 − ∂2e

21 = ∂1

∂x2

∂x2 − ∂2

∂x2

∂x1 = −bδ(2)(x)

S112 = ∂1e

12 − ∂2e

11 = ∂1

∂x1

∂x2 − ∂2

∂x2

∂x1 = 0 (12.56)

We may write this result with the Burgers vector ba = (0, b) in the form

Saµν = baδ(2)(x). (12.57)

Let us now calculate the curvature tensor for this defect which is

Rµνλκ = eaκ(

∂µ∂ν − ∂ν∂µ)

eaλ. (12.58)

Since eaµ in (12.52) is single-valued, derivatives in front of it commute. Hence Rµνλκ

vanishes identically,

Rµνλκ ≡ 0. (12.59)

A pure dislocation gives rise to torsion but not to curvature.

12.2.2 Disclination

As a second example for a multivalued mapping, we generate curvature by thetransformation

xi = δiµ[xµ + Ωεµνx

νφ(x)], (12.60)

with the multi-valued function (12.49). The symbol εµν denotes the antisymmetricLevi-Civita tensor. The transformed metric

gµν = δµν −2Ω

xσxσεµνε

µλενκx

λxκ. (12.61)

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12.3 Differential-Geometric Properties of Affine Spaces 305

Figure 12.2 Edge disclination in a crystal associated with a missing semi-infinite section

of atoms of angle Ω. The multivalued mapping from the ideal crystal to the crystal with

the disclination introduces a δ-function type curvature in the image space.

is single-valued and has commuting derivatives. The torsion tensor vanishes since(∂1∂2 − ∂2∂1)x

1,2 is proportional to x2,1δ(2)(x) = 0. The local rotation field ω(x) ≡12(∂1x

2 − ∂2x1), on the other hand, is equal to the multi-valued function −Ωφ(x),

thus having the noncommuting derivatives:

(∂1∂2 − ∂2∂1)ω(x) = −2πΩδ(2)(x). (12.62)

To lowest order in Ω, this determines the curvature tensor, which in two dimensionsposses only one independent component, for instance R1212. Using the fact that gµνhas commuting derivatives, R1212 can be written as2

R1212 = (∂1∂2 − ∂2∂1)ω(x). (12.63)

In defect physics, the mapping (12.60) is associated with a disclination whichcorresponds to an entire section of angle α missing in an ideal atomic array (seeFig. 10.2).3

12.3 Differential Geometric Properties of Affine Spaces

Up to now we have studied only such affine spaces which were obtained from aMinkowski spacetime by introducing an infinitesimal amount of defects. In reality,defects can pile up and the spacetime must be described by the full non-linearformulation of affine spaces. At the linear level we have learned how dislocationsand disclinations manifest themselves by certain non-vanishing contour integralsaround Burgers circuits. In this section we would like to discuss these geometricaspects at the non-linear level.

The general affine spacetime will be characterized by the same type of integra-bility conditions as the spacetime with infinitesimal defects which were:

2Ibid., Eq. 2.86 on p. 1359.3Ibid., Fig. 2.2 on p. 1366.

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306 12 Torsion and Curvature from Defects

The metric and the connection are single-valued twice-differentiable functionswhich satisfy the integrability conditions

(

∂µ∂ν − ∂ν∂µ)

gλκ = 0, (12.64)(

∂µ∂ν − ∂ν∂µ)

Γκσλ = 0. (12.65)

Remember that the first condition ensures the antisymmetry of the curvature tensorin the last two indices [see (11.134)]. By antisymmetrizing the second condition inσλ it can also be replaced by an integrability condition for the torsion

(

∂µ∂ν − ∂ν∂µ)

Sσλκ = 0. (12.66)

Moreover, using the decomposition (11.119), the Christoffel symbol is seen to beintegrable as well:

(

∂µ∂ν − ∂ν∂µ)

κσλ

= 0. (12.67)

Since ∂µgλκ can be expressed in terms of products of Christoffel symbols and metrictensors, and since products of integrable functions are integrable4 derivatives of gλκsatisfy

(

∂µ∂ν − ∂ν∂µ)

∂σgλκ = 0. (12.68)

Conversely, with the Christoffel symbol consisting of products of gλκ and ∂µgλκ, thiscondition implies (12.67) and thus is completely equivalent to it due to the necessaryvalidity of (12.64).

12.3.1 Local Parallelism

In order to understand the geometric properties of such a general affine spacetimelet us first introduce the concept of local parallelism.

Consider a vector field v(x) = eava(x) which is parallel in the inertial frame

in the naive sense that all vectors point in the same direction. This simply means∂bv(x) = ea∂bv

a = 0. But when changing to the coordinates xµ we find

∂bva = ∂be

aµv

µ = ebν∂ν

(

eaµvµ)

= eνbeaµDνv

µ = 0. (12.69)

Thus parallel vector fields have their local components vµ change in such a way thattheir covariant derivatives vanish:

Dνvµ = ∂νv

µ + Γνλµvλ = 0. (12.70)

4This follows form the chain rule of differentiation

(∂µ∂ν − ∂ν∂ν) (fg) = [(∂µ∂ν − ∂ν∂µ) f ] g + f (∂µ∂ν − ∂ν∂µ) g

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12.3 Differential-Geometric Properties of Affine Spaces 307

Similarly we find:

Dνvµ = ∂νvµ − Γνµλvλ = 0. (12.71)

Note that the basis tetrads eνa, eaν are parallel vector fields, by construction [see

(11.95)].Let us study this type of situation in general: Given an arbitrary connection Γµν

λ

we first ask the question under what condition it is possible to find a parallel vectorfield in the whole space. For this we consider the vector field vµ(x) at a point x0

where it has the value vµ(x0). Let us now move to the neighboring position x0 +dx.There the field has components

vµ (x0 + dx) = vµ(x0) + ∂νvµ(x0)d

νx. (12.72)

If vµ(x) is a parallel vector field with Dνvµ = 0 the derivative satisfies

∂νvµ = −Γνκ

µvκ. (12.73)

This differential equation is integrable over a finite region of spacetime if and onlyif Schwarz’s criterion is fulfilled which says

(∂λ∂ν − ∂ν∂λ) vµ = 0 (12.74)

terms of second order in dx.If we calculate

(∂λ∂ν − ∂ν∂λ) vµ = −∂λ (Γµνκv

κ) + ∂ν (Γλκµvκ) (12.75)

we find

− (∂λΓνκµ − ∂νΓλκ

µ) vκ − Γµνκ∂λvκ + Γµλκ∂νv

κ (12.76)

and thus, using once more (12.73),

(∂λ∂ν − ∂ν∂λ) vµ = −Rλνκ

µvκ. (12.77)

Thus the parallel field vµ(x) exists in the whole spacetime if and only if the curvaturetensor vanishes everywhere.

If Rλνκ is non-zero, the concept of parallel vectors cannot be carried over fromMinkowski space to the general affine spacetime over any finite distance. Such spacesare called curved . One says that in curved spaces there exists no teleparallelism.

We have illustrated before, that this is the case in the presence of disclinations.Disclinations generate curvature, i.e., a crystal containing disclinations is curved inthe differential geometric sense.

This is in accordance with the previous observation that the disclination densityθij coincides with the Einstein curvature tensor Gij.

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308 12 Torsion and Curvature from Defects

In the illustration we have also seen that even in the presence of a disclinationit still is meaningful to define a vector field as locally parallel . The condition forthis is that the covariant derivatives vanish at that point x0 : Dνv

µ(xa) = 0. If thiscondition is satisfied, the neighboring vector vµ(x), close to x0 differs from vµ(x0) atmost by terms of the order (x − x0)

2 rather than (x − x0) for non-parallel vectors.In order to see this in more detail let us draw an infinitesimal quadrangle ABCDin the coordinate frame xµ spanned by AB = dxµ = DC and BC = dxµ2 = AD (seeFig. 12.3). Now we compare the directions of when going around the circumference.When passing from A at xµ to B at xµ + dxy1 the vector components change fromvµ1 = vµ(x) to

Figure 12.3 Illustration of parallel transport of a vector around a closed circuit ABCD.

vµB = vµ (x+ x) = vµ1 + ∂νvµd x

1

νvAµ− A

Γνλµvλd x

1

ν (12.78)

When continuing to C at x0µ + d

µx1+d

µx2

we have

vcµ = vB

µ− B

ΓτκµvB

κd x2

τ

= vAµ− A

Γνλµvλd x

1

ν− B

ΓτκµvA

κd x2

τ +

+B

Γτκµ A

ΓνλκvAd

νx1d x

2

τ

= vµA−A

Γνλµvλ

(

d x1

ν + d x2

ν)

− ∂νA

ΓτκµvκAd x1

νd x2

τ +

+A

Γτκµ A

ΓνλκvλAd x1

νd x2

τ + O(

dx3)

. (12.79)

We can now repeat the same procedure, but along the line ADC and find the sameresult with d x

1↔ d x

2interchanged. The difference between the two results is

vµABC − vµADC = −1

2Rντκ

µvaκdsντ + O

(

dx3)

(12.80)

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12.3 Differential-Geometric Properties of Affine Spaces 309

where dsντ =(

dνx1dτx2−d τ

x1dνx2

)

is the infinitesimal surface element of the quadran-

gle.There exists a similar geometric illustration of the torsion property Sµν

λ 6= 0.Consider a crystal with an edge dislocation (see Fig. 12.2). Let us focus attentionupon a closed circuit with the form of a parallelogram in the ideal reference crystal(i.e., in the coordinate frame ea) and suppose its image in the eµ-frame encloses thedislocation line (see Fig. 12.4).

Figure 12.4 Illustration of non-closure of a parallelogram after inserting an edge dislo-

cation.

Volterra process of constructing the dislocation, the reference crystal was cutopen, and a layer of atoms was inserted. In this process, the original parallelogramis opened such that the dislocation crystal has a gap between the open ends. Thegap vector is precisely the Burgers vector. To be specific, let the parallelogram in the

ideal reference crystal be spanned by the vectors AB = dax1

= DC,AD = d2xa

= BC.

In the defected spacetime xµ these become AB = dµx1, AD = d

µx2, D′C = d

′x1

µ, BC =

d′x2

µ. Since d′x1

µ, d′x2

µ are parallel in the ideal reference crystal, they are parallel

vectors, i.e., the vectors vµ(x) = dµx2, vµ

(

xµ + dµx1

)

satisfy (12.73) when going from

A to Bj, i.e., ∂νdµx2

= −Γνλµd

λx2

and hence

dx′2µ = d x

2

µ − Γνλµd x

2

νd x2

λ. (12.81)

Similarly the vectors dµx1

and d′x1

µ are parallel and therefore related by

d′x1

µ = d x1

µ − Γµνλdνx2dλx1. (12.82)

From this it follows that

bµ =(

d′x2+d x

1

−(

d′x1+d x

2

= −Sνλλdsνλ. (12.83)

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310 12 Torsion and Curvature from Defects

In a Minkowski space, the torsion vanishes and the image is again a closed parallel-ogram. Einstein assumed the vanishing of torsion in gravitational spacetime.

12.4 Circuit Integrals in Affine Spaces with Curvatureand Torsion

In order to establish contact with the circuit definitions of disclinations and dislo-cations in crystals, let us phrase the differential results (12.80) and (12.83) in termsof contour integrals. Given a vector field vµ(x) which is locally parallel , i.e., whichhas Dνv

µ(x) = 0, consider the change of vµ(x) while going around a closed contourwhich is

∆vµ =∮

dxν∂νvµ(x). (12.84)

By decomposing C into a large set of infinitesimal surface elements we can apply(12.80) and find

∆vµ =∮

C(xµ)dxν∂νv

µ = −1

2

S(xµ)dsτνRτνκ

µ(x)vκ(x). (12.85)

Note that the tetrad fields eµa are locally parallel by definition such that they satisfy

∆eaµ = −

C(xµ)dxν∂νea

µ = −1

2

S(xµ)dsτνRτνκ

µ(x)eκa(x). (12.86)

Actually, this relation follows directly from Stokes’ theorem:

∆eµa =∮

C(xµ)dxν∂νea

µ =∮

C(xµ)dsτν∂τ∂νea

µ =

= −1

2

S(xµ)dsτνRτυκ

µeaκ. (12.87)

For an infinitesimal circuit, we can remove the tetrad from the integral and have

∆eaµ ≈

−1

2

S(xµ)dsτνRτυκ

µ

eaκ ≡ ωµκe

κa. (12.88)

The matrix ωµκ has the property that ωµκ = gµλωλκ is antisymmetric, due to the

antisymmetry of Rενκµ in κµ. Hence ωµκ can be interpreted as the parameters ofan infinitesimal local Lorentz transformation. In three dimensions, this is a localrotation in agreement with what we observed previously:Curvature is a signal for disclinations and these are rotational defects.

Let us now give an integral characterization of torsion. For this we consider anarbitrary closed contour C(xa) in the inertial frame (which generalizes the paral-lelogram used in the previous discussion). In the defected spacetime this contour

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12.4 Circuit Integrals in Affine Spaces with Curvature and Torsion 311

has an image C ′(xa) which does not necessarily close. In order to find how much ismissing we form the integral

C(xa)dxµ =

C(xa)dxa

∂xµ

∂xa=∮

C(xa)dxaea

µ(xa). (12.89)

By Stokes’ theorem, this becomes

1

2

C(xa)dsab (∂aeb

µ − ∂beaµ) =

1

2

S(xa)dsab (ea

ν∂νebµ − (a↔ b)) = −

dsabSabµ.

(12.90)

The quantity

Sabµ = −1

2eaν [∂νeb

µ − (a↔ b)] (12.91)

is called anholonomity of the mapping xa → xµ. It is related to the torsion Sλκµ

conversion of the lower indices from the local to the inertial form

Sabµ = ea

λebκSλκ

µ

= −1

2

eaλeb

κ [ecκ∂λecµ − (a↔ b)]

≡ −1

2

[

eaλ∂λeb

µ − (a↔ b)]

. (12.92)

If the tetrad vectors are known as functions of the external coordinates xa, we canalso use ea

λ∂λ = ∂a and write the anholonomity in the form

Sabµ ≡ −1

2[∂aeb

µ − (a↔ b)] (12.93)

Sometimes one also converts the upper Einstein index µ into a local Lorentz indexc and works with

Sabc = ecµSab

µ = −1

2

[

ecµ∂aebµ − (a↔ b)

]

. (12.94)

If there is no torsion, the integral (12.90) vanishes. Otherwise the image of theclosed contour C(xa) has a gap and thus is defined as the Burgers vector

bµ =∫

C′(xµ)dxµ = −

C(xa)dsabSab

µ. (12.95)

It should be mentioned that often the circuit integrals measuring curvature andtorsion are executed in the opposite way by forming closed circuits C(xµ) aroundthe defect in the space xµ and studying the properties of the image circuit C ′(xa)in the ideal reference crystal. In the case of torsion, one measures by how much theimage c′(xa) fails to close. This gives the Burgers vector

ba =∫

c′(xa)dxa =

c(x′µ)dxµ

∂xa

∂xµ=∫

c(xµ)dxµeaµ (12.96)

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312 12 Torsion and Curvature from Defects

which, by Stokes’ theorem can be rewritten as

ba∫

S(xµ)dsνµ∂νe

aµ =

S(xµ)dsνµSνµ

λeaλ(x). (12.97)

The tensor Sµa ≡ Sλνµe

aλ = 1/2

(

∂µeaν − ∂νe

)

is obviously a converse form of the

anholonomity (12.95), with Einstein indices exchanged by local Lorentz indices.There is an analogous circuit integral characterizing the curvature from the

standpoint of the coordinates xa. For this one introduces the local Lorentz ten-sor related to Rµνρ

κ:

Rabcd ≡ ea

µebνec

λedκRµνλκ (12.98)

Then th circuit integral reads

∆eaµ = −1

2

S(xµ)dsedReda

bebµ. (12.99)

If one wants to calculate Rabcd directly in xa spacetime using differentiations one has

to keep in mind that under the anholonomic mapping xa → xµR, is not a tensor.In fact, a simple manipulation shows

Rµνλκ = ea

κ(

∂µ∂ν − ∂ν∂µ)

edλ

= edκ[

eaµ∂aebν∂b − (µ↔ ν)

]

edλ

= eaµebνed

κ (∂a∂b − ∂b∂a) edλ +

[

eaµedκ(

∂aebν

) (

∂bedλ

)

− (µ↔ ν)]

= eaµebνRabλ

κ +[

eaµedκea

σΓσνbeb

τΓτλd − (µ↔ ν)

]

= eaµebνRabλ

κ +[

ΓµνσΓκσλ − (µ↔ ν)

]

= eaµebνRabλ

κ + 2SµνσΓσλ

κ, (12.100)

where

Rabλκ ≡ ed

κ (∂a∂b − ∂b∂a) edλ (12.101)

is evaluated in the same way ad Rµνλκ in Eq. (11.129), but by forming ∂a derivatives

rather than ∂µ. Expressing also the torsion Sµνσ in terms of derivatives ∂/∂xa = ∂a

as in (12.93) we can write

Sµνσ = eaµe

bνec

σSabc. (12.102)

For the affine connection we may define, similarly,

Γµνσ ≡ eaµe

bνec

σΓabc (12.103)

with

Γabc = ea

µebνecλΓµν

λ = −eaµebνecλedν∂µedλ = −eaµecλ∂µebλ

= −ecλ∂aebλ ≡ ecλΓabλ = eb

λ∂aecλ. (12.104)

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12.5 Bianchi Identities for Curvature and Torsion Tensors 313

Then Rabcd of (12.98) can be written as

Rabcd = Rabd

d + 2SabeΓec

d. (12.105)

It should be pointed out that a nonvanishing curvature tensor has the consequencethat covariant derivatives no longer commute. If we form

DµDµvλ −DνDµvλ (12.106)

we find after some algebra

DνDµvλ −DµDνvλ = −Rνµλκvκ − 2Sνµ

ρRρvλ,

DνDµvκ −DµDνv

κ = Rνµλκvλ − 2Sνµ

ρDρvκ. (12.107)

12.4.1 Parallelism in World Crystal

From the standpoint of a world crystal with defects, parallelism has a simple mean-ing. Consider Fig. 11.1b. We identify the dashed curves xa = const. with indicatethe crystal planes of an elastically distorted crystal as seen from the local frame xµ,which may be thought of as the coordinates of an undistorted reference crystal. Anobserver within the distorted crystal orients himself by the planes xa = const. Hemeasures distances and directions by counting atoms along the crystal directions.The above definition of parallelism amounts to vectors being defined as parallel ifthey are so from his point of view, i.e., if they correspond to parallel vectors in theundistorted crystal. Thus the normal vectors to the dashed coordinate planes xa =const. are parallel to each other. Indeed, they form the vector fields eµa(x), whichalways satisfy Dνe

µa = 0 [see (11.95].

If the mapping xµ(xa) contains defects it is, in general, impossible to find aglobal definition of parallelism. Consider, for example, a wedge disclination whichis shown in Fig. 12.2, say the −900 one. The crystal has been cut from the left,and new crystalline material has been inserted in the Volterra construction process.The crystalline coordinate planes define parallel lines. With the right-hand piecestemming from the original crystal, there exists a completely consistent definitionof parallelism. For example, the almost horizontal planes are all parallel. The linescutting these vertically are also parallel by definition. On the left-hand side, thevertical lines continue smoothly into the inserted new crystalline material. In themiddle, however, they meet and turn suddenly out to be orthogonal. Still, thecoordinate planes define parallelism in any small region inside the original as wellas the inserted material except on the disclination line.

12.5 Bianchi Identities for Curvature and TorsionTensors

Because of their physical importance, let us derive a few important properties ofcurvature and torsion tensors. As noted before, the curvature tensor is antisym-metric in µν, by construction, and in λκ, due to the integrability condition (11.19)

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314 12 Torsion and Curvature from Defects

for the metric tensor. In addition, it satisfies so-called fundamental identity . Thisfollows directly from the representation (11.128), by adding terms in which µνλ areinterchanged cyclically:

(12.108)

where the symbol abbreviates a sum of cyclic permutations of the indicatedsubscripts. The derivation of the fundamental identity requires commuting deriva-tives in front of the metric tensor, i.e., it requires that the metric gµν satisfies theintegrability condition

(∂µ∂ν − ∂ν∂µ)gλκ = 0. (12.109)

It is therfore a Bianchi identity.In symmetric spaces where Sµνλ = 0 and Rµνλκ = Rµνλκ, the fundamental iden-

tity implies the additional symmetry property of the Riemann tensor

Rµνλκ + Rνλµκ + Rλµνκ = 0. (12.110)

Using the antisymmetry in µν and λκ leads once more to the property (11.143):

Rµνλκ = Rλκµν , (12.111)

which holds in symmetric spaces for the full curvature tensor.Another important identity is the original Bianchi identity which has given the

name to all similar identities in this book which are based on the intergrability con-dition of observable fields. The original Bainchi identity follows from the assumptionof the single-valuedness of the affine connection which implies that it satisfies theintegrability condition

(

∂µ∂ν − ∂ν∂µ)

Γλκρ = 0. (12.112)

Consider the vector

Rσνµ ≡ (∂σ∂ν − ∂ν∂σ) eµ, (12.113)

which determines the curvature tensor Rσνµλ via the scalar product with eλ [recall

(11.129)]. Applying the covariant derivative gives

DτRσνµ = ∂τRσνµ − ΓτσκRκνµ − Γτυ

κRσνκ. (12.114)

Performing cyclic sums over τσν and using the antisymmetry of Rσνµ in σν leadsto

DτRσνµ = ∂τRσνµ − ΓτµκRσνκ + 2Sτσ

κRνκµ. (12.115)

Now we use

∂σ∂νeµ = ∂σ(

Γνµαeα

)

= Γνµκeκ, (12.116)

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12.5 Bianchi Identities for Curvature and Torsion Tensors 315

to derive

∂τ∂σ∂νeµ = ∂τΓνµκ∂σeκ + (τσ) + ∂τ∂σΓνµ

κeα + Γνµκ∂τ∂σeκ. (12.117)

Antisymmetrizing in στ gives

∂τ∂σ∂νeµ − ∂σ∂τ∂νeµ = ΓνµαRτσα +

[

(∂τ∂σ − ∂σ∂τ ) Γνµα]

eα. (12.118)

This is the place where we make use of the integrability condition for the con-nection (12.112), dropping the last term. Hence

(12.119)

Inserting this into (12.115) and multiplying by en we obtain an expression involvingthe covariant derivative of the curvature tensor

(12.120)

This is the Bianchi identity , which guarantees the integrability of the connection.Within the defect interpretation of torsion and curvature, we are now prepared

to demonstrate that these two identities have a simple physical interpretation. Theyare the non-linear versions of the conservation laws for dislocation and disclinationdensities. These read5

∂iαij = −εjklθkl, (12.121)

∂iθij = 0. (12.122)

They state that disclination lines never end while dislocation lines can end at mostat a disclination line.

Consider now Eq. (12.120). Linearizing this gives

∂τRσνµλ + ∂σRντµ

λ + ∂νRτµσλ = 0. (12.123)

Contracting ν and µ and τ with λ we obtain

∂τRσνντ + ∂σRνλ

νλ + ∂νRτνστ = 2∂τRσ

τ + ∂σR = 2∂τGστ = 0. (12.124)

Since in three dimensions, the Einstein tensor Gµν corresponds to the disclinationdensity θµν in Eq. (12.38), we see that (12.124) indeed coincides with the defectconservation law (12.122).

The fundamental identity (12.108) has the linearized form

2(

∂νSµλκ + ∂µSλν

κ + ∂λSνµκ)

= Rνµλκ +Rµλν

κ +Rλνµκ. (12.125)

5See Eqs. (11.92) and (11.93) in Part III of the textbook [4].

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316 12 Torsion and Curvature from Defects

Contracting ν and κ gives

2(

∂νSµνν + ∂µSλν

ν − ∂λSµνν)

= Rνµλν +Rµλκ

κ +Rλνµν = Rµλ − Rλµ,(12.126)

where we have used the antisymmetry of Rνµλκ in the last two indices which is aconsequence of the integrability condition for the metric tensor. The right-hand sideis the same as Gµλ −Gλµ.

In three dimensions we can contract this equation with the ε-tensor and find

εjkl (∂iSkli + ∂kSlnm − ∂lSknn) = εijklGkl. (12.127)

Inserting here Sklj = (1/2)εkliαij from (12.38), and Eq. (12.44) for Glk, this becomesthe conservation law (12.121) for the dislocation density.

12.6 Special Coordinates in Riemann Spacetime

12.6.1 Geodesic Coordinates

To a local observer, curved spacetime looks flat in his immediate neighborhood.After all, this is why men believed for a long that the earth has the form of a flatdisc. In four-dimensional spacetime the equivalent statement is that, in a freelyfalling elevator cabin, people would not experience any gravitational force as longas the cabin is small enough to make higher non-linear effects negligible. The cabinconstitutes an inertial frame of reference for the motion of a mass point. FromEq. (11.22) we see that its coordinates in an arbitrary geometry can be determinedfrom the requirement of a vanishing Christoffel symbol µ′λ′, κ′ = 0, which amountsto

∂λ′gµ′λ′(x′) = 0, (12.128)

∂λ′gµ′λ′(x′) = −gµ′σ′gλ′τ ′∂λ′gσ′τ ′(x′) = 0 (12.129)

Given an arbitrary set of coordinates x, the derivatives are connected by

∂λ′gµ′λ′(x′) = ∂λ

[

gµν(x)αµµ′αν

ν′]

αλλ′

= ∂λgµν(x)αµ

µ′ανν′αλλ′

+ gµν∂λαµµ′αν

ν′αλλ′ + gµναµµ′∂λαν

ν′αλλ′. (12.130)

Recall that derivative symbols ∂µ are meant to act only on the first function behindit. Equations (12.128) or (12.129) provide us with D2(D + 1)/2 partial differentialequations for the D coordinates x′µ

(x) which do not, in general, have a solutionover a finite region. If ∂λ′g

µ′λ′ were to vanish over a finite region, the spacetimewould necessarily be euclidean. So we can, at best, achieve

∂λ′gµ′ν′(x′0) = 0 (12.131)

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12.6 Special Coordinates in Riemann Spacetime 317

at some point x0. This implies, via (12.128), that also ∂λ′gσ′τ ′(x′0) = 0 and thus the

vanishing of the Christoffel symbols at that point. Then a mass point will moveforce-free at x0. In any neighborhood of x0 there are gravitational forces of orderO(x− x0).

Let us try and solve (12.131) by an expansion

x′µ′

= x0µ + aµλ(x− x0)

λ +1

2!aµλκ(x− x0)

λ(x− x0)κ

+1

3!aµλκδ(x− x0)

λ(x− x0)κ(x− x0)

δ + . . . . (12.132)

The associated transformation matrix αµµ′ ≡ ∂x′µ

/∂xµ satisfies

αµµ′ = aµ

µ′ + aµλµ′(x− x0)

λ +1

2!aµλκ

µ′(x− x0)λ(x− x0)

κ + . . . ,

∂λαµµ′ = aµλ

µ′ + aµλκµ′(x− x0)

κ + . . . ,

∂κ∂λαµµ′ = aµλκ

µ′ + . . . . (12.133)

Inserting this into (12.130) we find

∂λgµ′ν′ = ∂λg

µν(x0)aµµ′aν

ν′ + gστ (x0)[

aσλµ′aτ

ν′ +(

µ′ ↔ ν ′)]

= 0 + O(x− x0). (12.134)

This is solved by

aµµ′ = gµ

µ′(x0), aλκµ =

1

2Γλκ

µ(x0), (12.135)

in accordance with (11.99). Hence the coordinates which are locally geodesic at x0

are given by

x′µ = x0µ + (x− x0)

µ +1

2Γλκ

µ(x− x0)λ(x− x0)

κ + O(

(x− x0)3)

. (12.136)

Note that while the Christoffel symbols vanish in the geodesic frame at x0, theirderivatives are nonzero if the curvature is nonzero at x0.

In order to complete the construction of a freely falling coordinate system wejust note that in the neighborhood of the point, the geodesic coordinates can alwaysbe brought to a Minkowski-form by a further linear transformation

(

x′ − x0

)µ → Lµα (xµ − x0)α (12.137)

which transforms gµν into gαβ, i.e.,

gαβ = LµαLνβgµν = gαβ. (12.138)

Such a linear transformation does not change the geodesic property of the coordi-nates such that the coordinates (x′′ − x0)

α are a local inertial frame, which is whatwe wanted to find.

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318 12 Torsion and Curvature from Defects

As far as the crystalline defects are concerned, the possibility of constructinggeodesic coordinate is related to the fact that, in the regions between defects, thecrystal can always be distorted elastically to form a regular array of atoms. In thecontinuum limit, these regions shrink to zero but so do the Burgers’ vectors of thedefects. Therefore even though any small neighborhood does contain some defects,these themselves are infinitesimal such that the perfection of the crystal is disturbedonly infinitesimally.

12.6.2 Canonical Geodesic Coordinates

The condition of being geodesic determined the coordinates transformation (12.132)up to the coefficients of the quadratic terms.

x′µ = x0µ + (x− x0)

µ +1

2Γλκ

µ (x− x0)λ (x− x0)

κ

+1

3!aµλκδ (x− x0)

λ (x− x0)κ (x− x0)

δ + . . . (12.139)

By construction, the transformation matrix

αµν =∂x′µ

∂xν= δµν + Γλν

µ (x− x0)λ +

1

2aµλκν (x− x0)

λ (x− x0)κ + . . . (12.140)

has the property of making the Christoffel symbol of the point x0 vanish. It isobvious that the higher coefficients aµλκδ must have an influence upon the derivativesof the Christoffel symbols. In general, these cannot be made zero since the curvaturetensor at the point x0 where Gammaµ′ν′

λ′ vanishes is

Rµ′ν′λ′κ′ = ∂µ′Γν′λ′

κ′ −(

µ′ ↔ ν ′)

. (12.141)

This implies that only in a flat spacetime can one find aµλκν to have also ∂µ′Γν′λ′κ′ = 0.

Even though the derivatives cannot be brought to zero, there is a most conve-nient coordinate system referred to as canonical , in which the derivatives satisfy thefollowing relation

∂µ′ Γν′λ′κ′ + ∂λ′Γν′µ′

κ′ = 0. (12.142)

Before we show how to find such a system device, let us first see what its advantagesare. The canonical condition allows us to invert the relation (12.141) for Rκ′

µ′ν′λ′ andexpress (always at a geodesic coordinate point) the derivatives of the Christoffelsymbols uniquely in terms of the curvature tensor

∂ν′Γµ′λ′κ′ = −1

3

(

Rµ′ν′λ′κ′ +Rλ′µ′ν′

κ′)

. (12.143)

This has the consequence, the metric gµ′ν′(x′) can in the neighborhood of the point

x0µ, be expanded uniquely up to second order in terms of the curvature tensor. In

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12.6 Special Coordinates in Riemann Spacetime 319

order to see this we recall Eqs. (11.99). Differentiating these once more we find thesecond-order derivatives

∂κ∂λgµν = ∂κΓλµσgσν + ∂κΓλν

σ + Γλµσ∂κgσν + Γλν

σ∂κgµσ. (12.144)

At a point where the coordinates are geodesic, this becomes simply

∂κ′∂λ′gµ′ν′ = −1

3

(

Rλ′κ′µ′σ′ +Rµ′κ′λ′

σ′)

gσ′ν′ −1

3

(

Rλ′κ′ν′σ′ +Rν′κ′λ′

σ′)

gµ′σ′

=1

3

(

Rκ′µ′λ′υ′ +Rκ′ν′λ′µ′

)

. (12.145)

Hence the metric has the expansion

gµ′ν′(x′) = gµ′ν′(x0) +

1

2∂κ′∂λ′gµ′ν′(x0)

(

x′ − x0

)κ′ (

x′ − x0

)λ′

+ . . .

= gµ′ν′ (x0) +1

3Rz′µ′λ′ν′

(

x′ − x0

)κ′ (

x′ − x0

)λ′

+ . . .

= gµ′ν′(x0) +1

3Rz′µ′λ′ν′

(

x′ − x0

)κ′

(x− x0)λ′ + . . . (12.146)

Let us now turn to the construction of these canonical coordinates. For this we takethe transformation law for the Christoffel symbols (11.103)

Γµ′ν′λ′ = αµ′

µαν′ναλ

λΓµνλ − αµ′

µαν′ν∂µα

λ′

ν , (12.147)

and differentiate once more by

∂κ′ =∂xκ

∂xκ′∂κ = ακ′

κ∂κ. (12.148)

This gives

∂κ′Γµ′ν′λ′ = ακ′

καµ′µαν′

ναλ′

λ∂κΓµνλ + ακ′

κ∂καµ′µαν′

ναλ′

λΓµνλ

+ ακ′καµ′

µ∂καν′ναλ

λΓµνλ + ακ′

καµ′µαν′

ν∂καλ′

λΓµνλ (12.149)

− ακ′κ∂καµ′

µαν′ν∂µα

λ′

ν− ακ′καµ′

µ∂καν′ν∂µα

λ′

ν + ακ′καµ′

µαν′ν∂κ∂µα

λ′

ν .

Besides the known transformation coefficient αµν, this formula also involves theinverse coefficients ακ

λ. Since for x ∼ x0, αµν are close to unity, the inverse is

simply (recall (11.68)

αµν =

∂xν

∂x′µ= δµ

ν − Γλνµ (x− x0)

λ + . . . (12.150)

Indeed,

αµναµ

ν = δµµ′ + Γλν

µ (x− x0)λ − Γλν

µ (x− x0)λ + . . . = δµ

µ′ . (12.151)

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320 12 Torsion and Curvature from Defects

Inserting αµν and ανν into the above transformation law gives

∂κ′Γµ′ν′λ′ =

[

∂κΓµνλ + Γµν

σΓσκλ − aλκµν

]

κ′=κ,λ′=λ,µ′=µ,ν′=ν. (12.152)

Note the appearance of the coefficients aλκµν of the cubic expansion terms. Inter-changing on the left-hand side κ′µ′υ′ cyclically and adding the three expansionsgives

∂κ′Γµ′ν′λ′ + 2 cyclic perms of

(

κ′µ′ν ′)

(12.153)

+[

∂κΓµνλ + Γµν

σΓσκλ +2 cyclic perms of (κµν) − 3aλκµν

]∣

κ′=κ,λ′=λ,µ′=µ,ν′=ν.

By setting the left-hand side equal to zero we obtain the desired equation for abκµν ..

Thus given an arbitrary coordinate frame xµ, the coefficients abκµν can indeedbe chosen such as to make the geodesic coordinate frame x′µ in the neighborhood ofthe point x0 canonical, and thereby determining gµν(x

′) in this neighborhood up toquadratic order uniquely in term of the curvature tensor as stated in Eq. (12.146).

12.6.3 Harmonic Coordinates

While geodesic properties of coordinates can be enforced at most at one point thereexists a way of fixing the choice of coordinates in the entire spacetime by choos-ing what are called harmonic coordinates. These were introduced first by T. De-Donder and C. Lanczos and extensively used by V. Fock in his gravitational work[2]. Given an arbitrary set of coordinates xµ, one asks for d independent scalarfunction fa(x) (a = 0, 1, 2, d) which satisfy the Laplace equation in curved space

D2fa(x) = gµνDµDνfa(x) = 0. (12.154)

Since fa(x) are supposed to be scalar functions, we calculate

DµDνf = Dµ∂νf =(

∂µ∂ν − Γµνλ∂λ

)

f (12.155)

and hence

D2f = gµνDµDνf =(

gµν∂µ∂ν − Γλ∂λ)

f (12.156)

where we have introduced the contracted affine connection

Γλ ≡ Γµµλ, (12.157)

for brevity. In a symmetric space

Γλ =1

2gµνgλκ

(

∂µgνκ + ∂νgµκ − ∂κgµν)

= gµνgλκ∂µgνκ −1

2gλκgµν∂κgµν

= − 1√−g∂κ(√−ggλκ

)

(12.158)

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12.6 Special Coordinates in Riemann Spacetime 321

and

D2f =

[

gµν∂µ∂ν +1√−g∂κ

(√−ggλκ)

]

f = ∆f, (12.159)

where

∆ ≡ 1√−g(

∂µgµν√−g∂ν

)

(12.160)

is the Laplace-Beltrami operator ∆ in curved space. The Laplace operator in aspacetime with torsion is related to the Laplace-Beltrami operator by

D2f = ∆f − Sµµν∂λf. (12.161)

Suppose we have found D functions fa(x) which satisfy (12.154), then we introducethe harmonic coordinates Xa as

Xa = fa(x). (12.162)

When transforming the Laplace equation (12.154) from coordinates xµ to the har-monic coordinates Xa, we obtain

(

gbc∂b∂c − Γc∂c)

Xa = −Γcδca = 0. (12.163)

Thus, harmonic coordinates are characterized by vanishing Γa (a = 1, . . . , d).

12.6.4 Coordinates with det(gµν) = 1

A further choice of coordinates which was favored by Einstein, since it simplifiessome formulas, is one in which the determinant of the metric is constant and hasthe Minkowski value −1 in all space. Since

Γµνµ =

1

2gµν

(

∂µgνλ + ∂νgµλ − ∂λgµν)

=1

2gµν∂νgµν =

1√−g∂ν√−g = ∂ν log

√−g (12.164)

we can state this condition also in the form

Γµνµ = 0. (12.165)

Given an arbitrary coordinate system xµ, the special ones xµ are found by a trans-formation

xµ = αµνxν (12.166)

which fulfills the condition

√−g = |det (αµν) |√−g = |det (αµν) |. (12.167)

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322 12 Torsion and Curvature from Defects

Taking the logarithm and differentiating gives

Γµνµ = ∂ν log det

(

αλκ)

= ∂νtr log(

αλκ)

= tr(

α−1∂να)

= αλκ∂να

λκ = 0. (12.168)

These are D differential equations which determine D new coordinate functionsx(x).

Note the difference with respect to harmonic coordinates which have Γµ =Γλ

λµ = 0, i.e., the first two indices contracted, on the general connection, whilethe present condition has the first (or the second index) contracted with the third,on the Christoffel symbol.

12.6.5 Orthogonal Coordinates

For many calculations, it is useful to employ orthogonal coordinates, in which gµνhas only diagonal elements. Then many entries of the Christoffel symbols vanish

Γµλ,κ = 0, Γµλκ = 0, µ 6= λ, κ 6= µ, κ 6= λ, (12.169)

and the calculation of the others is greatly simplified. In a symmetric space, we mayuse formula (12.1) for the Riemann tensor Rµνλ

κ and find that it vanishes wheneverall its indices are different. The non-vanishing elements can be calculated as follows:

ν 6= κ, λ 6= µ, ν 6= λ

Rνκλµ = −1

2

(

∂λ∂κgµν + ∂µ∂νgκλ − ∂λ∂νgµκ)

+1

2∂µ(

log√gνν)

∂νgλκ −1

2∂ν

(

log√

gµµ

)

(

∂λgµκ − ∂µgλκ)

(12.170)

−1

2∂λ(

log√gνν) (

∂νgµκ − ∂κgµν)

− 1

2∂ν(

log√gλλ

) (

∂λgκµ − ∂µgκλ)

+1

2Γκλ

ρ∂ρgνµ,

ν 6= κ, λ 6= µ, ν 6= λ, ν 6= µ

Rνκλµ = −1

2

(

∂µ∂νgκλ − ∂λ∂νgµκ)

+1

2∂µ(

log√gνν)

∂νgλκ −1

2∂λ(

log√gνν)

∂αgµκ (12.171)

−1

2∂ν

(

log√

gµµ

)

(

∂λgµκ − ∂µgλκ)

− 1

2∂ν log

√gλλ

(

∂λgµκ − ∂µgκλ)

,

ν 6= κ, λ 6= µ, ν 6= λ, ν 6= µ, κ 6= λ

Rνκλµ =1

2∂λ∂νgµκ −

1

2∂ν

(

log√

gµµgλλ

)

∂λgµκ −1

2∂λ(

log√gνν)

∂νgµκ. (12.172)

The Ricci tensor becomes

Rµν =∑

λ

1

gλλRµλλν (12.173)

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12.7 Number of Independent Components of Rµνλκ and Sµν

λ 323

giving the off-diagonal elements

µ 6= ν

Rµν =∑

λ6=µ,λ6=ν

[

∂µ∂ν(

log√gλλ

)

− ∂µ(

log√gνν)

∂ν(

log√gλλ

)

− ∂µ(

log√gλλ

)

∂ν

(

log√

gµµ

)

+ ∂µ(

log√gλλ

)

∂ν(

log√gλλ

)

]

= −∂µ∂ν log√−g + ∂µ

(

log√gνν)

∂ν(

log√−g

)

+ ∂µ log√−g∂ν log

gµµ (12.174)

+ ∂m∂ν log√

gµµgνν − 2∂µ(

log√gνν)

∂ν log√

gµµ −D∑

λ=1

∂µ(

log√gλλ

)

∂ν(

log√gλλ

)

,

and the diagonal elements

Rµµ = −∂2 log√−g + 2∂2

µ

(

log√

gµµ

)

− 2(

∂µ log√

gµµ

)2

+2∂µ(

log√−g

)

∂µ

(

log√

gµµ

)

−D∑

λ=1

(

∂µ log√gλλ

)2

−gµµD∑

λ=1

1

gλλ

[

∂2λ

(

log√

gµµ

)

+ ∂λ(

log√−g

)

∂λ

(

log√

gµµ

)

(12.175)

−2∂λ(

log√gλλ

)

∂λ(

log√gλλ

)]

.

The curvature scalar reads

R =D∑

λ

1

gλλ

2∂λ2(

log√−g

)

− 2∂λ2 log

√gλλ

+2(

∂λ log√gλλ

)2 − 4∂λ(

log√gλλ

)

∂λ(

log√−g

)

+(

∂λ log√−g

)2+

D∑

κ

(

∂λ√gκκ

)2

. (12.176)

12.7 Number of Independent Components of Rµνλκ and

Sµνλ

With the antisymmetry in µν and λκ, there are in d dimensions, N Rd = [d(d− 1)/2]2

components of Rµνλκ and NSd = d2(d− 1)/2 components of Sµν

λ. In symmetric

spaces, the further symmetry of Rµνλκ between µν and λκ. Thus Rµνλκ may be

viewed as an 12d(d − 1) × 1

2d(d − 1) matrix in the first and second pairs R(µν) (λκ)

This matrix is symmetric due to the absence of torsion, so that it has

1

2

1

2d(d− 1) ×

[

1

2d(d− 1) − 1

]

=1

8d(d− 1)(d2 − d+ 2) (12.177)

components. Now, the first fundamental identity (12.110) not only leads to thesymmetry, it contains also the information that the completely antisymmetric part of

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324 12 Torsion and Curvature from Defects

Rµνλκ+Rνλµκ+Rλνκ vanishes. This gives d(d− 1)(d− 2)(d− 3)/4! further relations,and one is left with

N Rd =

1

8d(d− 1)(d2 − d+ 2) − 1

24d(d− 1)(d− 2)(d− 3) =

1

12d2(d2 − 1)(12.178)

independent components of Rµνλκ. In four dimensions, this number is 20, in threedimensions it is 6.

12.7.1 Two Dimensions

In two dimensions, there is only one independent component, for instance R1221.Indeed, the most general tensor with the above symmetry properties is

Rµνλκ = −const × eµνeλκ, (12.179)

where eµν =√−gεµν is covariant version of the Levi-Civita symbol (ε12 = −ε21 = 1,

see Appendix 11A). Contracting this with gνλ gives the Ricci tensor

Rµκ = −const × gνλεµνελκ = −const ×(

gµκ − gλλgµκ

)

= const × gµκ, (12.180)

and the scalar curvature

R = const × gµνgµκ = 2 × const . (12.181)

Hence const = R/2 and the single independent element of Rµνλκ is

R1221 = gR

2(12.182)

while the full curvature tensor is given by

Rµνλκ = −eµνeλκR

2. (12.183)

The number N Rd = 6 of independent components of Rµνλκ in three dimensions

agrees with the number of independent components of the Ricci tensor Rµν , whose

knowledge must therefore be sufficient to calculate Rµνλκ. Indeed, we can easily seethat

Rµνλκ =1

4eµνδeλκτ

(

Rτδ − 1

2gτδR

)

(12.184)

where

eµνδ =√−gεµνδ (12.185)

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12.7 Number of Independent Components of Rµνλκ and Sµν

λ 325

is the three-dimensional covariant version of the Levi-Civita symbol in Eq. (11A.7).The proof of Eq. (12.184) follows by contraction with eµνδeλκτ , which gives via theidentity for products of two Levi-Civita tensors

Rτδ − 1

2gτδR = eµνδeλκτ Rµνλκ. (12.186)

This equation is trivially valid due to the identity (1A.18) and the definition (11.136)of the Ricci tensor.

Since Rµνλκ is a tensor, its N Rd = d2(d2 − 1)/12 components are different in

different coordinates frame. It is useful to find out how many independent invariantsone can form which do not depend on the frame. In two dimensions, the scalarcurvature R determined Rµνλκ completely. In general, the invariants of Rµνλκ can

all be constructed by suitable contractions with gµν. The tensors Rµνλκ and gµν

together have d2(d2 − 1)/12 + d(d+ 1)/2 matrix elements. There are d2 arbitrarycoordinate transformations matrices ∂x′µ/∂xλ which can be applied to these tensors.The number of invariants is equal to the number of independent components of bothRµνλκ and gµν have in a specific coordinate system. This number is

N invd =

1

12d2(d2 − 1) +

d(d+ 1)

2−N2 =

d

12(d− 1)(d− 2)(d+ 3). (12.187)

This formula is valid only for d 6= 2 since for d = 2 we have seen before that thereis N inv

2 = 1 invariant, the scalar curvature. The above counting breaks down sinceone of the N2 coordinate transformations subtracted in (12.187) happens to leaveR1234 and gµν invariant. For d = 3, 4 the numbers are N inv

3,4 = 3, 14 respectively.

12.7.2 Three Dimensions

In three dimensions, the invariants are the eigenvalue of the characteristic equation

det(

gµλRλκ − λδµκ)

= det(

g−1R−λ)

= −λ3 + λ2I1 − λI2 + I3, (12.188)

where

I1 = tr(

g−1R)

= gµνRλµ = R,

I2 =1

2

(

RµνR

νλ − Rλ

λRκκ)

, (12.189)

I3 = det(

g−1R)

= det(

gµλRλν

)

=det

(

Rµν

)

det(

gµν) .

12.7.3 Four or More Dimensions

In four or more dimensions, relation (12.184) generalizes to the Weyl decompositionof the curvature tensor

Rµνλκ = − 1

d − 2

(

gµλRνκ − gνλRµκ + gνκRµλ − gµκRνλ

)

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326 12 Torsion and Curvature from Defects

+R

(d− 1)(d− 2)

(

gµλgνκ − gνλgµκ)

+ Cµνλκ, (12.190)

where Cµνλκ is called the Weyl conformal tensor , which vanishes for d = 3, due to(12.184). Each of the three terms in this decomposition has the same symmetryproperties as Rµνλκ. In addition, Cµνλκ satisfies the d(d+ 1)/2 conditions

Cµκ = Cµν

νκ = 0, (12.191)

since the Ricci tensor comes entirely from the first two terms. Hence, the Weyltensor has

NCd =

1

12d2(d2 − 1) − 1

2d(d+ 1) =

1

12d(d+ 1)(d+ 2)(d− 3) (12.192)

independent elements which is the same as the number of invariants of Rµνλκ. Inmany cases, this makes it possible to find all invariants by going to a coordinateframe in which gµν = gµν and Rµν = diagonal, the first by going to a freely fallingframe, the second by performing an appropriate additional Lorentz transformation.This procedure works as long as Rµν does not have any degenerate eigenvalues.Otherwise the Lorentz transformations remain independent and the counting doesnot work [3].

The above results have interesting consequences as far as a possible geometrictheory of gravitation in lower-dimensional spaces is concerned. It turns out that a3+1-dimensional spacetime is impossible to have a theory which reduces to Newton’stheory in the weak coupling limit. As we shall see later in Chapter 15, the crucialgeometric quantity in Einstein’s theory is the Einstein tensor

Gµν = Rµν −1

2gµνR. (12.193)

Einstein’s theory postulates this tensor to be proportional to the symmetric energy-

momentum tensor of matter,m

T µν [see Eq. (15.62)]

Gµν = κTµν , (12.194)

with some constant κ. In the above discussion we have learned that the Ricci tensorin two spacetime dimensions can be expressed in terms of the scalar curvature as

Rµν = gµνR

2. (12.195)

But this implies that in two spacetime dimensions, the Einstein tensor Gµν vanishesidentically . Hence also the energy-momentum tensor vanishes and there is no Ein-stein theory of gravity. At first sight, there seems to be an escape by allowing forthe presence of a so-called cosmological term, in which case the Einstein equationreads,

Gµν = κTµν + Λgµν . (12.196)

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12.7 Number of Independent Components of Rµνλκ and Sµν

λ 327

However, even if this is added, the two-dimensional theory has a severe problem:The metric gµν is determined completely by the local energy-momentum tensor

gµν = −κ

Λ

m

T µν , (12.197)

and hence vanishes in the empty spacetime between mass points. Thus also thisversion of gravity is unphysical.

How about a geometric theory of gravitation in 2+1 dimensions? Here the Riccitensor is independent of scalar such that there does exist a nontrivial Einstein tensorGµν . Still, the tensor is almost trivial. In three dimensions there is no Weyl tensorCµνλκ and the full curvature tensor is determined in terms of the Ricci tensor byEq. (12.190) for d = 3

Rµνλκ = −(

gµλRνκ − gνλRµκ + gνκRµλ − gµκRνλ

)

+R

2

(

gµλgνκ − gνλgµκ)

. (12.198)

Inserting

Rµν = Gµν −gµνd− 2

G, (12.199)

this becomes

Rµνλκ = −(

gµλGνκ − gνλGµκ + gνκGµλ − gµκGνλ

)

+ G(

gµλgνκ − gνλgµκ)

. (12.200)

According to Einstein’s equation we have

Gµν = κTµν , (12.201)

and see that Rµνλκ is completely determined by the local energy distribution. In theempty spacetime between masses there is no curvature. As we shall understand later,this implies physically that interstellar dust would experience no relative acceleration(tidal forces).

12.7.4 Constant Curvature

For a space with constant curvature all these equations simplify. Consider a sphereof radius r embedded in D dimensions has an intrinsic dimension D′ ≡ D − 1 anda curvature scalar

R =(D′ − 1)D′

r2 . (12.202)

This is most easily derived as follows. Consider a line element in D dimensions

(dx)2 = (dx1)2 + (dx2)2 + . . .+ (dxD)2 (12.203)

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328 12 Torsion and Curvature from Defects

and restrict the motion to a spherical surface

(x1)2 + (x2)2 + . . .+ (xD)2 = r2, (12.204)

by eliminating xD. This brings (12.203) to the form

(dx)2 = (dx1)2 + (dx2)2 + . . .+ (dxD′

)2 +(x1dx1 + dx2 + . . .+ xD

dxD′

)2

r2 − r′2, (12.205)

where r′2 ≡ (x1)2 + (x2)2 + . . .+ (xD′

)2. The metric on the D′-dimensional surfaceis therefore

gµν(x) = δµν +xµxν

r2 − r′2. (12.206)

Since R will be constant on the spherical surface, we may evaluate it for small xµ

(µ = 1, . . . , D′) where gµν(x) ≈ δµν + xµxν/r2 and the Christoffel symbols (11.22)

are Γµνλ ≈ Γµνλ ≈ δµνxλ/r

2. Inserting this into (11.128) we obtain the curvaturetensor for small xµ:

Rµνλκ ≈1

r2

(

δµκδνλ − δµλδνκ)

. (12.207)

This can be extended covariantly to the full surface of the sphere by replacing δµλby the metric gµλ(x):

Rµνλκ(x) =1

r2

[

gµκ(x)gνλ(x) − gµλ(x)gνκ(x)]

, (12.208)

so that Ricci tensor is [recall (11.136)]

Rνκ(x) = Rµνκµ(x) =

D′ − 1

r2 gνκ(x). (12.209)

Contracting this with gνκ [recall (11.137)] yields indeed the curvature scalar (12.202).

Notes and References

[1] R. Eotvos, Math. Nat. Ber. Ungarn 8, 65 (1890).See alsoJ. Renner, Hung. Acad. Sci. 53, Part II (1935).

[2] In general relativity there have been theories based on spaces in which this isnot satisfied. Then the object Qλµν ≡ −Dλgµν becomes a dynamical field tobe determined from field equations. SeeT. DeDonder, La gravitation de Weyl-Eddington-Einstein, Gauthier-Villars,Paris, 1924;H. Weyl, Phys. Z. 22, 473 (1921); Ann. Phys. 59, 101 (1919); 65, 541 (1921);A.S. Eddington, Proc. Roy. Soc. 99, 104 (1921) and The Mathematical Theory

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Notes and References 329

of Relativity , Springer, Berlin 1925.In such spaces, the correction is defined as

Γµνλ ≡ ea

λ(

∂µ −Dµ

)

eaν

and can be decomposed as

Γµνλ ≡ ea

λ(

∂µ−Dλ

)

eaν = Γµνλ−

(

Sλµν − Sλνµ + Sλνµ)

+1

2(Qµ

λν −Qλ

νµ +Qλνµ),

where Sµνλ ≡ 1

2(Γµνλ − Γνµ

λ) as defined in (11.113).

[3] More on the counting of independent components and invariants can be foundin the textbook:S. Weinberg, Gravitation and Cosmology Principles and Applications of theGeneral Theory of Relativity, John Wiley & Sons, New York, 1972.

[4] The physics of defects is explained in the textbookH. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects,World Scientific, Singapore, 1989 (kl/b2), where kl is short for the wwwaddress http://www.physik.fu-berlin.de/~kleinert).

[5] For the introduction of harmonic coordinates see:T. DeDonder, La gravifique Einsteinienne, Gauthier-Villars, Paris, 1921;C. Lanczos, Phys. Z. 23, 537 (1923).

[6] J.A. Schouten, Ricci-Calculus. An Introduction to tensor analysis and its ge-ometrical applications, Springer, Berlin, 1954.

[7] see, e.g., C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation, Freemanand Co., New York, 1973, and references therein.

[8] V.N. Ponomorev, Bull. Acad. Pol. Sci. (Math., Astr., Phys.) 19, 545 (1971);This work was criticized on the basis of a general class of gravitational fieldtheories with torsion byF.W. Hehl, Phys. Lett. A 36, 225 (1971).

[9] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, anmd Financial Markets, World Scientific, Singapore, 4th edition,2006 (kl/b5);P. Fiziev and H. Kleinert, Europhys. Lett. 35, 241 (1996) (hep-th/9503074);H. Kleinert and A. Pelster, Autoparallels From a New Action Principle, Gen.Rel. Grav. 31, 1439 (1999) (gr-qc/9605028);H. Kleinert, Nonholonomic Mapping Principle for Classical and Quantum Me-chanics in Spaces with Curvature and Torsion, Gen. Rel. Grav. 32, 769 (2000)(kl/258).

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330 12 Torsion and Curvature from Defects

[10] H. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects,World Scientific, Singapore, 1989 (kl/b2). See in particular pp. 1338–1377(kl/b1/gifs/v1-1338s.html).

[11] T.W. Kibble, J.Math.Phys. 2 (1961) 212.

H. Kleinert, MULTIVALUED FIELDS

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Do it big or stay in bedLarry Kelly

Opera producer in movie “Callas Forever”

13

Curvature and Torsion from Embedding

It is also possible to construct spaces with curvature and torsion by an embeddingprocedure in a flat spacetime [1]. This is done by imposing imposing suitable con-straints. We shall do this for spaces with only positive signatures of the metric, sowe can talk about spaces instead of spacetimes everywhere.

13.1 Curvature

Instead of mappings from the space xa to xµ with rotational defects, there is an-other way to obtain curvature. This is by embedding the space xµ into a higher-dimensional “Minkowskian” spacetime xA, A = 1, . . . , N with a metric gAB consist-ing only of diagonal elements ±1. The mapping xA(xµ) is smooth but cannot beinverted to xµ(xA). Thus there are N basis vectors eA in the embedding space and

eλ(xµ) = eAe

Aλ(x

µ) = eA∂xA

∂xλ(13.1)

form four local tangent vectors in the 4-dimensional submanifold xA(xµ). Theyinduce a metric

gλπ(xµ) = eλ (xµ) eκ (xµ) (13.2)

which can be used to define

eAλ (xµ) = gλλ′

(xµ)eAλ′ (xµ) (13.3)

but these vectors are no longer reciprocal to eAλ (xµ), i.e.,

eAλeBλ 6= δAB (13.4)

which is obvious since there are not enough of them. They do fulfill, however,

eAλeAκ = δλ

κ. (13.5)

331

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332 13 Curvature and Torsion from Embedding

Let us take an example: the surface of a sphere of radius a in three dimensions withthe mapping

xA =(

x1, x2, x3)

= a (sin θ cosϕ, sin θ, sinϕ, cos θ) (13.6)

and vectors

eA1 = a (cos θ cosϕ, cos θ, sinϕ,− sin θ) = eA1

eA2 = a (− sin θ sinϕ, sin θ cosϕ, 0) = eA2 (13.7)

where we have set xµ=1 = θ, xµ=2 = ϕ. The metric is

gµν = a2

(

1 00 sin2 θ

)

, gµν =1

a2

(

1 00 sin−2 θ

)

(13.8)

such that

e1A = eA1 = eA1, e2A =1

a

(

−sinϕ

sin θ,sinϕ

sin θ, 0)

. (13.9)

The connection is symmetric:

Γ221 = eA1∂2eA

2 = a eA1 (− sin θ cosϕ,− sin θ sinϕ, 0)

= −a2 sin θ cos θ = −Γ212 = −Γ122. (13.10)

All other elements vanish. By raising the last index we obtain

Γ221 = − sin θ cos θ, Γ21

2 = cotθ. (13.11)

The curvature tensor becomes

R1221 = ∂1Γ22

1 − ∂2Γ121 − Γ12

1Γ211 − Γ12

2Γ221 + Γ22

1Γ111 + Γ22

2Γ121

= − cos2 θ + sin2 θ + cot θ sin θ cos θ = sin2 θ, (13.12)

implying that

R1221 =

1

a2 . (13.13)

All other elements can be obtained using antisymmetry of Rµνλκ in µ → ν, λ → κ

and symmetry under µν ↔ λκ, which is a consequence of the symmetry of Γµνλ in

µν [recall the derivation of (12.111)]. Thus we can form the Ricci tensor

Rµνλµ = Rν

λ =1

a2

(

1 00 1

)

, (13.14)

and the curvature scalar

R = Rµν =

2

a2 . (13.15)

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13.2 Torsion 333

Note that for non-invertible vectors eAλ, the curvature has to be calculated from(11.127). The formula (11.129) can no longer be used since in the derivation of thisformula one would need

∂µeaν = Γµν

λeaλ (13.16)

which no longer follows from the correct relation

Γµνλ = eA

λ∂µeAν (13.17)

due to the non-invertibility

eAλeBλ 6= δAB (13.18)

In general, it is possible to generate any curved spaces by embedding in a higher-dimensional Minkowski space. If the curved space has d dimension, the embeddingspace has to have at least d(d+ 1)/2 dimension. This is seen by looking at themetric

gµν(x) =∂xA

∂xµ∂xB

∂xνgAB, (13.19)

and noticing that for a given d dimensional matrix, this converts to d(d+ 1)/2differential equations for the functions xA (xµ).

13.2 Torsion

It is also possible to construct spaces with curvature and torsion by an embeddingprocedure in a flat space. This is done by imposing imposing nonholonomic con-straints. Parallel transport in the embedded space is determined as an inducedparallel transport on the surface of constraints.

13.2.1 Strategy

In the last section we have defined a geometry by selecting a subspace from a eu-clidean space with the help of constraints. This is not the only possible procedure.We may equally well impose constraints only on the velocity space of all possibletrajectories in the euclidean space. A metric is naturally induced in the embeddedspace by restricting the scalar product in the space of all curves in the bigger em-bedding space to the smaller embedded space. A connection is induced uniquely bycompatibility conditions for the embedding of the velocity space with the parallel-transport law in the embedding space. This means the following: Take a curve inthe original space connecting points 1 and 2. This curve is embedded into a biggereuclidean space by specifying the velocity vectors of the image curve. A vector fromthe velocity space at point 1 is parallel-transported along the curve to point 2, andthen it is embedded in the bigger space. We require that the resulting vector must

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334 13 Curvature and Torsion from Embedding

be the same as the one obtained by running in the opposite direction: The vector atpoint 1 is first embedded into a bigger euclidean space and then parallel-transportedalong the image of the curve connecting points 1 and 2. This compatibility condi-tion ensures that the connection in the original space is uniquely determined by theembedding law.

Constraints imposed on the velocity space can be nonholonomic, and this is thesource of torsion. In this context, the notion of ”holonomic” and ”nonholonomic”constraints is exactly the same as in classical mechanics. For a mechanical system,generalized velocities are elements of the velocity space of its configuration space.Let the motion be subject to constraints linear in velocities. According to the Hertzclassification [2], constraints are said to be holonomic if they are integrable (i.e.,equivalent to some constraints on the configuration space only), and nonholonomicif they are non-integrable. Sometimes dynamical systems with nonholonomic con-straints are simply called nonholonomic systems. It is important to realize that themotion of nonholonomic systems does not occur on any submanifold of the config-uration space, nonetheless it is described by a smaller number of parameters thanthe corresponding unconstrained motion.

Upon an embedding via nonholonomic constraints on the velocity space, anycurve that parallel-transports its velocity vector along itself has an image with sameproperty. Therefore straight lines in the euclidean space have natural images n theembedded space which are autoparallels. Using Gauss’ principle of least constraintit sis possible to show that autoparallels describe a constrained motion such thatthe acceleration (or the force) induced by the constraints has a least deviation fromthe acceleration of the corresponding unconstrained motion, while geodesics play nospecial role in the constrained dynamics. See Section 14.2 for this derivation.

13.2.2 Nonholonomic Embedding

We denote the coordinates of the embedding euclidean space by xA, A = 1, 2, ... , andthose of the embedded space of smaller dimension by qµ, µ = 1, 2, ..., N . Supposewe specify a set of transformation functions εAµ(q) which map curves qµ(t) into

euclidean curves xA(t) by a relation

vA(t) = εAµ(q(t))vµ(t) , vµ = qµ(t) , vA = xA(t) , (13.20)

This implies an integral relation for the associated curves

xA(t) = xA(0) +

t∫

0

dt′ qµ(t)εAµ(q(t)) . (13.21)

For any curve qµ(t), this determines a a curve in the euclidean space up to a globaltranslation. Thus it determines an embedding of the space of all paths in q-spaceinto the space of all paths in x-space.

The acceleration along a curve in euclidean space is defined by

dvA(t)

dt= qA(t). (13.22)

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13.2 Torsion 335

By analogy with (11.108), we define the covariant acceleration in the embeddedspace by another mapping of the type (13.20):

dvA(t)

dt≡ εAµ(q(t))

Dvµ

dt. (13.23)

Let us seek for an affine connection Γµνλ in q-space which describes this covariant

acceleration intrinsically, i.e., without referring back to the embedding euclideanspace. It is given by

Dvµ(t)

dt= vν(t) + Γκν

µvκ(t)vν(t). (13.24)

The important observation is that the property (13.23) fixed Γµνλ uniquely.

In order to prove this statement, let us introduce some useful notations. Anytwo tangent vectors vA, vB in euclidean space have a scalar product

v · v = δij vAvB . (13.25)

If the vectors vA and vA are the images of two different velocities vµ and vµ at thespace space point q, then the embedding coefficients εAµ(q) determine an inducedmetric in q-space

v · v = gµν(q)vν(q)vµ(q) , gµν(q) ≡ εAµ(q) ε

Aν(q) . (13.26)

It is useful to introduce the quantity

εiµ(q) ≡ εAν(q)gνµ(q) , (13.27)

where gµλ(q)gλν(q) = δµν . From (13.26) follows that

εAµ(q) εAν(q) = gµν(q) , εi µ(q) εAν(q) = δνµ . (13.28)

Using the metrics gµν(q) and δij to lower or raise indices in q- and x-spaces, respec-tively, the embedding condition (13.23) can be written in a more general form

dvij···kn···

dt=

d

dt

(

εAµεBν · · · εkλεnβ · · · vµν···λβ···

)

= εAµεBν · · · εkλεnβ · · ·

Dvµν···λβ···

dt, (13.29)

where the covariant derivative reads

Dvµν···λβ···

dt= vµν···αβ··· +

(

Γµλµvµν···αβ··· + · · · − Γαλ

λvµν···λβ···

− · · ·)

vλ . (13.30)

Performing the time derivatives on the left-hand side of (13.29) and applying rela-tions (13.28) we find

Γνλµ(q)vλ = εiµ(q) εAν(q) = −εAν(q) εiµ(q). (13.31)

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336 13 Curvature and Torsion from Embedding

This equation ensures that along any curve qµ(t), the fields εAµ(q(t)) and εiµ(q(t))

are transported parallel, as expressed by the relations

D

dtεA

µ(q(t)) = 0,D

dtεiµ(q(t)) = 0. (13.32)

Applying the covariant derivative D/dt to the metric (13.26) we obtain from thechain rule of differentiation

D

dtgµν(q(t)) = 0. (13.33)

Equation (13.31) can be rewritten as

Γνλµ(q)vλ = εiµ(q) ∂λε

Aν(q)v

λ = −εAν(q) ∂λεiµ(q)vλ. (13.34)

This must hold for any velocity vλ(t) along the curve qλ(t), implying that

Γµνλ(q) = εAα(q) ∂νε

Aµ(q) g

αλ(q) . (13.35)

Thus we have succeeded in determining metric and parallel transport by anembedding of all paths in q-space in the bigger euclidean x-space.

13.2.3 Torsion

Let us now turn to the analysis of the affine connection (13.35). First of all, weobserve that the torsion tensor

Sνκµ =

1

2gµλ

[

εAλ(q) ∂κεAν(q) − εAλ(q) ∂νε

Aκ(q)

]

(13.36)

is, in general, nonzero. The torsion induced by the embedding is zero if and only if

εAν,µ(q) = εAµ,ν(q) . (13.37)

If this integrability condition is satisfied, the matrix elements εAλ(q) are the deriva-tives of functions εA(q)

εAµ(q) = ∂µεA(q) , (13.38)

and the integral in Eq. (13.21) can be performed trivially to yield a point-to-pointembedding

xA = εA(q) . (13.39)

The metric tensor gµν has N(N + 1)/2 independent components. The torsiontensor Sνκ

µ has N2(N − 1)/2 independent components. To embed a general metricspace with torsion, the number NM , being the number of independent embeddingcoefficients εAµ, should be greater or equal to N(N2+1)/2. This leads to the relationbetween the dimensions of q- and x-spaces:

(dim[q])2 + 1 ≤ 2 dim[x] . (13.40)

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 337

Notes and References

[1] H. Kleinert and S.V. Shabanov, Spaces with Torsion from Embedding, andthe Special Role of Autoparallel Trajectories, Phys. Lett B 428 , 315 (1998)(quant-ph/9503004) (www.physik.fu-berlin.de/~kleinert/259).

[2] V.I. Arnold, V.V. Koslov and A.I. Neishtadt, in: Encyclopedia of Mathemat-ical Sciences, Dynamical Systems III, Mathematical Aspects of Classical andCelestial Mechanics, Springer-Verlag, Berlin, 1988;L.A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965;G. Hamel, Theoretische Mechanik – Eine einheitliche Einfuhrung in diegesamte Mechanik, Springer Series in Mathematics, Vol. 57, Springer, Berlin,1978.

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Those are my principles,and if you don’t like them... well, I have others

Groucho Marx (1890 - 1977)

14Nonholonomic Mapping Principle

The multivalued, nonholonomic mappings from flat to curved spacetime with torsionenable us to reformulate Einstein’s equivalence principle in a new more powerfulway. Whereas Einstein postulated, that equations written down in flat spacetimewith curvilinear coordinates remain valid in curved spacetime, we may sharpen thispostulate to the form:

The physical laws in curved spacetime are the direct images the flat-spacetime lawsunder multivalued mappings.

If the space has only curvature and no torsion, this new equivalence principle leadsto the same conclusions as Einstein’s.

If the space has torsion, however, the new equivalence principle makes new pre-dictions, and it is interesting to investigate these.

14.1 Motion of Point Particle

The derivation of the geodesic trajectories of point particles in curved space wasperformed in Section 11.2. Only minor modifications will be necessary to follow thenew equivalence principle. As observed when going from Eq. (11.14) to (11.15), we

simplify the discussion by considering the nonrelativistic actionm

A of Eq. (11.16) ifwe use the proper time τ to parameterize the paths.

14.1.1 Classical Action Principle for Spaces with Curvature

Instead of performing a an ordinary coordinate transformation in flat space fromMinkowski coordinates xa to curvilinear coordinates xµ via Eq. (11.7), we performa multivalued coordinate transformation

dxa = eaµ(x)dxµ, (14.1)

where the basis vectors eaµ describe coordinate transformations in which

∂µeaν(x) − ∂νe

aµ(x) 6= 0. (14.2)

338

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14.1 Motion of Point Particle 339

This implies that second derivatives in front of the multivalued functions xa(xµ) donot commute as in Eq. (11.31):

(∂λ∂κ − ∂κ∂λ)xa(x) 6= 0, (14.3)

thus violating the Schwarz integrability criterion. If the space has also torsion, thenthe functions eaν(x) have also no commuting derivatives [recall (11.32)]:

(∂µ∂ν − ∂ν∂µ)eaλ(x) 6= 0. (14.4)

In either case, the metric in the image space has the same form as in Eq. (11.9),and the derivation of the extremum of the action seems, at first, to follow the samepattern as in Section 11.2, leading to the equation of motion (11.25) for geodesictrajectories. The nonholonomically tranformed action (11.2) is independent of thetorsion fields Sµν

λ, and for this reason also the equation of motion (11.25) is indif-ferent to the presence of torsion.

This result would be perfectly acceptable, were it not for an apparent incon-sistency with another result obtained by applying the new variational principle.Instead of transforming the action and varying it in the usual way, we may trans-form the equation of motion of a free particle (11.1) in flat space nonholonomicallyinto a space with curvature and torsion.

14.1.2 Autoparallel Trajectories in Spaces with Torsion

In the absence of external forces, the equation of motion (11.1) in flat space statesthat the second derivative of qi(τ) vanish. In spacetime, the free equations of motionread qa(τ) = 0, where the dot denotes the derivative with respect to the invariantlength s as parametrizing the straight lines, as in Eq. (11.25). These are transformednonholonomically by Eq. (14.1) as follows:

d2qa

dτ 2 =d

dτ(eaµq

µ) = eaµqµ + eaµ,ν q

µqν = 0, (14.5)

or asqµ + ea

µeaκ,λqκqλ = 0. (14.6)

The subscript λ separated by a comma denotes the partial derivative: f,λ(x) ≡∂λf(x). The quantity in front of qκqλ is the affine connection (11.93). Thus wearrive at the transformed flat-space equation of motion

q µ + Γκλµqκqλ = 0. (14.7)

The solutions of this equation are called autoparallel trajectories. They differ fromthe geodesic trajectories described by (11.25) by an extra torsion term. Inserting thedecomposition (11.119) and using the antisymmetry of Sµν

λ in the first two indices,we may rewrite (14.7) as

q µ + Γµκλqκqλ − 2Sµκλq

κqλ = 0. (14.8)

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340 14 Nonholonomic Mapping Principle

Note the index positions of the torsion tensor, which may be written more explicitlyas Sµκλ ≡ gµσgλκSσκ

κ. This is not antisymmetric in the last two indices so that itpossesses a symmetric part which contributes to Eq. (14.7).

How can we reconcile this result with an application of the new equivalence prin-ciple applied to the action. Since the tranformed action is independent of the torsionand carries only information on the Riemann part of the space geometry, torsion canenter the equations of motion only via some overlooked feature of the variation pro-cedure. Indeed, a moment’s thought convinces us that this was applied incorrectlyin the previous section incorrectly. According the the new equivalence principlewe must also transform the variational procedure nonholonomically to spaces withcurvature and torsion. We must find the image of the flat-space variations δqa(τ)under the multivalued mapping

qµ = eaµ(q)qa. (14.9)

The images are quite different from ordinary variations as illustrated in Fig. 14.1(a).The variations of the cartesian coordinates δqa(τ) are done at fixed endpoints of thepaths. Thus they form closed paths in the x-space. Their images, however, lie in aspace with defects and thus possess a closure failure indicating the amount of torsionintroduced by the mapping. This property will be emphasized by writing the imagesδSqµ(τ) and calling them nonholonomic variations. The superscript indicates thespecial feature caused by torsion.

Let us calculate them explicitly. The paths in the two spaces are related by theintegral equation

qµ(τ) = qµ(τa) +∫ τ

τadτ ′ea

µ(q(τ ′))qa(τ ′). (14.10)

For two neighboring paths in x-space differing from each other by a variation δqa(τ),equation (14.15) determines the nonholonomic variation δSqµ(τ):

δSqµ(τ) =∫ τ

τadτ ′δS[ea

µ(q(τ ′))qa(τ ′)]. (14.11)

A comparison with (14.14) shows that the variation δS and the derivatives withrespect to the parameter s of qµ(τ) commute with each other:

δS qµ(τ) =d

dτδSqµ(τ), (14.12)

just as for ordinary variations δqa in Eq. (2.7):

δqa(τ) =d

dτδqa(τ). (14.13)

Let us also introduce auxiliary nonholonomic variations of the paths qµ(τ) in xµ-space: Indeed, a moment’s thought convinces us that this was applied incorrectlyin the previous section incorrectly. According the the new equivalence principlewe must also transform the variational procedure nonholonomically to spaces with

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14.1 Motion of Point Particle 341

curvature and torsion. We must find the image of the flat-space variations δqa(τ)under the multivalued mapping

qµ = eaµ(q)qa. (14.14)

The images are quite different from ordinary variations as illustrated in Fig. 14.1(a).The variations of the cartesian coordinates δqa(τ) are done at fixed endpoints of thepaths. Thus they form closed paths in the x-space. Their images, however, lie in aspace with defects and thus possess a closure failure indicating the amount of torsionintroduced by the mapping. This property will be emphasized by writing the imagesδSqµ(τ) and calling them nonholonomic variations. The superscript indicates thespecial feature caused by torsion.

Let us calculate them explicitly. The paths in the two spaces are related by theintegral equation

qµ(τ) = qµ(τa) +∫ τ

τadτ ′ea

µ(q(τ ′))qa(τ ′). (14.15)

For two neighboring paths in x-space differing from each other by a variation δqa(τ),equation (14.15) determines the nonholonomic variation δSqµ(τ):

δSqµ(τ) =∫ τ

τadτ ′δS[ea

µ(q(τ ′))qa(τ ′)]. (14.16)

A comparison with (14.14) shows that the variation δS and the derivatives withrespect to the parameter s of qµ(τ) commute with each other:

δS qµ(τ) =d

dτδSqµ(τ), (14.17)

just as for ordinary variations δqa in Eq. (2.7):

δqa(τ) =d

dτδqa(τ). (14.18)

Let us also introduce auxiliary nonholonomic variations of the paths qµ(τ) inxµ-space:

-δqµ ≡ eaµ(q)δqa. (14.19)

In contrast to δSqµ(τ), these vanish at the endpoints,

-δq(τa) = -δq(τb) = 0, (14.20)

just as the usual variations δqa(τ), i.e., they form closed paths with the unvariedorbits.

Using (14.17), (14.18), and the fact that δSqa(τ) ≡ δqa(τ), by definition, wederive from (14.16) the relation

d

dτδSqµ(τ) = δSea

µ(q(τ))qa(τ) + eaµ(q(τ))

d

dτδqa(τ)

= δSeaµ(q(τ))qa(τ) + ea

µ(q(τ))d

dτ[eaν(τ)

-δqν(τ)]. (14.21)

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342 14 Nonholonomic Mapping Principle

After inserting

δSeaµ(q) = −Γλν

µδSqλeaν ,

d

dτeaν(q) = Γλν

µqλeaµ, (14.22)

this becomesd

dτδSqµ(τ) = −Γλν

µδSqλqν + Γλνµqλδqν +

d

dτ-δqµ. (14.23)

It is useful to introduce the difference between the nonholonomic variation δSqµ andan auxiliary closed nonholonomic variation δqµ:

δSbµ ≡ δSqµ − -δqµ. (14.24)

Then we can rewrite (14.23) as a first-order differential equation for δSbµ:

d

dτδSbµ = −Γλν

µδSbλqν + 2Sλνµqλ -δqν . (14.25)

After introducing the matrices

Gµλ(τ) ≡ Γλν

µ(q(τ))qν(τ) (14.26)

and

Σµν(τ) ≡ 2Sλν

µ(q(τ))qλ(τ), (14.27)

equation (14.25) can be written as a vector differential equation:

d

dτδSb = −GδSb+ Σ(τ) -δqν(τ). (14.28)

Although not necessary for the further development, we solve this equation by

δSb(τ) =∫ τ

τadτ ′ U(τ, τ ′) Σ(τ ′) -δq(τ ′), (14.29)

with the matrix

U(τ, τ ′) = Ts exp[

−∫ τ

τ ′dτ ′′G(τ ′′)

]

, (14.30)

where Ts denotes the time-ordering operator for the parameter s. In the absenceof of torsion, Σ(τ) vanishes identically and δSb(τ) ≡ 0, and the variations δSqµ(τ)coincide with the auxiliary closed nonholonomic variations δqµ(τ) [see Fig. 14.1(b)].In a space with torsion, the variations δSqµ(τ) and -δqµ(τ) are different from eachother [see Fig. 14.1(c)].

We now calculate the variation of the action (11.11) under an arbitrary nonholo-nomic variation δSqµ(τ) = δqµ + δSbµ. Since s is the invariant path length, we mayjust as well use the auxiliary action (11.16) to calculate this quantity (it differs onlyby a trivial factor 2):

δSA = M∫ τb

τadτ(

gµν qνδS qµ +

1

2∂µgλκδ

Sqµqλqκ)

. (14.31)

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14.1 Motion of Point Particle 343

Figure 14.1 Images under holonomic and nonholonomic mapping of fundamental δ-

function path variation. In the holonomic case, the paths q(τ) and q(τ)+δq(τ) in (a) turn

into the paths q(τ) and q(τ) + -δq(τ) in (b). In the nonholonomic case with Sλµν 6= 0, they

go over into q(τ) and q(τ) + δSq(τ) shown in (c) with a closure failure bµ at tb analogous

to the Burgers vector bµ in a solid with dislocations.

After a partial integration of the δq-term we use (14.20), (14.17), and the identity∂µgνλ ≡ Γµνλ + Γµλν , which follows directly form the definitions gµν ≡ eaµe

aν and

Γµνλ ≡ ea

λ∂µeaν , we obtain

δSA = M∫ τb

τadτ[

− gµν(

qν + Γλκν qλqκ

)

-δqµ +

(

gµν qν d

dτδSbµ + Γµλκδ

Sbµqλqκ)

]

.

(14.32)To derive the equation of motion we first vary the action in a space without

torsion. Then δSbµ(τ) ≡ 0, and (14.32) becomes

δSA = −M∫ tb

tadτgµν(q

ν + Γλκν qλqκ) -δqν . (14.33)

Thus, the action principle δSA = 0 produces the equation for the geodesics (11.25),which are the correct particle trajectories in the absence of torsion.

In the presence of torsion, δSbµ is nonzero, and the equation of motion receivesa contribution from the second parentheses in (14.32). After inserting (14.25), the

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344 14 Nonholonomic Mapping Principle

nonlocal terms proportional to δSbµ cancel and the total nonholonomic variation ofthe action becomes

δSA = −M∫ τb

τadτgµν

[

qν +(

Γλκν + 2Sνλκ

)

qλqκ]

-δqµ

= −M∫ τb

τadτgµν

(

qν + Γλκν qλqκ

)

-δqµ. (14.34)

The second line follows from the first after using the identity Γλκν = Γλκ

ν+2Sνλκ.The curly brackets indicate the symmetrization of the enclosed indices. SettingδSA = 0 and inserting for -δq(τ) the image under (14.19) of an arbitrary δ-functionvariation δqa(τ) ∝ εaδ(τ − s0) gives the autoparallel equations of motions (14.7),which is what we wanted to show.

The above variational treatment of the action is still somewhat complicatedand calls for a simpler procedure [1, 2]. The extra term arising from the secondparenthesis in the variation (14.32) can be traced to a simple property of the auxiliaryclosed nonholonomic variations (14.19). To find this we form the time derivativedt ≡ d/dt of the defining equation (14.19) and find

dt-δqµ(τ) = ∂νea

µ(q(τ)) qν(τ)δqa(τ) + eaµ(q(τ))dτδq

a(τ). (14.35)

Let us now perform variation -δ and s-derivative in the opposite order and calculatedτ

-δqµ(τ). From (14.14) and (11.41) we have the relation

dτqλ(τ) = e λ

i (q(τ)) dτqi(τ) . (14.36)

Varying this gives

-δdτqµ(τ) = ∂νea

µ(q(τ)) -δqνdtqa(τ) + ea

µ(q(τ)) -δdτqa. (14.37)

Since the variation in xa-space commute with the s-derivatives [recall (14.18)], weobtain

-δdτqµ(τ) − dτ

-δqµ(τ) = ∂νeaµ(q(τ)) -δqνdtq

a(τ) − ∂νeaµ(q(τ)) qν(τ)δqa(τ). (14.38)

After re-expressing δqa(τ) and dtqa(τ) back in terms of -δqµ(τ) and dtq

µ(τ) = qµ(τ),and using (11.93), this becomes

-δdτqµ(τ) − dτ

-δqµ(τ) = 2Sνλµqν(τ) -δqλ(τ). (14.39)

Thus, due to the closure failure in spaces with torsion, the operations dτ and -δ donot commute in front of the path qµ(τ). In other words, in contrast to the openvariations -δqµ(τ) (and of course the usual δqµ(τ)), the auxiliary closed nonholonomicvariations -δ of velocities qµ(τ) no longer coincide with the velocities of variations.

This property is responsible for shifting the trajectory from geodesics to autopar-allels. Indeed, let us vary an action

A =

τb∫

τa

dτL (qµ(τ), qµ(τ)) (14.40)

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14.1 Motion of Point Particle 345

directly by -δqµ(τ) and impose (14.39), we find

-δA =

τb∫

τa

∂L

∂qµ-δqµ +

∂L

∂qµd

dτ-δqµ +2Sµνλ

∂L

∂qµqν -δqλ

. (14.41)

After a partial integration of the second term using the vanishing -δqµ(τ) at theendpoints, we obtain the Euler-Lagrange equation

δAδqµ

=∂L

∂q µ− d

dt

∂L

∂qµ=

-δA-δqµ

− 2Sµνλqν

∂L

∂qλ= −2Sµν

λqν∂L

∂qλ. (14.42)

This differs from the standard Euler-Lagrange equation by the additional torsionforce. For the action (11.11), we thus obtain the equation of motion

q µ +[

gµκ(

∂νgλκ −1

2∂κgνλ

)

+ 2Sµνλ]

q ν qλ = 0, (14.43)

which is once more the equation (14.7) for autoparallels.Thus a consistent application of the new equivalence principle yields consistently

autoparallel trajectories for point particles in space with curvature and torsion.

14.1.3 Special Properties of Gradient Torsion

A torsion tensor which consists of an antisymmetric combination of gradients of ascalar field θ(q) as follows:

Sµνλ(q) =

1

2

[

δ λν ∂µθ(q) − δ λ

µ ∂νθ(q)]

, (14.44)

is called gradient torsion. If spacetime possesses only gradient torsion, its effectupon the equations of motion of a point particle can be simulated in a purely Rie-mannian spacetime. by adding the scalar field θ(q) in a suitable way to the action.Then gradient torsion appears as a nongeometric external field. By extremizingthe transformed action in the usual way, the resulting equation of motion coincideswith the autoparallel equation derived in the initial spacetime with torsion from themodified variational principle in Eqs. (14.8):

q µ + Γµκλqκqλ − 2Sµκλq

κqλ = 0. (14.45)

For a pure gradient torsion (14.44), this becomes

qλ(s)+Γ λµν (q(s))qµ(s)qν(s) = −θ(q(s))qλ(s)+gλκ(q(s))∂κθ(q(s)), (14.46)

with the extra terms on the right-hand side reflecting the closure failure of parallel-ograms caused by the torsion.

The same trajectory is found from the following alternative action in a purelyRiemannian spacetime

m

A = −mc∫ σb

σadσ eθ(q)

[

gµν(q(σ))qµ(σ)qν(σ)]

12 . (14.47)

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346 14 Nonholonomic Mapping Principle

The extra factor eθ(q) has precisely the same effect in a Riemannian spacetime asthe gradient torsion (14.44) in a Riemann-Cartan spacetime. Indeed, the extremumof this action can be derived from the geodesic trajectory without the σ-field byintroducing, for a moment, an auxiliary metric

gµν(q) ≡ e2θ(q) ≡ gµν(q). (14.48)

The invariant line element remains, of course,

ds =[

gµν(q(σ))qµ(σ)qν(σ)]

12 = e−θ(q)

[

gµν(q(σ))qµ(σ)qν(σ)]

12 = e−θ(q)ds. (14.49)

By varying the action as in Eqs. (11.14)–(11.19), we obtain the modified equationof motion (11.21):

gλνd2q ν(σ)

dσ2 +(

∂µgλν −1

2∂λgµν

)

dqµ(σ)

dqν(σ)

dσ= 0. (14.50)

Inserting (14.48) and (14.49), this becomes

gλν

(

d2q ν(σ)

dσ2 − θdq ν(σ)

)

+(

∂µgλν −1

2∂λgµν

)

dqµ(σ)

dqν(σ)

+ 2θ(q)dqν(σ)

dσ− ∂λθ(q)gµν

dqµ(σ)

dqν(σ)

dσ= 0. (14.51)

This coincides with the autoparallel trajectory (14.46).

14.2 Autoparallel Trajectories from Embedding

There exists another way of deriving autoparallel trajectories. Instead of usingmultivalued mappings to carry physical laws from flat space to spaces with curvatureand torsion, we may use the embedding of Section 13.1 to do so.

14.2.1 Special Role of Autoparallels

Let us first remark that apart from extremizing a length between two fixed end-points, geodesics in a Riemann space can be obtained by embedding the Riemannspace in a euclidean space of a higher dimension. This is done by imposing certainconstraints on the cartesian coordinates spanning the euclidean space. The pointson the constraint surface constitute the embedded Riemann space. Straight lines inthe euclidean space, which are geodesic and autoparallel and also determine a freemotion in that space, become geodesics when the motion is restricted to the con-straint surface. The restriction of the free motion to the constraint surface is donein a conventional way, i.e., by adding the equations of constraints to the equationsof motion. When the constraint force is removed, geodesic trajectories turns intostraight lines in the embedding space.

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14.2 Autoparallel Trajectories from Embedding 347

For curved space with torsion the embedding procedure was described in Chap-ter 13. The consequences for the trajectories were worked out in Ref. [3]. It turnsout that also from this point of view, autoparallel curves are specially favored ge-ometric curves in the embedded space. The are the images of straight lines in theembedding space by Eq. (13.23).

14.2.2 Gauss Principle of Least Constraint

There is also a mechanical argument favoring autoparallel over geodesic motion.This is intrinsically linked with the concept of inertia. Inertia favors trajectorieswhose acceleration deviates minimally from the acceleration of the correspondingunconstrained motion. This property can be formulated mathematically by meansof Gauss’ principle of least constraint [4, 5].

Consider a Lagrangian system in the space [x] with a Lagrangian L = L(x, v).At each moment of time, a state of the system can labeled by a point in phasespace (xi(t), vi(t)). Let Hij(x

i(t), vi(t)) ≡ ∂2L/∂vi∂vB be the Hessian matrix of thesystem. Consider two paths xi1(t) and xi2(t). Gauss’ deviation function (sometimesalso called Gauss’ constraint) for two paths reads

G =1

2

(

vi1 − vi2)

Hij

(

vB1 − vB2)

. (14.52)

It measures the deviation of two possible motions from one another [4, 5].Now, let the motion in x-space be subject to constraints. All paths xi(t) allowed

by the constraints are called conceivable motions. A path xi(t) is called releasedmotion if it satisfies the Euler-Lagrange equations for the Lagrangian L withoutconstraint. Gauss’ principle of least constraint says that the deviation of conceivablemotions from a released motion takes a stationary value for the actual motion.

In our case, the released motion is a free motion with zero acceleration ¨xi= 0.

Accelerations of the conceivable motions satisfy

vi = εiµvµ + εiµ,νv

µvν . (14.53)

Since Hij = δij for the euclidean Lagrangian, Gauss’ deviation function (14.52)assumes the form

G =1

2

[

vi]2

=1

2[vµ + εµ εν v

ν ]2 , (14.54)

where an infinitesimal factor dτ 2 has been removed. Remembering Eq. (13.31), wemay also write

G =1

2

[

Dvµ

dt

]2

. (14.55)

This has a minimum at G ≥ 0, which is reached for trajectories satisfying theautoparallel equation of motion

Dvµ

dτ= 0 . (14.56)

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348 14 Nonholonomic Mapping Principle

Another derivation of the autoparallel equation (14.56) rests on the d’Alembert-Lagrange principle [4, 5]. In theoretical mechanics, one defines a Lagrange derivative

[L]A ≡ d

∂L

∂vA− ∂L

∂xA. (14.57)

The d’Alembert-Lagrange principle asserts that motion of a system with the La-grangian L proceeds such that

vA [L]A = 0 (14.58)

for all velocities allowed by the constraints. Taking the free Lagrangian L = vAvA/2with [L]A = δABv

B, and the constraint (13.20) we find that only the autoparallelequation (14.56) satisfies this principle.

Finally we point out that the motion of a holonomic system is completely de-termined by the restriction of the Lagrangian to the constraining surface [4]. Thus,holonomic constrained systems are indistinguishable from ordinary unconstrainedLagrangian systems. This is not true for nonholonomic systems, meaning that theEuler-Lagrange equations for the Lagrangian restricted on the constraining sur-face do not coincide with the original equations for the constrained motion. Thisdifficulty prevents us from applying a conventional Hamiltonian formalism to theautoparallel motion, and subjecting it to a canonical quantization. In other words,Dirac’s method of quantizing constrained systems [6] does not apply to nonholo-nomic systems since these do not follow the conventional Lagrange formalism [4].

14.3 Maxwell-Lorentz Orbits as Autoparallel Trajectories

It is rather straightforward to set up the Maxwell-Lorentz equations for the motionof a charged particle in curved space. We rewrite the flat-spacetime equation ofmotion (1.165) as

d2qa(τ)

dτ 2 =e

cF a

b(q(τ))d

dτqb(τ), (14.59)

and subject this to a multivalued mapping. This adds the tidal forces to the accel-eration term and leads to

d2qλ(τ)

dτ 2 + Γµνλqµqν =

e

cmF a

b(q(τ)) qb. (14.60)

It is now interesting to observe that this equation of motion may be viewed a anautoparallel motion in an affine geometry with torsion. The torsion is determinedby the particle from the equation [7]

Sµνλ =

e

mc3Fµν q

λ. (14.61)

Indeed, if we insert this torsion into the autoparallel equation (14.8), we obtain theMaxwell-Lorentz equation (14.60) in curved spacetime.

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14.4 Bargmann-Michel-Telegdi Equation from Torsion 349

14.4 Bargmann-Michel-Telegdi Equation from Torsion

Interestingly enough, also the (14.62) can be understood as a purely geometric equa-tion in a space with torsion. If we transform the flat-spacetime equation (14.62) tocurved spacetime, it becomes

D

dτSa=

e

2mc

[

gF abSb +g − 2

m2c2paScF

cκpκ

]

= 0. (14.62)

For a classical particle for which g = 1, the this equation is the same as for a spinvector undergoing a parallel transport along the trajectory qµ(τ) according to thelaw (11.111):

DSµdτ

=DSµdτ

+ SµνλSν qλ − Sνλ

µSν qλ + SλµνS

ν qλ. (14.63)

Inserting (14.61) yields

DSµdτ

=DSµdτ

+e

mc3

(

F µν qλS

ν qλ − Fνλ qµSν qλ + Fλ

µ qνSν qλ

)

. (14.64)

Recalling the transversality (1.299) of the spin vector, the last term vanishes. andwe arrive at

DSµdτ

=DSµdτ

+e

mc

(

F µνS

ν − 1

c2qµ SνFνλq

λ)

, (14.65)

which is indeed the same as (14.62) for g = 1.

Notes and References

[1] H. Kleinert and A. Pelster, Gen. Rel. Grav. 31, 1439 (1999) (gr-qc/9605028);H. Kleinert, Mod. Phys. Lett. A 4, 2329 (1989) (kl/199), where kl is shortfor the www address http://www.physik.fu-berlin.de/~kleinert;H. Kleinert, Quantum Equivalence Principle for Path Integrals in Spaceswith Curvature and Torsion, Lecture at the XXV International SymposiumAhrenshoop on Elementary Particles in Gosen/Germany, CERN report 1991,ed. H. J. Kaiser (quant-ph/9511020);H. Kleinert, Quantum Equivalence Principle, Lecture presented at the Sum-mer School Functional Integration: Basics and Applications in Cargese/France(1996) (kl/199).

[2] See the discussion in Chapter 10 of the textbookH. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets, 4th ed., World Scientific, Singapore, 2006(kl/b5/psfiles/pthic10.pdf).

[3] H. Kleinert and S.V. Shabanov, Spaces with Torsion from Embedding, andthe Special Role of Autoparallel Trajectories, Phys. Lett B 428 , 315 (1998)(quant-ph/9503004) (kl/259)

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350 14 Nonholonomic Mapping Principle

[4] V.I. Arnold, V.V. Koslov and A.I. Neishtadt, in: Encyclopedia of Mathemat-ical Sciences, Dynamical Systems III, Mathematical Aspects of Classical andCelestial Mechanics, Springer, Berlin, 1988;L.A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965;G. Hamel, Theoretische Mechanik – Eine einheitliche Einfuhrung in diegesamte Mechanik, Springer Series in Mathematics, Vol. 57, Springer, Berlin,1978.

[5] L.S. Polak (ed.), Variational principles of mechanics. Collection of papers,Moscow, Fizmatgiz, 1959.

[6] P.A.M. Dirac, Lectures on Quantum Mechanics, Yeshiva University Press, NY,1964.

[7] H.I. Ringermacher, Class. Quant. Grav. 11, 2383 (1994).

H. Kleinert, MULTIVALUED FIELDS

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A field that has rested gives a bountiful crop

Ovid (43 BC - 17 AD)

15Field Equations of Gravitation

In the previous chapter, we have shown that a particle subject to a gravitational fieldfollows equations of motion which look precisely the same as those of a particle inMinkowski space if these are expressed in curvilinear coordinates. The informationon the gravitational field is contained in certain properties of the metric. We maynow ask the question how to calculate such a metric associated with a gravitationalmassive object. For this, the ten components of the metric tensor gµν(x) have to beconsidered as dynamical variables and we need an action principle for it [1, 2, 3, 4].

15.1 Invariant Action

The equation of motion for gµν(x) must be independent of the general coordinatesemployed. This is guaranteed if the action is invariant under Einstein transforma-tions xµ → x′µ

(xµ), whose infinitesimal form is

dxµ → dx′µ′

= αµ′

µ(x)dxµ. (15.1)

We want to set up a local action for the gravitational field, which by definition givenafter Eq. (2.83) should have the form of an integral over a Lagrangian density

A =∫

d4xL(x), (15.2)

where L(x) is a function of the metric and its derivatives, depending at most quadrat-ically on the derivatives ∂λgµν(x) (after a possible integration by parts). Under thecoordinate transformations (15.1), the volume element transforms as

d4x→ d4x′ = d4x detα. (15.3)

The simplest Lagrangian density L(x) which leaves the action (15.2) invariant canbe formed from the determinant of the metric

g = det(gµν). (15.4)

Since the metric changes under (15.1) to g′µ′ν′(x′) = gµν(α

−1)µµ′(α−1)νν′ , we see that

g → g′ = g detα−2, (15.5)

351

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352 15 Field Equations of Gravitation

implying that an action

Ac =Λ

κ

d4x√−g (15.6)

is invariant under coordinate transformations (15.1). Such an action by itself is,however, not capable of giving equations of motion for the gravitational field sinceit contains no derivatives of gµν . The gµν-field cannot propagate. We must find somescalar Lagrangian density L containing gµν and ∂λgµν in a local way. Then

A =∫

d4x√−gL(g, ∂g) (15.7)

would be an invariant action which could govern the gravitational forces.Now, the only fundamental scalar quantity which occurred in the previous geo-

metric analysis and which involves the derivatives ∂λgµν is R, the scalar curvature.Therefore, Einstein postulated the following gravitational field action

f

A= − 1

d4x√−g R. (15.8)

Here κ is a constant related to Newton’s gravitational coupling GN ≈ 6.673 · 10−8

cm3 g−1 s−2 of Eq. (1.3) by

1

κ=

c3

8πGN

. (15.9)

It can be expressed in terms of the Planck length (12.46) as

1

κ=

h

8πl2P. (15.10)

For a system consisting of a set of mass points m1, . . . , mN , we add the particleaction (11.2) and obtain a total action

A =f

A −N∑

n=1

mnc∫

dsn ≡f

A +m

A . (15.11)

In the following formulas it will be convenient to set κ = 1 since κ can always bereintroduced as a relative factor between field and matter parts in all field equationsto be derived.

Variation of the particle paths xn(sn) at fixed gµν(x) gives the equations of motionof a point particle in an external gravitational field as discussed in the beginning. Inaddition, the action (15.11) permits to find out which gravitational field generatedby the presence of these points. They are obtained from the variational equation

δAδgµν(x)

= 0 (15.12)

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15.2 Energy-Momentum Tensor and Spin Density 353

There are 10 independent components of gµν . Four of them are unphysical, repre-senting merely reparametrization degrees of freedom.

Equations (15.12) are not sufficient to determine the geometry of spacetime.The curvature tensor Rµνλ

κ also contains torsion tensors Sµνλ combined to a con-

tortion tensor Kµνλ. It has 24 independent components, which are determined by

the equation of motion

δAδKµν

λ(x)= 0. (15.13)

Einstein avoided this problem by considering only symmetric (Riemannian) spacesfrom the outset. For spinning matter, however, this may not be sufficient, anda determination of torsion fields from the spin densities may be necessary for acomplete dynamical theory.

15.2 Energy-Momentum Tensor and Spin Density

It is useful to study separately the derivatives of the different pieces of the actionwith respect to gµν and Kµνλ separately. In view of the physical interpretations tobe given later we introduce

δm

Aδgµν

≡ −1

2

√−g m

Tµν , (15.14)

δf

Aδgµν

≡ −1

2

√−gf

Tµν , (15.15)

respectively, as the symmetric energy-momentum tensors of matter and field, and

δm

AδKµν

λ ≡ −1

2

√−g m

Σνλ,µ, (15.16)

δf

AδKµν

λ ≡ −1

2

√−gf

Σνλ,µ. (15.17)

as the spin current density of matter and field, respectively.Let us calculate these quantities for a point particle. For a specific world line

qµ(σ) parameterized by an arbitrary timelike variable σ, the action reads [recall(11.11), (11.12)]

m

A = −mc∫

ds = −mc2∫

dσ√

gµν(q(σ))qµ(σ)qν(σ)

= −mc∫

dσ∫

d4x√

gµν(x)qµ(σ)qν(σ) δ(4)(x− q(σ)). (15.18)

Variation with respect gµν(x) and Kµνλ(x) gives

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354 15 Field Equations of Gravitation

δm

Aδgµν(x)

≡−1

2mc∫

dσ1

gµν(q(σ))qµ(σ)qν(σ)qµ(σ)qν(σ)δ(4)(x−q(σ)) (15.19)

=−1

2m∫

dτ qµ(τ)qν(τ)δ(4)(x−q(τ)),

where τ = s/c is the proper time (11.4). The functional derivative with respect toKµν

λ(x) vanishes identically:

δm

AδKµν

λ(x)≡ 0. (15.20)

Thus we identify energy-momentum tensor and spin current densities

m

Tµν(x) ≡ m qµ(τ)qν(τ)δ(4)(x− q(τ)), (15.21)

Σνλ,µ(x) ≡ 0. (15.22)

We now determine these quantities for the gravitational field sing the action(15.8) First we perform the variation of

√−g with respect to δgµν . For this we write

δ√−g = − 1

2√−g δg, (15.23)

and observe that by varying the matrix elements gµν in the determinant we obtainthe cofactors which, in turn, are g times the inverse metric gµν [see Eq. (11A.22)].From this follows

δg = ggµνδgµν , (15.24)

as derived in Eq. (11A.24). Moreover, due to gµνgνλ = δµλ, we have

gλµδgµν = −gνκδgλκ, (15.25)

so that δgλκ = −gλµgκνδgµν and

δg = ggµνδgµν = −ggµνδgµν

δ√−g =

1

2

√−ggµνδgµν = −1

2

√−ggµνδgµν . (15.26)

Note that (15.25) implies a change of sign [compared with (15.14) and (15.15] if wecalculate the energy-momentum tensors from a variation δgµν with

δm

Aδgµν(x)

=1

2

√−g m

T µν (x). (15.27)

After writing the action (15.8) as

f

A= −1

2

d4x√−ggµνRµν

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15.2 Energy-Momentum Tensor and Spin Density 355

we find

δf

A = −1

2

d4x√−g

−1

2gµνδg

µνR + δgµνRµν + gµνδRµν

= −1

2

d4x√−g

[

δgµν(Rµν −1

2gµνR) + gµνδRµν

]

. (15.28)

The factor accompanying δgµν is known as the Einstein tensor

Gµν ≡ Rµν −1

2gµνR. (15.29)

Note that this tensor is symmetric only in symmetric spaces. The variation in δgµν ,however, picks out only the symmetrized part of it.

Consider now the variation of the Ricci tensor in (15.28)

δRµν = ∂κδΓµνκ − ∂µδΓκν

κ − δΓκντΓµτ

κ − ΓκντδΓµτ

κ + δΓµντΓκτ

κ + ΓµντδΓκτ

κ.

(15.30)

The left-hand side is a tensor. It is then useful to express also the right-hand sidevia covariant forms. For this we observe that contrary to the affine connection Γµν

κ

itself, the variation δΓµνκ is a tensor1 This follows directly from the transformation

law (11.106). The last, non-holonomic piece ∂µ∂νξκ in Γµν

κ disappear in δΓµνκ since

it is the same for Γµνκ and Γµν

κ + δΓµνκ. We therefore rewrite (15.30) in terms of

covariant derivatives as

δRµν = DκδΓµνκ −DµδΓκν

κ + 2SκµτδΓτν

κ. (15.31)

Indeed, this is equal to

δRµν = −∂κδΓµνκ − ∂µδΓκνκ − Γκµ

τδΓτνκ − Γκν

τδΓµτκ

+ ΓκτκδΓµν

τ + ΓµκτδΓτν

κ + ΓµντδΓκτ

κ − ΓµτκδΓτκν + 2Sκµ

τδΓτνκ, (15.32)

which is the same as (15.30). In symmetric spaces, the covariant relation (15.30)was first used by Palatini.

We now have to express δRµν in terms of δgµν and δKµνλ. It is useful to perform

all operations underneath the integral in (15.28):

−1

2

d4x√−ggµνδRµν . (15.33)

Due to the tensor nature of δΓµνκ we can take gµν through the covariant derivative

and write (15.33) as

−1

2

d4x√−g

(

DκδΓµµκ −DµδΓκ

µκ + 2SκµτδΓτµ

κ)

(15.34)

1In contrast, the difference δΓµνκ ≡g+δg

Γ µνκ now is not a tensor.

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356 15 Field Equations of Gravitation

The covariant derivatives can now be removed by a partial integration. In a spacewith torsion, partial integration has some particular features which requires a specialdiscussion.

Take any tensors Uµ...ν , V...ν... and consider an invariant volume integral

d4x√−g Uµ...ν...DµV...ν.... (15.35)

A partial integration gives

−∫

d4x

[

∂µ√−g Uµ...ν...V...ν... +

i

Uµ...νi...ΓµνiλiV...λi...

]

+ surface terms, (15.36)

where the sum over i runs over all indices of V...λi..., linking them via the affineconnection with the corresponding indices of Uµ...νi.... Now we use the relation

∂µ√−g =

√−g Γµκκ =

√−g Γµκκ =

√−g(

2Sµ + Γκµκ)

(15.37)

and (15.36) becomes

−∫

d4x√−g

[

(∂µUµ...λi... − Γκµ

κUµ...λi... +∑

i

ΓµνiλiUµ...νi...)V...λi...

+ 2Sµ∑

i

Uµ...λi...V...λi...

]

+ surface terms. (15.38)

Now, the terms in parentheses are just the covariant derivative of Uµ...νi... such thatwe arrive at the rule∫

d4x√−g Uµ...ν...DµV...ν... = −

d4x√−gD∗µUµ...ν...V...ν... + surface terms,(15.39)

where D∗µ is defined as

D∗µ ≡ Dµ + 2Sµ, (15.40)

where we have abbreviated:

Sκ ≡ Sκλλ, Sκ ≡ Sκλ

λ. (15.41)

It is easy to show that the operators Dµ and D∗µ can be interchanged, i.e., thereis also the rule∫

d4x√−g V...ν...DµU

µ...ν... = −∫

d4x√−gUµ...ν...D∗µV...ν... + surface terms.(15.42)

For the particular case that V...ν... is equal to 1, the second rule yields

d4x√−gDµU

µ = −∫

d4x√−g 2SµU

µ + surface terms. (15.43)

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15.2 Energy-Momentum Tensor and Spin Density 357

With the help of the last rule we can replace the covariant derivatives Dµ inEq. (15.34) by −2Sµ, and obtain

−1

2

d4x√−ggµνδRµν = −1

2

d4x√−g

(

−2SκδΓννκ + 2SµδΓκ

µκ + 2SκντδΓτν

κ)

.

(15.44)

The result can also be stated as follows:

−1

2

d4x√−ggµνδRµν = −1

2

d4x√−gSµκ,τδΓτµκ (15.45)

where Sµκ,τ is the following combination of torsion tensors:

1

2Sµκ

,τ ≡ Sµκτ + δµ

τSκ − δκτSµ. (15.46)

This tensor is referred to as the Palatini tensor. The relation can be inverted to

Sµνλ =1

2

(

Sµν,λ +1

2gµλSνκ

,κ − 1

2gνλSµκ

,κ)

. (15.47)

We now proceed to express δΓτµκ in terms of δgµν and δKµνλ. For this purpose

we note that the varied metric gµρ + δgµρ certainly satisfies the identity (11.97),

DτΓ+δΓ

(

gµρ + δgµρ)

= 0, (15.48)

where DΓ+δΓ is the covariant derivative formed with the varied connection Γµνλ +

δΓµνλ. For variations δgµρ this implies

Γ

Dτ δgµρ = δΓτµρ + δΓτρµ (15.49)

where we have introduced

δΓµτρ ≡ gρλ δΓµτλ. (15.50)

This gives

1

2

(

Γ

Dτ δgµρ+Γ

Dµ δgτρ−Γ

Dρ δgτµ

)

= δΓτµρ − δSτµρ + δSµρτ − δSρτµ

= δΓτµρ − δKτµρ, (15.51)

where

δSτµρ ≡ gρλ δSτµλ ≡ gρλ

1

2

(

Γτµλ − Γµτ

λ)

(15.52)

andδKτµρ ≡ δSτµρ − δSµµρ − δSµρτ + δSρτµ, (15.53)

are the results of a variation of Sµνλ at fixed gµν . Note that even though Γµν

λ =

Γµνλ −Kµν

λ, the left-hand side of (15.51) cannot be identified with gρκδΓτµλ since

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358 15 Field Equations of Gravitation

δKµνλ contains contribution from δSµν

λ at fixed δgµν and from δgµν at fixed Sµνλ.

The first term in (15.51), in fact, equal to gρκδΓτµκ + δKτµ

κ|Sλµν=fixed. Using (15.51),

we rewrite (15.45) as

−1

2

d4x√−ggµνδRµν = (15.54)

−1

2

d4x√−gSµκ,τ

[

δKτµ,κ +1

2

(

Dτδgµκ +Dµδgτκ −Dκδgτµ)

]

.

The first term shows that the Palatini tensor Sµκ,τ plays the role of the spin current

of the gravitational field [recall the definition (15.17) up to a factor 1/κ]

Σµκ,τ = −1

κSµκ

,τ . (15.55)

The second term can now be partially integrated, leading to

1

4

d4x√−g

D∗µSµρ,εδgµρ +D∗µS

µρ,τδgτρ −D∗qSµρ,τδgµτ

+ surface term. (15.56)

After relabeling the indices in (15.51), we arrive at the following variation of theaction with respect to δgµν , using the identity δgµνGµν = −δgµνGµν following from(15.25),

−1

2

d4x√−g

[

Gµν − 1

2D∗λ

(

Sµν,λ − Sνλ,µ + Sλµ,ν)

]

δgµν , (15.57)

so that the complete energy-momentum tensor of the field reads

f

Tµν = −1

κ

[

Gµν − 1

2D∗λ

(

Sµν,λ − Sνλ,µ + Sλµ,ν)

]

. (15.58)

Actually, the variation δgµν yields only the symmetrized part off

T µν . This spec-ification is, however, unnecessary. We shall demonstrate later that total angular

momentum conservation [see Eq. (18.13)] makesf

T µν symmetric as it stands (eventhough Gµν is not).

Thus we arrive at the following field equations

−κf

Σµκ,τ = Sµκ

,τ = κm

Σµκ,τ , (15.59)

−κf

Tµν = Gµν − 1

2D∗λ

(

Sµν,λ − Sνλ,µ + Sλµ,ν)

= κm

Tµν , (15.60)

which for a set of spinless point particles reduce to

Sµκ,τ = 0 , (15.61)

Gµν = κm

Tµν . (15.62)

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15.3 Total Energy-Momentum Tensor and Defect Density 359

15.3 Total Energy-Momentum Tensor and Defect Density

In defect physics, the total energy-momentum tensor obtained in (15.58) has a directphysical interpretation. In three euclidean dimensions, the linearized version of(15.58) reads

−κf

Tij = Gij −

1

2∂κ(

Sij,k − Sjk,i − Ski,j)

, (15.63)

with the spin density (15.46)

−1

f

Σij,k= Sij,k = Sijk + δikSj − δjkSi. (15.64)

Let us insert the dislocation density according to

Sijk =1

2

(

∂i∂j − ∂j∂i)

uk =1

2εijαlk. (15.65)

Then the spin density reads

Sij,k = εijlαlk + δikεjplαlp − δjkεiplαlp. (15.66)

Since both sides are antisymmetric in ij, we can contract them with εijn,

εijnSij,k = 2αnk + εkjnεjplαlp − εiknεiplαlp = 2αnk − 2(

δkpδnl − δklδnp)

αlp

= 2αkn, (15.67)

and see that Sij,k becomes simply

Sij,k = εijlαkl. (15.68)

Thus the spin density is equal to the dislocation density.The spin density has a vanishing divergence

∂kSij,k = εijl∂kαkl = 0. (15.69)

In terms of the derivatives of the displacement field ui(x), the spin density reads

Sij,k = εijlεkmn∂m∂nul. (15.70)

In this expression, the conservation law (15.69) is trivially fulfilled.Let us now form the three combinations of ij, k appearing in (15.63)

1

2

(

Sij,k − Sjk,i + Ski,j)

=1

2

(

εijlαkl − εjklαij + εkilαjl)

. (15.71)

By contracting the identity

εijlδkm + εjklδim + εkilδjm = εijkδlm (15.72)

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360 15 Field Equations of Gravitation

with αml, we find

εijlαkl + εjkklαil + εkilαjl = εijkαll (15.73)

so that

1

2

(

Sij,k − Sjk,i + Ski,j)

= −εjklαil +1

2εijkαll. (15.74)

The right-hand side is recognized to be

εjklKli (15.75)

where

Klj = −αjl +1

2δljKkk

is Nye’s contortion tensor. With this notation, equation (15.63) becomes

−κf

T ij = Gij − εjhl∂nKli. (15.76)

Now we recall that the Einstein tensor Gij for a metric gij = δij+∂iuj+δjui coincideswith the disclination density Θji. But then, comparison with Eq. (12.45) shows thatthe total energy-momentum tensor limes −κ is nothing but the total defect densityηij :

−κf

T ij = ηij (15.77)

Notes and References

[1] R. Utiyama, Phys. Rev. 101, 1597 (1956).

[2] T.W.B. Kibble, J. Math. Phys. 2, 212 (1961).

[3] F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys.48, 393 (1976).

[4] H. Kleinert, Gauge fields in Condensed Matter , Vol. II: Stresses and De-fects, Differential Geometry, Crystal Defects, World Scientific, Singapore, 1989(http://www.physik.fu-berlin.de/~kleinert/b2).

H. Kleinert, MULTIVALUED FIELDS

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The more minimal the art, the more maximum the explanation.

Hilton Kramer

16Minimally Coupled Fields of Integer Spin

So far we have discussed the gravitational field interacting with classical relativisticmassive point particles. If we want to include quantum effects, we must describethese particles by the Klein-Gordon equation (2.59). For a consistent interpretationof the negative-energy wave functions, this field has to be quantized, so that incomingnegative-energy wave functions describe outgoing antiparticles (recall p. 55).

Photons are described by Maxwell’s equations [recall (1.194) and (2.86)].For electrons, muons, and neutrinos, and other particles with spin 1/2, the fields

follow Dirac’s equation.All these equation can be coupled minimally to gravity by means of the multival-

ued mapping principle. We simply transform the flat-spacetime action to spacetimeswith curvature and torsion by means of a multivalued coordinate transformation.

In this text we shall not discuss the quantum aspect of the relativistic fields andtreat only their classical limit.

16.1 Scalar Fields in Riemann-Cartan Space

The action (2.25) of a charged scalar field in flat spacetime is transformed non-holonomically to a general affine spacetime. The partial derivative ∂a is equal viaEq. (14.1) to

∂a = eaµ(x)∂µ, (16.1)

and the volume element d4xa in flat spacetime becomes

d4xa = d4xµ |det eaµ(x)|. (16.2)

Since eaµ(x) is the square root of the metric gµν(x) by Eq. (11.39), the determinantsare also related by a square root, so that (16.2) without the indices amounts to thereplacement rule for the flat-spacetime volume:

d4x→ d4x√−g. (16.3)

With these rules, the action (2.25) becomes

A =∫

d4x√−g

[

h2eaµ(x)∂µφ∗(x)ea

ν(x)∂νφ(x) −M2c2φ∗(x)φ(x)]

. (16.4)

361

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362 16 Minimally Coupled Fields of Integer Spin

This expression cannot yet be used for field-theoretic calculations since the fieldseaν(x) are multivalued. We can, however, use Eq. (11.42) to rewrite the action as

A =∫

d4x√−g

[

h2gµν(x)∂µφ∗(x)∂νφ(x) −M2c2φ∗(x)φ(x)

]

. (16.5)

This expression contains only the single-valued metric tensor. The equation of mo-tion can be derived most simply by applying an integration by parts to the gradientterm. Ignoring a boundary term we obtain

A =∫

d4x√−g

[

−h2φ∗(x)∆φ(x) −M2c2φ∗(x)φ(x)]

, (16.6)

where

∆ ≡ 1√−g(

∂µ√−ggµν∂ν

)

(16.7)

is the well-known Laplace-Beltrami differential operator. From the action (16.6) weobtain directly the equation of motion as in (2.38):

δAδφ∗(x)

=∫

d4x′√

−g′[

−h2δ(4)(x′ − x)∆′φ(x′) −m2c2δ(4)(x′ − x)φ(x′)]

= (−h2∆ −M2c2)φ(x) = 0. (16.8)

This equation of motion contains an important prediction. There is no extraR−term in the wave equation, which would be allowed by covariance. In manytextbooks [1], the Klein-Gordon equation is therefore written as

(−h2∆ − ξh2R−M2c2)φ(x), (16.9)

with a parameter ξ for which several numbers have been proposed in the literature:1/6, 1/12 1/8. If present, the same R-term would of course appear in the nonrela-tivistic limit of (16.9). This is obtained by setting φ(x) = e−iMc2t/hψ(x) and lettingc → ∞. Assuming that g0i = 0 and choosing g00 = 1, this leads to the Schrodingerequation:

(

− 1

2Mh2∆ − ξh2Rs

)

ψ(x) = ih∂tψ(x), (16.10)

where Rs is curvature scalar of space. On a sphere of radius r in D dimensions it isequal to (D − 1)(D − 2)/r2.

The number ξ = (D − 2)/4(D − 1), makes the massless equation (16.9) confor-mally invariant in D spacetime dimensions [2], so that ξ = 1/6 is a preferred value insome theories. When DeWitt set up a time-sliced path integral in curved space, hefound ξ = 1/6 from his particular slicing assumptions [3]. A slightly different slicingled to ξ = 1/12 [4]. In more recent work, DeWitt prefers the value ξ = 1/8 [5]. Thevalue ξ = 0 was deduced from the multivalued mapping principle in Ref. [6].

So far there is no direct experimental confirmation of this prediction. There ishowever, indirect evidence. To see this we must assume that the extra R-term isuniversal, i.e., that the number ξ holds for all point particles and irrespective of the

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16.2 Electromagnetism in Riemann-Cartan Space 363

coordinates in which the Schrodinger equation is expressed. This is a reasonablephysical assumption for otherwise each particle would carry two gravitational pa-rameters, mass and something else. The point is now that the hydrogen atom inmomentum space is equivalent to a particle on a sphere in four dimensions of ra-dius pE =

√−2ME. The spectrum of the Schrodinger equation without an R-term

would be given by the eigenvalue equation

pEn/Mc = α, n = 1, 2, 3, . . . , (16.11)

where α ≈ 1/137 is the fine structure constant. This equation yields Rydberg levels

E = −Mc2α2

2n2 . (16.12)

If there was an R-term as in (16.10), this spectrum would be determined by

pE(n+ 6ξ)/Mc = α, n = 1, 2, 3, . . . . (16.13)

The parameter ξ would directly appear in the denominator of (16.12) as [7]

E = − Mc2α2

2(n+ 6ξ)2 . (16.14)

Such a distortion of the Rydberg spectrum would certainly have been detected forall the above candidates for ξ.

It is sometimes useful to express the Laplace-Beltrami operator (16.7) in termsof the Riemann connection Γµν

λ. For this we use Eq. (11A.24) to calculate thederivative

1√−g(

∂µ√−g

)

=1

2gλκ(∂µgλκ) = Γµλ

λ. (16.15)

By differentiating the inverse metric (11.42) we have, furthermore,

∂µgµν = −Γµ

µν − Γµνµ. (16.16)

Using the chain rule of differentiation on the right-hand side of (16.7), we then findthe alternative expression

∆ = gµν∂µ∂ν − Γµµν∂ν . (16.17)

16.2 Electromagnetism in Riemann-Cartan Space

Let us go through the same procedure for the electromagnetic action (16.20). Thevolume clement is again mapped according to the rule (16.3). The covariant curl istreated as follows. First we introduce vector fields transforming like the coordinatedifferentials dxµ

Aµ(x) = eaµ(x)Aa(x), jµ(x) = eaµ(x)ja(x), (16.18)

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364 16 Minimally Coupled Fields of Integer Spin

and rewrite the field strengths with the help of (11.86) as

Fab(x) = ∂aAb(x) − ∂bAa(x) = eaµ(x)∂µeb

ν(x)Aν(x) − ebµ(x)∂µea

ν(x)Aν(x)

= eaµ(x)eb

ν(x)[

DµAν(x) −DνAµ(x)]

≡ eaµ(x)eb

ν(x)Fµν(x). (16.19)

Then the action (16.20) becomes

em

A =∫

d4x√−g em

L (x) ≡∫

d4x√−g

[

− 1

4cF µν(x)Fµν(x) −

1

c2jµ(x)Aµ(x)

]

,(16.20)

Expressing the covariant derivatives as in (11.87), the field strength take the form

Fµν(x) = ∂µAν(x) − ∂νAµ(x) − 2SµνλAλ(x) (16.21)

The last term destroys gauge invariance as noted first by Schrodinger [8] (see theremarks in the Preface) This is why he derived upper bounds for the photon massfrom experimental observations. The present upper bound is

mγ < 3 × 10−27eV. (16.22)

This estimate comes from observations of the range of magnetic fields emanatinginto spacetime from pulsars. The range corresponding to the above number is theCompton wavelength

lγ =h

mγc> 6952 light years. (16.23)

In order to be invariant under the usual electromagnetic gauge transformations

Aµ → Aµ + ∂µΛ, (16.24)

the electromagnetic action

Aem = −1

4

d4x√−gFµνF µν , (16.25)

must contain the same field strengths in spacetimes with curvature and torsion asin flat spacetime:

Fµν = ∂µAν − ∂νAµ, (16.26)

This covariant curl is compatible with invariance under general coordinate trans-formation since it can be rewritten as a covariant curl involving Riemann covariantderivatives Dν :

DµAν − DνAµ = ∂µAν − ΓµνλAλ − ∂νAµ + Γνµ

λAλ = ∂µAν − ∂νAµ. (16.27)

Thus the field strength (20.45) is gauge invariant under electromagnetic and Einsteintransformations. For this reason, many authors have postulated that the photondoes not couple to torsion. Later we shall argue that this is not really necessary:torsion will be seen to be nonpropagating field, so that empty space where photonsare observed to propagate with light velocity cannot carry any torsion.

H. Kleinert, MULTIVALUED FIELDS

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Notes and References 365

Notes and References

[1] N.D. Birell and P.C.W. Davies, Quantum Fields in Curved Space, CambridgeUniversity Press, Cambridge, 1982.

[2] See Eq. (13.241) inH. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets , World Scientific, Singapore 2004, 4th ed.kl/b5).

[3] B.S. DeWitt, Rev. Mod. Phys., 29, 337 (1957).

[4] K.S. Cheng, J. Math. Phys. 13, 1723 (1972).

[5] B.S. DeWitt, Supermanifolds, Cambridge Univ. Press, Cambridge, 1984.

[6] See Chapters 10 and 11 in the textbook [2] (kl/b5/psfiles/pthic10.pdf).

[7] See Section 13.10 in the textbook [2]. (kl/b5/psfiles/pthic13.pdf).

[8] E. Schrodinger, Proc. R. Ir. Acad. A 49, 43, 135 (1943); 49, 135 (1943); 52, 1(1948); 54, 79 (1951).

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We can lick gravity, but sometimes the paperwork is overwhelming

Wernher von Braun (1912 - 1977)

17Particles with Half-Integer Spin

Let us now see how electrons and other particles of half-integer spin are coupled togravity [1].

17.1 Local Lorentz Invariance and AnholonomicCoordinates

Spin is defined in Lorentz-invariant theories as the total angular momentum in therest frame of the particle. To measure the spin s of a particle moving with velocityv, we go to a comoving frame by a local Lorentz transformation. Then its quantummechanical description requires 2s+1 states |s, s3〉 with s3 = −s, . . . , s which, uponrotations, transform according to an irreducible representation of the rotation groupwith angular momentum s.

For a particle of spin s = 1/2 such as an electron, a muon, or any other massivelepton in Minkowski spacetime, this transformation property is automatically ac-counted for by the states that can be created by a quantized Dirac field ψα(x) withthe action (2.140):

m

A=∫

d4xa ψ(xa) (iγa∂a −m)ψ(xa), (17.1)

where the matrices γa satisfy the Dirac algebra (1.218):

γa, γb

= 2gab. (17.2)

The Dirac equation is obtained by extremizing this action as in Eq. (2.142):

δm

Aδψ(xa)

= (iγa∂a −m)ψ(xa) = 0. (17.3)

17.1.1 Nonholonomic Image of Dirac Action

By complete analogy with the treatment of the action of a scalar field in Section 16.1we can immediately write down the action in a spacetime with curvature and torsion:

m

A=∫

d4x√−gψ(x)

[

iγaeaµ(x)∂µ −m

]

ψ(x), (17.4)

366

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17.1 Local Lorentz Invariance and Anholonomic Coordinates 367

where x are the physical coordinates xµ. In contrast to the scalar case, however, thistransformed action contains the multivalued tetrad fields for which the field-theoreticformalism is invalid. We must find a way of transforming away the multivaluedcontent in ea

µ(x). This is done by the introduction, at each point xµ, of infinitesimalcoordinates dxα associated with a freely falling Lorentz frame. We may simplyimagine infinitesimal freely falling elevators inside of which there is no gravity. Theremoval of the gravitational force holds only at the center of mass of a body. At anydistance away from it there are tidal forces where either the centrifugal force or thegravitational attraction becomes dominant. At the center of mass, the coordinatesdxα are Minkowskian, but the affine connections are nonzero and have in general anonzero curvature which cause the tidal forces.

Intermediate Theory

We proceed as in Section 4.5 and observe that the modified Dirac Lagrangian density

m

L= ψ(xα)

iγα[

∂α −D(Λα(xα))−1∂αD(Λ(xα))]

−m

ψ(xα) (17.5)

describes electron just as well as the original one, where Λ(xα) is an arbitrary localset of Lorentz transformations which connects the flat-spacetime coordinates xa in(17.1) with new coordinates xα:

dxa = Λaα(x)dx

α, dxα = (Λ−1)αa(x)dxa ≡ Λa

α(x)dxa. (17.6)

Here and in the sequel, we shall suppress the superscript of x in the arguments ofΛ(x) whenever it is not necessary to be explicit. The same will be done for ψ(x) Themetrics in the two coordinate systems are Minkowskian for any choice of Λa

α(x)dxα:

gαβ(x)=Λaα(x)Λ

bβ(x)gab = (ΛT )α

a(x) gab Λbβ(x)≡gab, (17.7)

in accordance with Eq. (1.28).The solutions of the associated Dirac equation ψ′(x) are obtained from the orig-

inal solutions ψ(x) by the spinor representation of the local Lorentz transformation:ψ′(x) = D(Λ(x))ψ(x). This reflects the freedom of solving Dirac’s anticommutationrules (1.218) by the x-dependent γ-matrices D(Λ(x))−1γαD(Λ(x)) [recall (1.229)and (1.28)]:

D(Λ(x))−1γαD(Λ(x)), D(Λ(x))−1γβD(Λ(x)) = γa, γbΛaα(x)Λ

bβ(x)

= gabΛaα(x)Λ

bβ(x) = gαβ. (17.8)

We now recall Eq. (1.254) according to which, in a slightly different notation,

D(Λ(x))−1∂αD(Λ(x)) = −i 1

2ωα;δσ(x)

(

Σδσ)

B

C . (17.9)

The right-hand side may be defined as the spin connection for Dirac fields:

D

ΓαBC(x) ≡ i

1

2ωα;δσ(x)

(

Σδσ)

B

C . (17.10)

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368 17 Particles with Half-Integer Spin

Here ωα;βγ are the generalized angular velocities obtained by relations of the type

(1.251) from the tensor parameters ωβγ of the local Lorentz transformations Λ(x) =

e−iωβγ(x)Σβγ .

According to Eq. (1.253), the generalized angular velocities ωα;βγ appear also in

the derivatives of the local Lorentz matrices Λaα(x) as

Λ−1γa(x)∂αΛ

aβ(x) = ωα;

γβ(x) = −ωα;β

γ(x). (17.11)

Thus, if we defineΛ

Γαβγ ≡ Λa

γ∂αΛaβ = −Λa

β∂αΛaγ . (17.12)

we can write the Dirac spin connection as

D

ΓαBC(x) ≡ −i 1

2

Λ

Γαδσ(x)

(

Σδσ

)

B

C . (17.13)

The transformation has produced a Lagrangian density

m

L= ψ(x) (iγαDα −m)ψ(x), (17.14)

with the covariant derivative matrix

(Dα)BC = δB

C∂α−D

ΓαBC(x), (17.15)

which is completely equivalent to the original Dirac Lagrangian density in Eq. (17.4),as long as the spin connection is given by (17.12) with single-valued Lorentz trans-formations Λa

γ. This is the analog of the Schrodinger Lagrangian (4.80) which wasthe starting point for the introduction of electromagnetism by multivalued gaugetransformations.

We may now proceed in the same way as before by postulating the local param-eters the tensor fields ωβγ(x) to be multivalued so that the components of the spinconnection are no longer generalized angular velocities (17.11) but new independentfields, which cannot be calculated from ωβγ(x) by gradient equations like (1.251).Then the Lagrangian density (17.14) with the covariant derivative (17.15) describesa nontrivial theory coupled to torsion. The affine connection (17.12) will be seen inEq. (17.66) to coincide with the contortion in the locally Minkowskian coordinatesdxα. The intermediate spacetime dxα has no Riemannian curvature, so that the theRiemann-Cartan curvature tensor is determined completely by the contortion ten-sor via Eq. (11.141). The covariant derivative formed with the Christoffel symbolsallows for the definition of parallel vector fields over any distance. This theory is acounterpart of the famous teleparallel theory developed by Einstein since 1928 andinfluenced by a famous letter exchange with Cartan (recall the Preface). There thesituation is the opposite: the Riemann-Cartan curvature tensor vanishes identically,and the Riemann curvature is given via Eq. (11.141) by

−Rµνλκ = DµKνλ

κ − DνKµλκ +

(

KµλρKνρ

κ −KνλρKµρ

κ)

, (17.16)

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17.1 Local Lorentz Invariance and Anholonomic Coordinates 369

17.1.2 Vierbein Fields

In order to describe the correct gravitational forces, we must go one step further.

Following the standard procedure of Section 4.5 we first perform the analog ofa single-valued gauge transformation which is here an ordinary coordinate transfor-mation from xα to xµ:

dxα = dxµhαµ(x). (17.17)

The transformation has an inverse

dxµ = dxαhαµ(x), (17.18)

and the matrix elements hαµ(x) and hαµ(x) satisfy, at each x, the orthonormality

and completeness relations

hαµ(x)hβµ(x) = δα

β, hαµ(x)hαν(x) = δµ

ν . (17.19)

The 4 × 4 transformation matrices hαµ(x) and hαµ(x) are called vierbein fields and

reciprocal vierbein fields, respectively. As in the case of the multivalued basis tetradseaµ(x), ea

µ(x) we shall freely raise and lower the indices α, β, γ, . . . with the metric

gαβ and its inverse gαβ:

hαµ(x) ≡ gαβhβµ(x), hαβ(x) ≡ gαβh

βµ(x). (17.20)

Since the transformation functions are, for the moment, single-valued, they sat-isfy

∂µhαν(x) − ∂νh

αµ(x) = 0, ∂µhα

ν(x) − ∂νhαµ(x) = 0. (17.21)

For spinor fields as functions depending on the final, physical coordinates xµ, thecovariant derivative (17.15) becomes

(Dα)BC = δB

Chαµ(x)∂µ−

D

ΓαBC(x). (17.22)

The flat spacetime xa coordinates are now related to the physical coordinates xµ bythe equation

dxa = Λaα(x)dx

α = Λaα(x)h

αµ(x)dx

µ, (17.23)

where the Lorentz transformation Λ(x) are with multivalued, but the action

m

A=∫

d4x√−g ψ(x)

(

iγαhαµ(x)Dµ −m

)

ψ(x), (17.24)

contains only the single-valued geometric fields hαµ(x) and Aα;δσ(x).

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370 17 Particles with Half-Integer Spin

17.1.3 Local Inertial Frames

This is now the place where we can easily introduce curvature by allowing thecoordinates xα to be multivalued functions of the physical coordinates of xµ. Thenthe vierbein fields satisfy no longer the relation (17.21). Since they, themselves, aresingle-valued functions the action is perfectly suited to describe electrons and otherelementary spin-1/2 Dirac particles in spacetimes with curvature and torsion.

Since dxα is related to dxa by a Lorentz transformation (17.23), the length of dxα

is measured by of the nonholonomic coordinates dxα is Minkowskian at the point x:

ds2 = gαβdxαdxβ, (17.25)

where

gαβ = Λaα(x)Λ

bβ(x)gab ≡

1−1

−1−1

αβ

. (17.26)

is the Minkowski metric, due to (1.28).Combining (17.25) with (17.17), we see that the vierbein fields hα

µ(x) transformthe metric gµν(x) to the constant Minkowski metric

gαβ = hαµ(x)hβ

ν(x)gµν(x) (17.27)

at each x. The inverse of this relation shows that the metric gµν(x) is the square ofthe matrices hαµ(x),

gµν(x) = hαµ(x)hβν(x)gαβ ≡ hαµ(x)hβν(x). (17.28)

just as it was the square of the multivalued basis tetrads eaµ(x) in Eq. (11.39).There is a simple physical relation between the physical coordinates xµ, and

the infinitesimal coordinates dxα. The latter are associated with small freely fallingLorentz frames at each xµ. Such frames are also called inertial frames. They may beimagined as small freely falling elevators in which there is no gravity. The removalof the gravitational force holds only at the center of mass of the elevators. At anydistance away from it there are tidal forces where either the centrifugal force or thegravitational attraction becomes dominant. Let us verify this explicitly. In a smallneighborhood of an arbitrary point Xµ we solve the differential equation (17.18) bythe functions

xα(X; x) = aα+hαµ(X)(xµ−Xµ)+1

2hαλ(X)Γµν

λ(X)(xµ−Xµ)(xν−Xν)+. . . . (17.29)

The derivatives

∂xα(X; x)

∂xµ= hαµ(X) + hαλ(X)Γµν

λ(X)(xν −Xν) ≡ h(X; x) (17.30)

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17.1 Local Lorentz Invariance and Anholonomic Coordinates 371

fulfill (17.18) at x = X. Consider now a point particle satisfying the equation ofmotion (14.7). In the coordinates (17.29), the trajectory satisfies the equation

qα = hαµ(X)qµ + hαλ(X)Γµνλ(X) qµ(qν −Xν) + . . . . (17.31)

and

qα = hαµ(X)qµ+hαλ(X)Γµνλ(X) qµ(qν−Xν)+hαλ(X)Γµν

λ(X) qµqν+. . . . (17.32)

Inserting here (14.7), the first and third terms cancel each other, and the trajec-tory experiences no acceleration at the point X. In the neighborhood, there aretidal forces. Thus the infinitesimal constitute an inertial frame in an infinitesimalneighborhood of the point X.

The metric in the coordinates xα(X; x) is

gαβ(X; x) =∂xα(X; x)

∂xµ∂xβ(X; x)

∂xµgµν(x) = hαµ(X)hβν(X)gµν(x) (17.33)

+[

hαλ(X)hβν(X)Γµκλ(X)(xκ −Xκ) + (α↔ β)

]

gµν(x).

We now expand the metric gµν(x) in the neighborhood of X as

gµν(x) = gµν(X) + ∂λgµν(X)(xλ −Xλ) + . . .

= gµν(X) −[

gµλΓκλν(X) + gνλΓκλ

µ(X)]

(xκ −Xκ) + . . . . (17.34)

Inserting this into (17.33) and using (17.27), we obtain

gαβ(X; x) = gαβ + O(x−X)2). (17.35)

This ensures, that the affine connection formed from gαβ(X; x) vanished at x = X,so that there are no forces at this point. In any neighborhood of X, however, therewill be tidal forces.

In the coordinates dxa, there are no tidal forces at all. This is possible everywhereonly due to defects which make Λa

α(x) multivalued.The coordinates xα(X; x) are functions of x which depend on X. There exists

no single function xα(x), so that derivatives in front of xα(x) do not commute:(

∂µ∂ν − ∂ν∂µ)

xα(x) 6= 0, (17.36)

implying that

∂µhαν(x) − ∂νh

αµ(x) 6= 0. (17.37)

The functions hαµ(x) and hµα(x), however, which describe the transformation tothe freely falling elevators are single-valued. They obey the integrability condition

(

∂µ∂ν − ∂ν∂µ)

hαλ(x) = 0,(

∂µ∂ν − ∂ν∂µ)

hαλ(x) = 0. (17.38)

This condition has the consequence that if we construct a tensorh

Rµνλκ(x) from the

transformation matrices hαµ(x) in the same way as Rµνλ

κ(x) was made from eaµ(x)

in Eq. (11.129), we find an identically vanishing result:

h

Rµνλκ = hα

κ(∂µ∂ν − ∂ν∂µ)hαλ ≡ 0. (17.39)

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372 17 Particles with Half-Integer Spin

17.2 Difference between Vierbein and MultivaluedTetrad Fields

Note that hαµ(x) and eaµ(x) are completely different mathematical objects withdifferent integrability properties. While hαλ(x) satisfies (17.38), eaλ(x) does not,since the commutator of the derivatives in front of eaλ(x) determine the curvaturetensor via Eq. (11.129).

Both, the multivalued basis tetrads eaµ(x) and the vierbein fields hαµ(x) are“square roots” of the metric gµν(x). They differ by a local Lorentz transformationΛa

µ(x). This is precisely the freedom one has in defining such a “square root”.By introducing Lorentz transformations with rotational defects dxα = dxaΛa

α(x),where Λα

a (x) are non-integrable functions of x, we have achieved that the vierbeinfields have commuting derivatives [recall (17.38)].

From Eq. (17.23) we see that the relation between the vierbein and multivaluedbasis tetrad fields is given by

eaµ(x) = Λaα(x)h

αµ(x). (17.40)

whose inverse isΛa

α(x) ≡ eaµ(x)hαµ(x), (17.41)

Since hαµ and hαµ are single-valued functions with commuting derivatives, the

curvature tensor Rµνλκ in (11.129) may be expressed completely in terms of the

noncommuting derivatives of the local Lorentz transformations Λaα(x) and Λa

α(x).To see this we insert Eq. (17.40) into (11.129), and use (17.38) to find for thecurvature tensor the alternative expression

Rµνλσ = hγ

σ[

Λaγ(

∂µ∂ν − ∂ν∂µ)

Λaα

]

hαλ ≡ hγσRµνα

γ hαλ. (17.42)

From the defect point of view, the single-valued matrices hαµ(x) create an inter-

mediate coordinate system dxα which, by the integrability condition (17.38), has thesame disclination content as the coordinates xµ, but is completely free of disloca-tions. The metric in the new coordinate system xα is Minkowski-like at each pointin spacetime. Still, the coordinates xα do not form a Minkowski spacetime sincethey differ from the inertial coordinates dxa by the presence of disclinations, i.e.,there are wedge-like pieces missing with respect to an ideal reference crystal. Thecoordinates xα cannot be defined globally from xµ. Only the differentials dxα areuniquely related to dxµ at each spacetime point by Eqs. (17.17) and (17.18). Thelocal Lorentz transformations Λa

α(x) have noncommuting derivatives on account ofthe disclinations residing in the coordinates dxα. The coordinate system dxα canonly be used to specify derivatives with respect to xα, or the directions of vectors(and tensors) with respect to the intermediate local axes

eα(x) ≡ eµ(x)∂xµ

∂xα= eµ(x)hα

µ(x)

≡ eaeaµ(x)hα

µ(x) ≡ eaΛaα(x). (17.43)

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17.2 Difference between Vierbein and Multivalued Tetrad Fields 373

We can go back to the local basis via the reciprocal vierbein fields

eµ(x) = ea(x)∂xα

∂xµ= eα(x)h

αµ(x). (17.44)

Thus, an arbitrary vector may be transformed as follows,

v(x) ≡ eava(x) = eae

aµ(x)v

µ(x) = eaΛaα(x)h

αµ(x)v

µ(x)

= eaΛaα(x)h

αµ(x)vµ(x) = eaΛaα(x)v

α(x) = eaΛaα(x)vα(x), (17.45)

where we have introduced the co- and contravariant components

vα(x) ≡ vµ(x)hαµ(x), vα(x) ≡ vµ(x)hαµ(x). (17.46)

The orthogonality relations (17.19) imply the inverse relations

vµ(x) = vα(x)hαµ(x), vµ(x) ≡ vµ(x)hα

µ(x). (17.47)

For vector fields vβ, vβ whose components refer to the intermediate basis eα(x), the

covariant derivatives are

Dαvβ = ∂αvβ −Λ

Γαβγvγ , Dαv

β = ∂αvβ +

Λ

Γαγβvγ, (17.48)

whereΛ

Γαβγ is the spin connection for vector fields to be calculated from the local

multivalued Lorentz transformations Λaβ(x) of Eq. (17.41) rather than from eaµ

[compare also (11.93)]:

Λ

Γαβγ ≡ Λa

γ∂αΛaβ = −Λa

β∂αΛaγ . (17.49)

Thus, the spin connection for local Lorentz vector fields is precisely the object foundin the covariant derivatives of the spinor fields (17.22).

By analogy with the pure gradient (4.50) of a single-valued gauge function in theabelian gauge theory of magnetism, the spin connection (17.49) reduces to trivialgauge field for single-valued local Lorentz transformations Λ(x). Indeed, it is easilyverified that the field strength associated with this gauge field, the covariant curl

Fµναγ ≡ ∂µ

Λ

Γναγ − ∂ν

Λ

Γµαγ − [

Λ

Γµ,Λ

Γν ]αγ, (17.50)

vanishes. As in Eqs. (11.128), (11.125), the commutator is defined by consideringΛ

Γναγ as matrices (

Λ

Γν)αγ. For multivalued Lorentz transformations, the covariant curl

is in general nonzero, andΛ

Γναγ(x) is a nonabelian gauge field, whose field strength

Fµναγ ≡ is a tensor under single-valued local Lorentz transformations.

For multivalued transformations Λ(x), the a nonabelian gauge field (17.49) hasnonzero field strengths (17.50).

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374 17 Particles with Half-Integer Spin

From Eq. (17.49) it follows that Λaα(x) and Λa

α(x) satisfy identities like (11.94),(11.95):

DαΛaβ = 0, DαΛa

β = 0. (17.51)

It is instructive to rewrite the spin connection in matrix notation using thenotation (17.17) for the local Lorentz transformations. Then the relation (17.45)shows that

va(x) = Λaαv

α(x), va(x) = Λaαvα(x)=(gΛg)a

αvα(x)=(

ΛT−1)

a

αvα(x), (17.52)

and therefore

∂αva(x) = Λa

βDαvβ(x) = Λa

β

[

∂αδβα + (Λ−1∂αΛ)βγ

]

vγ(x). (17.53)

∂αva(x) = ΛaβDαvβ(x) = Λa

β[

∂αδβγ + (ΛT∂αΛ

T−1)βγ]

vγ(x)

= Λaβ[

∂αδβγ − (Λ−1∂αΛ)γβ

]

vγ(x). (17.54)

From this we identify

Λ

Γαβγ = (Λ−1∂αΛ)βγ = −(Λ−1∂αΛ)γβ, (17.55)

which is the same as (17.49).If a field has several local Lorentz indices α, β, γ, . . ., each index receives an own

contribution proportional to the gauge field Aαβγ . If it has, in addition, Einstein in-

dices µ, ν, λ, . . ., there are also additional terms proportional to the affine connectionΓµν

λ. As an example, the covariant derivatives of the fields vµβ and vβµ with respectto the nonholonomic coordinates dxα are from (17.48) and (12.70), (12.71):

Dαvµβ = ∂αv

µβ −

Λ

Γαβγvγ + hα

κΓκνµvνβ, (17.56)

Dαvβµ = ∂αv

β +Λ

Γαγβvγ − hα

κΓκµνvβν . (17.57)

The covariant derivatives with respect to the physical coordinates xλ are

Dλvµβ = ∂λv

µβ − hαλ

Λ

Γαβγvγ + Γλν

µvνβ , (17.58)

Dλvβµ = ∂λv

β + hαλΛ

Γαγβvγ − Γλµ

νvβν . (17.59)

Let us express the spin connection (17.49) for vector fields in terms of eaµ andha

µ. With the help of (17.41), we calculate

Λ

Γαβγ = ea

λhγλhαµ∂µ(e

aνhβ

ν)

= hγλhαµhβ

νΓµνλ + hγλhα

µδλν∂µhβν

= hγλhαµhβ

ν(Γµνλ + hδν∂µhδ

λ). (17.60)

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17.2 Difference between Vierbein and Multivalued Tetrad Fields 375

Employing the covariant derivatives (17.56) and (17.57), this equation can be recastas

Dαhβµ = 0, Dαh

βµ = 0, (17.61)

so that hαµ satisfies similar identities as ea

µ in (11.95) and as Λaα in (17.51).

At this place it is useful to introduce the symbols

h

Γµνλ ≡ hα

λ∂µhαν ≡ −hαν∂µhαλ. (17.62)

They are defined in terms of hαµ in the same way as Γµνλ is defined in terms of eaµ

in Eq. (11.93). Then we may rewrite Eq. (17.60) as

Λ

Γαβγ = hγλhα

µ(

Γµνλ − hδ

λ∂µhδν

)

= hγλhαµhβ

ν(Γµνλ− h

Γµνλ). (17.63)

If we now decompose the two connections on the right-hand side into Christoffel partsand contortion tensors in the same way as in Eqs. (11.114)–(11.116), we realize thatdue to the identity

gµν(x) = eaµ(x)ebν(x)gab ≡ hαµ(x)h

βν(x)gαβ, (17.64)

the two Christoffel parts in Γµνλ and

h

Γµνλ are the same:

Γµνλ ≡

h

Γµνλ. (17.65)

As a consequence,Λ

Γαβγ becomes

Λ

Γαβγ = hγλhα

µhβν(

Λ

Γµνλ +Kµν

λ −h

Γµνλ− h

Kµνλ)

= hγλhαµhβ

ν(Kµνλ− h

Kµνλ). (17.66)

where Kµνλ is the contortion tensor (11.118), and

h

Kµνλ denotes the expression

(11.116) with eaµ, eaµ replaced by hαµ, hα

µ. Explicitly, these tensors are

Kµνλ = Sµν

λ − Sνλµ + Sλµν , (17.67)

h

Kµνλ =

h

Sµνλ− h

Sνλµ+

h

Sλµν , (17.68)

whereh

Sµνλ ≡ 1

2

(

hαλ∂µh

αν − hα

λ∂νhαµ

)

. (17.69)

This is the so-called object of anholonomy , often denoted by Ωµνλ. They are anti-

symmetric in the first two indices, which makes their combinations (17.68) antisym-metric in the last two indices, if the last index is lowered by a contraction with gλκ.

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376 17 Particles with Half-Integer Spin

The tensors (17.69) and (17.68) have therefore the same symmetry properties as thecontortion and torsion tensors. The spin connection (17.66) in which the last indexis lowered by a contraction with the Minkowski metric gαβ [recall (17.27)] is thenantisymmetric in the last two indices.

It will be helpful to freely use hαµ, hαµ for changing indices α into µ, for instance,

Kαβγ ≡ hγλhα

µhβνKµν

λ, (17.70)

h

Kαβλ = hγλhα

µhβν h

Kµνλ. (17.71)

17.3 Nonholonomic Image of Dirac Action

Inserting (17.66) into (17.22) we finally obtain single-valued image of the flat-spacetime action (17.1):

m

A=∫

d4x√−gψ(x)

[

iγαhαµ(x)Dµ −m

]

ψ(x), (17.72)

with the covariant derivative

Dµ ≡ ∂µ − i1

2hγλhβ

ν(Kµνλ− h

Kµνλ)Σβ

γ, (17.73)

Note that with the help of the gauge fields in (17.49), the curvature tensor Rµναγ

defined in Eq. (17.42) can be rewritten as

Rµναγ= Λa

γ(

∂µ∂ν − ∂ν∂µ)

Λaα= ∂µ

Λ

Γναγ − ∂ν

Λ

Γµαγ −

Λ

ΓµαδΛ

Γνδγ +

Λ

ΓναδΛ

Γµδγ . (17.74)

This follows directly by performing the derivatives successively and inserting (17.49),while using the pseudo-orthogonality of the Lorentz matrices Λ(x). On the right-hand side we recognize the standard covariant curl formed from the nonabelian gauge

fieldΛ

Γναγ in the same way as in Eq. (17.50). Thus we shall denote the right-hand

by

Fµναγ ≡ ∂µ

Λ

Γναγ − ∂ν

Λ

Γµαγ −

Λ

ΓµαδΛ

Γνδγ +

Λ

ΓναδΛ

Γµδγ = Rµνα

γ = hαλRµνλ

κhγκ. (17.75)

It is instructive to prove this equality in another way using Eq. (17.63). This leadsto the complicated expression

Fµνβγ =

∂µ

[

(Γ− h

Γ)νλκhβ

λhακ

]

− (µ ↔ ν)

(Γ− h

Γ)µλτ (Γ− h

Γ)ντκhβ

λhγκ − (µ↔ ν)

, (17.76)

which may be regrouped to[

∂µΓνλκ − (ΓµΓν)λ

κ − (µ↔ ν)]

hβλhγκ

+

Γνλκ∂µ

(

hββλhγκ

)

− ∂µ

(

h

Γνλκhβ

λhγκ

)

− (µ ↔ ν)

+(

Γµh

Γν +h

Γµ Γν−h

Γµh

Γν

)

λ

κ

hβλhγκ − (µ↔ ν)

. (17.77)

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17.3 Nonholonomic Image of Dirac Action 377

Recalling (11.129) we see that the equality (17.75) is verified if we demonstrate thevanishing of the terms in curly brackets. The first term inside these brackets is

Γνλκ∂µhβ

λhγκ+ Γνλκhβ

λ∂µhγκ−(µ↔ ν)=−Γνλ

γ h

Γµβλ + Γνβ

λΓνβκ h

Γµκγ+(µ↔ ν) ,

and the second contributes [using (17.62)]

−∂µ(

hβλ∂νh

γλ

)

− (µ↔ ν) =h

Γµβλ h

Γµβγ − (µ↔ ν) − hβ

λ(

∂µ∂ν − ∂ν∂µ)

hγλ.

Thus we find indeed, recalling (17.38) and (17.39),

Fµνβγ =

(

Rµνλκ− h

Rµνλκ)

hβλhγκ = Rµνλ

γhβλhγκ. (17.78)

The equality of the covariant curls of Fµναγ andRµνλ

κ up to a coordinate transfor-mation of last two indices is related to a fundamental algebraic property of covariantderivatives. Consider a vector field vλ and apply DµDν −DνDµ to it. We find

[

Dµ, Dν

]

vλ = ∂µ (∂νvλ − Γνλκvκ) − Γµ

τDτvλ − Γµλτ (∂νvτ − Γντ

κvκ) − (µ↔ ν) .

(17.79)

Since(

∂µ∂ν − ∂ν∂µ)

vλ = 0, we obtain the so-called Ricci identity

[

Dµ, Dν

]

vλ = −Rµνλκvκ − 2Sµν

τDτvλ. (17.80)

For a general tensor, Rµνλκ and Sµν

τ act additively on each index. Now, a similarrelation may be calculated for the components of the vector in the nonholonomicbasis ea

β:

[

Dµ, Dν

]

vβ = ∂µ

(

∂νvβ −Λ

Γνβγvγ

)

− ΓµντDτvβ −

Λ

Γµβγ

(

∂νvγ −Λ

Γνγδvδ

)

− (µ↔ ν)

= −Fµνβγvγ − 2SµντDτvβ. (17.81)

For a field of arbitrary spin, this generalizes to

[

Dµ, Dν

]

ψ =i

2Fµνβ

γΣβγψ − 2Sµν

τDτψ. (17.82)

Due to the complete covariance of (17.79) and (17.81), we may multiply (17.79) byhβ

λ and pass this factor through the covariant derivatives (which, in this process,change their connection since they are applied to different objects before and afterthe passage). The R-term in (17.80) and the F term in (17.82) remain simply relatedby (17.74).

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378 17 Particles with Half-Integer Spin

17.4 Alternative Form of Coupling

Let us compare the above derived minimal coupling of the vierbein filed hαµ(x)

to a spinning particle with the theory of Weyl, Fock, and Iwanenko [2, 3, 4] inRiemann spacetimes. These authors proposed to search for spacetime-dependentDirac matrices γµ(x) solving the Dirac algebra

γµ(x), γν(x) = gµν(x), (17.83)

which can obviously be expressed in terms of our vierbein fields as

γµ(x) = γαhαµ(x). (17.84)

In terms of these, the Dirac action can be written as

m

A=∫

d4x√−g ψ(x)

iγµ(x)Dµ −m

ψ(x), (17.85)

where Dµ is the covariant derivative (omitting the Dirac spin indices)

Dµ = ∂µδµ − Γµ(x), (17.86)

with the Dirac spin connection [compare (17.13)]

Γµ(x) ≡ −1

4γλ(x)Dµγ

λ(x) = −1

4γλ(x)

[

∂µγλ(x) + Γµν

λ(x)γν(x)]

. (17.87)

Let us verify that the action (17.85) is equivalent to the previous one in (17.72) ifthere is no torsion. Inserting (17.84) into (17.87) we find

Γµ = −(

hαλ∂µhβλ + hλαhβ

ν Γµνλ) 1

4γαγβ (17.88)

We now use the analog of (11.88) for hαν(x) [which follows from the completeness

relation (17.19)]

∂µhβλ = −hβν

(

hγλ∂µh

γν

)

. (17.89)

Then we can rewrite (17.88) as

Γµ =(

h

Γµνλ − Γµν

λ)

1

4γλγν , (17.90)

where we have used the definition (17.62). Comparison with (17.63) shows that

Γµ = −Λ

Γµβα1

4γαγβ. (17.91)

SinceΛ

Γµαβ is antisymmetric in αβ, this is the same as [recall (1.222)]

Γµ = − i

2

Λ

ΓµαβΣαβ , (17.92)

in agreement with (17.15), if the Dirac indices are added.

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17.5 Invariant Action for Vector Fields 379

17.5 Invariant Action for Vector Fields

Any theory which is invariant under general coordinate transformations way can berecast in such a way that its derivatives refer to the nonholonomic coordinates dxα.Since the metric in these coordinates is gαβ, the action has the same form as thosein a flat spacetime, except that derivatives of vector and tensor fields are replacedby covariant ones, for example

∂αvβ → Dαvβ = ∂αvβ −Λ

Γαβγvγ. (17.93)

For example,

A =∫

d4xαDαvβ(x)Dαvβ(x) (17.94)

is the nonholonomic form of a generally covariant action. As we said in the begin-ning, the specification of spacetime points must be made with the xµ coordinates.For this reason the action is preferably written as

A =∫

d4xµ√−gDαvβ(x

µ)Dαvβ(xµ). (17.95)

Under a general coordinate transformation a la Einstein, dxµ → dx′µ′

= dxµαµµ′ ,

the indices α are inert. For instance, hαµ itself transforms as

hαµ(x)−→

Ehα

µ′(x′) = hαµ(x)αµ

µ′ . (17.96)

Vectors and tensors with indices α, β, . . ., experience only changes of their argumentsx→ x− ξ so that their infinitesimal substantial changes are

δEvα(x) = ξλ∂λvα(x) (17.97)

δEDαvβ(x) = ξλ∂λDαvβ(x). (17.98)

The freedom in choosing hαµ(x) up to a local Lorentz transformation, when taking

the “square root” of gµν(x) in (2.50), implies that the theory should be invariantunder

δLdxα = ωαβ(x)dx

β , (17.99)

δLhαµ(x) = ωα

β(x)hβµ(x). (17.100)

Here ωαα′

(x) are the local versions of the infinitesimal angles introduced in (11.59)and (11.60).

Indeed the action (17.95) is automatically invariant if every index α is trans-formed accordingly.

δLvα(x) = ωαα′

vα′(x), (17.101)

δLDαvβ(x) = ωαα′

(x)Dα′vβ(x) + ωββ′

(x)Dαvβ′(x). (17.102)

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380 17 Particles with Half-Integer Spin

The variables xµ are unchanged since the local Lorentz transformations (17.99) affectonly the intermediate local directions defined by the differentials dxα. They leavethe physical coordinate xµ unchanged.

It is useful to verify explicitly how the covariant derivatives guarantee localLorentz invariance. Consider

δLvα = ωαα′

(x)vα′(x), δLvα = ωαα′

(x)vα′ . (17.103)

Then the derivative ∂αvβ transforms as

δL∂αvβ = (δL∂α)vβ + ∂α(δLvβ)

= ωαα′

∂α′vβ + ∂α(ωββ′

vβ′)

= ωαα′

∂α′vβ + ωββ′

∂αvβ′ + (∂αωββ′

)vβ′ . (17.104)

The spin connection behaves as follows: Due to the factors hλγhα

µhβν in (17.63),

the first piece ofΛ

Γαβγ, call it

Λ

Γ′

αβγ , transforms like a local Lorentz tensor:

δLΛ

Γ′

αβγ = ωα

α′Λ

Γ′

α′β + ωββ′

Λ

Γ′

αβ′

γ + ωγγ′Λ

Γ′

αβγ′ . (17.105)

But from the second pieceh

Γµνλ there is a non-tensorial derivative contribution,

δLh

Γµνλ = (δhδ

λ)∂δν + hδλ∂µ(δh

δν)

= ωδδ′hδ′

λ∂µhδν + hδ

λ∂µ(ωδδ′h

δ′

ν)

= ωδδ′hδ′

λ∂µhδν + ωδδ′hδ

λ∂µλ∂µh

δ′

ν + ∂µωδδ′(hδ

λhδ′

ν)

= ∂µωδδ′h

δ′

ν = −∂µωδ′δhδλhδ′

ν (17.106)

the cancellation in the third line being due to the antisymmetry of ωδδ′ = −ωδ′δ.

Thus we arrive at

δLh

Γµνλ = ∂µω

δδ′hδ

λhδ′

ν′ δLΛ

Γαβγ = δL0

Λ

Γαβγ + ∂αωβ

γ , (17.107)

where δL0denotes the proper Lorentz tensor transformation law satisfied by

Λ

Γαβγ in

(17.105). The last term is precisely what is required to cancel the last non-tensorialpiece of (17.104), when transforming Dανβ , so that we indeed obtain the covarianttransformation law (17.102).

17.6 Verifying Local Lorentz Invariance

Let us study the invariance under local Lorentz transformations in more detail.These serve to go at an arbitrary point xµ from one freely falling elevator to another.A spinor field ψ(x) transforms under them as follows:

δLψ(x) = − i

2ωαβ(x)Σαβψ(x), (17.108)

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17.6 Verifying Local Lorentz Invariance 381

Here Σαβ are the spin representation matrices of the local Lorentz group. They areantisymmetric in α, β and satisfy the commutation relations

[

Σαβ ,Σαγ

]

= −igααΣβγ . (17.109)

Recall that for vectors, the representation matrices were given by (1.51):(

Lαβ)

α′β′

= i[

gαα′gββ′ − (α↔ β)]

,(

Lαβ)

α′

β′

= i[

gαα′δββ′− (α ↔ β)

]

,(17.110)

by which the analog of the transformation law (17.108) reduces to (17.101):

δLvα = − i

2ωγδi

(

gγαδδβ − gδαδγ

β)

vβ = ωαβvβ . (17.111)

For Dirac spinors, the Lorentz generators Lαβ are replaced by Σαβ of Eq. (1.221):

Σαβ =i

4

[

γα, γβ]

. (17.112)

The infinitesimal Lorentz transformation of the derivative of ψ is

δL∂αψ = ωαα′

∂α′ψ + ∂αδLψ

= ωαα′

∂α′ψ − i

2∂α(

ωβγΣβγ

)

ψ

= ωαα′

∂α′ψ − i

2ωβγΣβγ∂αψ − i

2

(

∂αωβγ)

Σβγψ. (17.113)

The first two terms describe the usual Lorentz behavior of ∂αψ. The last term is dueto the dependence of the angles ωβγ(x) on x. It can be removed with the help of the

spin connectionΛ

Γαβγ and the covariant derivative (17.15) with the spin connection

(17.13)

Dαψ(x) ≡ ∂αψ(x) +i

2

Λ

ΓαβγΣβ

γψ(x). (17.114)

Indeed, if we calculate the variation of the second term in Dαψ(x) ≡ ∂αψ(x):

δLi

2

Λ

ΓαβγΣβ

γψ(x), (17.115)

we obtain two terms. There is a term with the regular Lorentz transformationproperty

δL0

i

2

Λ

ΓαβγΣβ

γψ = − i

2ωστΣστ

(

i

2

Λ

ΓαβγΣβ

γψ

)

. (17.116)

This follows fromi

2δL

Λ

ΓαβγΣβ

γψ +i

2

Λ

ΓαβγΣβ

γδLψ, (17.117)

and an application of the commutation rule (17.109). A second term arises from∂αωβ

γ, which isi

2∂αωβ

γΣβγψ (17.118)

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382 17 Particles with Half-Integer Spin

and cancels against the last term in (17.113). Thus Dαψ behaves like

δLDαψ = ωαα′

(x)Dα′ψ − i

2ωβγ(x)ΣβγDαψ, (17.119)

and represents, therefore, a proper covariant derivative which generalizes the stan-dard Lorentz transformation behavior to the case of local transformations ωα

β(x).

17.6.1 No Need for Torsion

A important observation is the following. The covariant derivative (17.15) does notneed the contortion field Kµν

λ to be covariant under local Lorentz transformations.

For this, the termh

Kµνλ is completely sufficient. It supplies the compensating

nontensorial term to make the derivative in front of a Dirac field a vector. Thus aconsistent theory which is invariant under local Lorentz transformations exists in aRiemann spacetime. Torsion is a pure luxury of the theory.

17.7 Field Equations with Gravitational Spinning Matter

Consider the action of a spin−1/2 field interacting with a gravitational field:

A[h,K, ψ] = − 1

2K

d4x√−gR+

1

2

d4x√−gψγαDαψ(x) + h.c.

=f

A [h,K] +m

A [h,K, ψ]. (17.120)

It is a functional of the vierbein field hαµ, the contortion Kµν

λ and the Dirac field

ψ(x). Varying A with respect to ψ we obtain the equation of motion

δm

Aδψ

=√−g(γαDα −m)ψ(x) = 0 (17.121)

of a Dirac particle in a general affine spacetime.To obtain the gravitational field equations we again define the spin current den-

sity, just as we did in (15.17), by differentiating with respect to Kµνλ at fixed hα

µ,and find for the gravitational field

δf

AδKµν

λ = −1

2

√−gf

Σνλ, µ, (17.122)

as given in (15.64).From the matter action (17.15) we obtain

√−g m

Σνλ, µ ≡ 2

δm

AδKµν

λ =√−g

[

− i

2ψ(x)γµ(x)γµ(x)Σν

λ(x)ψ(x) + h.c.]

= hγλhα

µhβν√−g

[

− i

2ψ(x)γαΣβ

γψΣβγψ(x) + h.c.

]

= hγλhα

µhβνhβ

ν√−g m

Σβγ,α. (17.123)

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17.7 Field Equations with Gravitational Spinning Matter 383

The expressionm

Σ βγ, α is recognized as the canonical spin current of a Dirac particle

in Minkowski spacetime andm

Σ νλ, µ is its generally covariant analog. Thus, for the

spin-1/2 field, the definition (15.16) of the spin current density is consistent withthe canonical definition,

m

Σνλ, µ ≡ −i

i

πiµΣν

λϕi = −i∑

i

∂m

L∂Dµϕi

Σνλϕi. (17.124)

where the sum over i covers all independent matter fields of the system. This is alsotrue, in general, by the fact that the general Einstein invariant matter action hasthe functional form [compare (17.15)]

m

A =m

A [h,K, ϕi] =∫

d4x√−g m

L(

hαµ, ϕi, Dµϕi

)

(17.125)

so that indeed, for fixed hαµ,

2δm

AδKµν

λ

hαµ

= 2√−g

i

∂m

L∂Dµϕi

i

2Σν

λϕi

≡ i√−g

i

πiµΣν

λϕi = − m

Σνλ, µ. (17.126)

The field equations associated with δKµνλ are therefore

−κf

Σνλ, µ = κ

m

Σνλ, µ, (17.127)

thus extending the field equation Eq. (15.59) to systems with spinning matter. To-gether with Eq. (15.55), this determines the Palatini tensor (15.46):

Sµν,λ = −κ m

Σµν,λ, (17.128)

and thus, via Eq. (15.47), the torsion of spacetime by the field equation:

Sµνλ =κ

2

(

m

Σµν,λ +1

2gνλ

m

Σµκ,κ − 1

2gνλ

m

Σµκ,κ)

. (17.129)

Let us now turn to the field equations arising from extremization with respectto hα

µ. We define the total energy-momentum tensor as

√−gTµα(x) ≡δA

δhαµ(x)

Sµνλ

, (17.130)

with the derivative formed at fixed Sµνλ. Due to the relation (17.89), we may use

the chain rule of differentiation to write alternatively

√−gTαµ(x) = − δAδhαµ(x)

Sµνλ

. (17.131)

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384 17 Particles with Half-Integer Spin

For the pure gravitational action which depends only on gµν = hαµhαν and Kµν

λ,this definition leads trivially to the same symmetric energy-momentum tensor asthat introduced earlier in (15.15), except that one index has that α-form. Thisfollows from the chain rule of differentiation together with (15.15):

√−gf

T µα ≡ δ

f

Aδhα

µ =δ

f

Aδgλκ

∂gλκ

∂hαµ =

√−gf

T µκ hακ. (17.132)

There is, of course, a similar rule involving the derivative with respect to hαµ as in(17.131):

√−gf

Tαµ ≡ − δ

f

Aδhαµ

= − δf

Aδgλκ

∂gλκ∂hαµ

=√−g

f

Tκµhακ. (17.133)

For matter, the actual calculation of the symmetric energy-momentum tensor ismost conveniently performed in two steps. Take, for instance, the Dirac field. As afirst step we differentiate

√−g and γαhαµ∂µ with respect to hα

µ while keeping, forthe moment, Dµ = const. The result is the so-called canonical energy-momentumtensor: √−g m

Θµα ≡ √−g1

2

(

ψγαiDµψ − hαµm

L)

+ h.c. (17.134)

This is a general feature of the formalism: The derivative of (17.125) with respectto the hα

µ fields contained in the covariant derivative Dµϕi = hαµDαϕi gives

δm

Aδhα

µ −→ √−g∑

i

∂m

L∂Dνϕi

Dµϕi hαν . (17.135)

The derivative of (17.125) with respect to the hαµ contained in the

√−g term addsto this

δm

Aδhα

µ −→ −√−ggµνm

L hαν . (17.136)

The sum of the two contributions yields

m

Θµα =

(

i

∂L

∂DνϕiDµϕi − gµν

m

L)

hαν , (17.137)

which is indeed the canonical energy-momentum tensor for an arbitrary Lagrangiancontaining covariant derivatives.

Applying this formalism to a pure gravitational field we can compare the firststep of differentiation at fixed Dµ with the variation (15.28) and find the symmetricpart of the equation

f

Θµα = −1

κGµνh

αν . (17.138)

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17.7 Field Equations with Gravitational Spinning Matter 385

We will see below that this holds, in fact, without symmetrization. Thus the canon-ical energy-momentum tensor of the gravitational field is equal to minus 1/κ timesthe Einstein tensor.

We now turn to the second step, the calculation of the remaining functionalderivative with respect to hα

µ. This is somewhat tedious. Let us write the additional

contribution tom

Θκδ as

√−g δ mΘκδ =

d4xδm

AδKµβ

γ

δΛ

Γµβγ

δhδκ

Sµνλ= −1

2

d4x√−g m

Σβγ, µ δ

Λ

Γµβγ

δhδκ

Sµνλ,

(17.139)and use for the spin connection the explicit form

Λ

Γµβγ = hγλhβ

ν(Γµνλ− h

Γµνλ) = −hβν

Γ

Dµ hγν = hγν

Γ

Dµ hβν (17.140)

whereΓ

Dµ denotes the part of the covariant derivative containing only the ordinary

connection Γµβλ. If we vary δhµβ

γ and hold Γµνλ fixed we have

δΛ

Γµβγ∣

Γµνλ= δhγν

Γ

Dµ hβν + hγν

Γ

D δhβν . (17.141)

Since Dµhγν = 0 [recall (17.61)], we see that

Γ

Dµ hβν =

Λ

Γµβλhλ

ν and we may write

δΛ

Γµβγ∣

Γµνλ= hγνDµδhβ

ν (17.142)

Inserting this into (17.139), a partial integration gives the first contribution

∆1

m

Θκδ = −(1/2)Dµ

m

Σκδ,µ. (17.143)

We now include the contribution from δΓµνλ. Using the decomposition (15.51) with

δSµνλ = 0, i.e. δKµνλ = 0, we find

∆2

m

Θ κδ =

1

4

[

(

m

Σνσ,µ− m

Σσµ,ν+

m

Σµν,σ

)]

∂gνσ∂hδ

κ . (17.144)

With

∂gνσ∂hδ

κ = gνκhδσ + (ν ↔ σ), (17.145)

this gives, altogether,

∆m

Θκδ(x) = −1

2D∗µ

(

m

Σκδ, µ− m

Σδµ, κ+

m

Σµκ, δ

)

. (17.146)

This is precisely the same type of correction ∆Θκδ = ∆Θκ

νhδν that had been addedto the canonical energy-momentum tensor Θκδ of the gravitational field in (15.58), in

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386 17 Particles with Half-Integer Spin

order to produce the symmetric one Tκδ. Here, it is obtained for arbitrary spinningmatter fields:

m

T κν =m

Θκν +∆m

Θκν +∆m

Θκν = −1

2D∗µ

(

m

Σκν, µ− m

Σνµ, κ+

m

Σµκ, ν

)

. (17.147)

For spin 12, this is the expression (3.229) found by Belinfante in 1939. We have

lowered the index ν on both sides which is permissible due to the covariant form ofthe equation.

In terms ofm

T µν , the field equations which follows from variations of the actionwith respect to δhα

µ have once more the simple form (15.62):

Gµν = κm

Tµν , (17.148)

with the energy-momentum tensors (17.147) of spinning matter.

[1] This Chapter follows largely the textbookH. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects,World Scientific, Singapore, 1989 (kl/b2), where kl is short for the wwwaddress http://www.physik.fu-berlin.de/~kleinert. See in particular pp.1338–1377 (kl/b1/gifs/v1-1338s.html),

[2] H. Weyl, Z. Phys. 56, 330 (1929);

[3] V. Fock, Z. Phys. 57, 261 (1929);

[4] V. Fock and D. Iwanenko, Phys. Z. 30, 648 (1929);

H. Kleinert, MULTIVALUED FIELDS

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Nothing endures but change

Heraclitus (540 BC - 480 BC)

18Covariant Conservation Law

According to Noether’s theorem derived in Chapter 3, the invariance of the actionand general coordinate transformations and local Lorentz transformations must beassociated with certain conservation laws. For the following considerations, we shallconsider hα

µ(x) and Γµβα(x) as independent variables. Then, from the derivation of

the canonical energy-momentum tensor in (17.132) it follows that varying the actionin hα

µ(x) at fixed Aµβγ(x) gives the canonical energy-momentum tensor

δA[hαµ, Aµβ

γ ]

δhαµ =

√−g Θµα . (18.1)

A functional derivative with respect to Aµβγ = Γµβ

γ at fixed hαµ, on the other

hand, is equivalent to a derivative with respect toKµβγ [recall (17.66)] and produces,

according to Eq. (17.122), the spin current density1

δA[hαµ, Aµβ

γ ]

δAµβγ = −1

2

√−g Σβγ,α hα

µ ≡ −1

2

√−g Σβγ,µ. (18.2)

These quantities will now be shown to satisfy covariant conservation laws.

18.1 Spin Density

Consider first local Lorentz transformations. Under these the vierbein fieldshα

α(x) (µ = 0, . . . , 3) behave like vectors in the index α,

δLhαα(x) = ωα

α′

(x)hα′

µ(x). (18.3)

Similarly, the field Aµβγ (µ = 0, . . . , 3) behave like tensors in β, γ, and receive, in

addition, a typical derivative term of gauge fields [see (17.111)]

δLAµβγ = ωβ

β′

(x)Aµβ′

γ + ωγγ′(x)Aµβγ′ + ∂µωβ

γ(x). (18.4)

1Recall that the field Aµβγ has the pure contortion form, Aµβ

γ = hγλhβ

ν(Kµνλ−

h

Kµνλ) and

thus is antisymmetric in β, γ, as is the case with Γαβγ .

387

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388 18 Covariant Conservation Law

The change of the action has to vanish. This gives

δLA =∫

d4x

δAδhα

µ(x)ωα

α′

(x)hα′

µ(x) (18.5)

+δA

δAµβγ(x)

(ωββ′

(x)Aµβ′

γ(x) + ωγγ′(x)Aµβγ′(x) + ∂µωβ

γ(x))

=∫

d4x√−g

Θµαωα

α′

hα′

µ − 1

2Σβ

γ,µ(ωβ

β′

Aµβ′

γ + ωγγ′Aµβγ′ + ∂µωβ

γ)

.

Partially integrating the last term gives∫

d4x√−g Θµ

αωαα′

hα′

µ +1

2∂µ(√−gΣβ

γ,µ)

ωβγ

−1

2

√−g Σβγ,µ(

ωββ′

Aµβ′

γ + ωγγ′Aµβγ′)

. (18.6)

Since ωβγ(x′) is an arbitrary antisymmetric function of x′ it can be chosen to be

zero everywhere except at some place x and we find

1

2

√−g(

Θµβhγ

µ − Θµγhβµ)

+1

2∂µ√−gΣβ

γ,µ

−1

2

√−g(

Σβδ′µAµγ

δ + Σδβ′

µAµδβ)

. (18.7)

Defining

Θγβ ≡ Θµ

βhγµ (18.8)

and raising the index γ with the Minkowski metric ηγγ′

, this reads

1

2

[

Θγβ − Θβγ]

+1

2Γµσ

σΣβγ,µ +1

2DLµΣβγµ = 0 (18.9)

whereL

Dµ is the covariant derivative for the local Lorentz index γ, i.e., for a vector

L

Dµ vα = ∂µva −Aµαβvβ = hβµDβvα, (18.10)

L

Dµ vα = ∂µv

α − Aµαβv

β = ∂µvα + Aµβ

αvβ = hβµDβvα. (18.11)

The derivativeL

Dµ σβγ,ν can be made completely covariant also in the Einstein index

µ, by going to

L

Dµ Σβγ,ν ≡ L

Dµ Σβγ,ν − ΓµλνΣβγ,λ. (18.12)

Now, the last term cancels part of to the middle on ein (18.9) and we have [recall(15.40)]

1

2D∗µΣ

βγ,µ =1

2

[

Θβγ − Θγβ]

. (18.13)

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18.2 Energy-Momentum Density 389

Being a covariant relation, this can be multiplied by hβλhγ

κ and the vierbeins canbe moved under the derivative, yielding

1

2hβ

λhγκDµΣ

βγ,µ − Θ[λ,κ] =1

2D∗µΣ

λκ,µ − Θ[λ,κ] = 0. (18.14)

For a vector this type of operation is demonstrated as follows:

hανDµv

α = hαν(

∂µvα + Aµβ

αvβ)

= ∂µ (hανvα) + hα

νAµβαvβ −

(

∂µhαν)

= ∂µvν + hα

νhαλhβν(

Γµνλ− h

Γµνλ)

vβ+h

Γµλνhα

λvα

= ∂µvν + Γµλ

νvλ ≡ Dµvν , (18.15)

and the extension to tensors is obvious.

18.2 Energy-Momentum Density

Let us now deduce the consequence of local Einstein invariance. In this case thespacetime coordinates must be transformed as well and the action is invariant in thefollowing sense:

A =∫

d4x√

−g(x)L(h(x), A(x)) =∫

d4x′√

−g′(x′)L(

h′(x′), A′(x′))

. (18.16)

If we change the variables x′ to x in the second integral we see that the difference

d4x

−g′(x)L(

h′(x), A′(x))

−√

−g(x)L (h(x), A(x))

(18.17)

must be concentrated in the neighborhood of the surface of the integration volume.This is because the original integrations

d4x′,∫

d4x covered the same volume sothat, after the change of variables x′ → x, the first integral runs through a slightlydifferent region. Infinitesimally this amounts to the statement that

δEA =∫

d4x δE

[

−g(x)L (h(x), A(x))]

(18.18)

is a pure surface term. The symbol δE denotes the substantial change under Einsteintransformations at fixed argument x [see (3.132),(11.80)], i.e.,

δEgµν(x) = Dµξν(x) + Dνξµ(x). (18.19)

Under Einstein transformations, the metric transforms as

δE√−g = −1

2

√−ggµνδEgµν =1

2

√−ggµνδEgµν , (18.20)

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390 18 Covariant Conservation Law

which, upon inserting (18.19), yields

1

2

√−ggµν[

ξλ∂λgµν +(

∂µξλ)

gλν +(

∂νξλ)

gµλ]

. (18.21)

Therefore

δE√−g = ξλ∂λ

√−g +√−g + ∂λξ

λ = ∂λ(

ξλ√−g

)

(18.22)

and

δE

d4x√−g =

d4x√gDλξ

λ =∫

d4x∂λ(

ξλ√−g

)

. (18.23)

This shows that the trivial action∫

d4x√−g indeed changes by a pure surface term.

There is complete invariance if we require ξλ(x) to vanish at the surface.The same result holds for a general action if L is a scalar Lagrangian density

satisfying

L′(x′) = L(x) (18.24)

and therefore

δEL(x) ≡ L′(x) − L(x) = L′(x′) −L(x′) = L(x) − L(x′) = ξλ∂λL(x). (18.25)

The variation of A is

δEA = δE

d4x(√−gL(x)

)

=∫

d4x[

δE√−g

]

L(x) +√−gδEL(x)

=∫

d4x

∂λ[

ξλ√−g

]

L(x) +√−gξλ∂λL(x)

=∫

d4x ∂λ(

ξλ√−gL(x)

)

. (18.26)

We can now derive the covariant conservation law associated with Einstein invarianceby using the substantial variations δEhα

µ and δEAµβγ and calculating δEA once more

as follows:

δEA =∫

d4x

(

δAdhα

µ δEhαµ +

dAδAµβ

γ δEAµβγ

)

=∫

d4x(√−gΘµ

αδEhαµ − 1

2

√−gΣβγ,µδEAµβ

γ)

. (18.27)

The substantial variations of the vierbein fields hαµ and Aµβ

γ are those of a vectorwith a super- or subscripts µ [recall (11.75), (11.76)]:

δEhαµ = ξλ∂λhα

µ − ∂κξµhα

κ, δEAµβγ = ξλ∂λAµβ

γ + ∂µξλAλβ

γ . (18.28)

Inserting these into (18.27), we find

δEA=∫

d4x√−gΘµ

α(

ξλ∂λhαµ− ∂κξ

µhακ)

−1

2

√−gΣβγ,µ(

ξλ∂λAµβγ+ ∂µξ

λAλβγ)

.

(18.29)

H. Kleinert, MULTIVALUED FIELDS

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18.2 Energy-Momentum Density 391

After partial integrations and letting ξλ be zero everywhere, except for a δ-functionsingularity at some place x, gives

∂κ(√−gΘλ

αhακ)

+√−gΘµ

α∂λhαµ

+1

2∂µ(√−gΣβ

γ,µAλβ

γ)

− 1

2

√−gΣβγ,µ∂λAµβ

γ = 0. (18.30)

The second line can be rewritten as

1

2∂µ(

−√−gΣβγ,µ)

Aλβγ +

1

2

√−gΣβγ,µ(

∂µAλβγ − ∂λAµβ

γ)

. (18.31)

If we introduce the covariant curl of the A field,

Fµλβγ ≡ ∂µAλβ

γ − ∂λAµβγ −

[

AµβγAλδ

γ − (µ↔ λ)]

, (18.32)

then (18.31) becomes

1

2∂µ(√−gΣβ

γ,µ)

Aλβγ +

1

2

√−gΣβγ,µ[

AµβδAµβ

γ − (µ ↔ λ)]

+1

2

√−gΣβγ,µFµλβ

γ.

(18.33)

The first three terms in (18.30) can now be collected into a covariant derivative D∗µdefined in (15.40), i.e.,

1

2

√−gD∗µ Σ βγ,µAλβ

γ, (18.34)

so that the second line in (18.30) becomes, after using the conservation law (18.13),

−√−gΘγβAλβ

γ +1

2

√−gΣβγ,µFµλβ

γ. (18.35)

In the first line of (18.32) we write

Θµα∂λhα

µ = ΘµαDL

λhαµ + Θµ

αAλαβhβ

µ (18.36)

and (18.30) takes the form

∂κ(√−gΘλ

κ)

+√−gΘµ

αDLλhα

µ − 1

2

√−gΣβγ,µFλµβ

γ = 0. (18.37)

This equation is covariant under local Lorentz transformations but not yet manifestlyso under Einstein transformations. In order to verify the latter we observe that thederivative DL of h can be rewritten as

DLλhα

µ = ∂λhαµ −Aλα

βhβµ

= − h

Γλκµhα

κ −(

Γλσµ− h

Γµ

λσ

)

hασ = −Γλσ

µhασ, (18.38)

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392 18 Covariant Conservation Law

in accordance with the identity Dλhαµ = 0. Then the second term is

−√−gΓλσµΘµσ, (18.39)

We now rewrite the first term as

√−g (D∗κΘλκ + Γκλ

τΘτκ) (18.40)

Then the completely covariant conservation law for the energy momentum tensor is[1, 2, 3, 4, 5].

D∗κΘλκ + 2Sκλ

τΘτκ − 1

2Σβ

γ,µFλµβ

γ = 0. (18.41)

18.3 Covariant Derivation of Conservation Laws

It should be noted that the conservation laws of energy, momentum and angularmomentum can be derived somewhat more efficiently, if some initial effort is spentin preparing the Einstein and local Lorentz transformations (18.28), (18.3), (18.4)of hα

µ and Aµαβ in covariant form. Take δEhα

µ. It can be rewritten as

δEhαµ = ξλ∂λhα

µ + Γλκµhα

λξκ −Dλξµhα

λ. (18.42)

Using the identity

∂λhαµ = − h

Γλνµhα

ν = Aλαβhβ

µ − Γλνµhα

ν , (18.43)

we can rewrite (18.42) in the covariant form

δEhαµ = −Dαξ

µ + (Aλαµ + 2Sλα

µ) ξβ. (18.44)

The reciprocal vierbein field hαµ transforms as

δEhαµ = Dµξ

α −(

Aβµα − 2Sβµ

α)

ξβ. (18.45)

Similarly, we find

δEAµαβ = ξλ∂λAµα

β +DµξλAλα

β − ΓµκλAλα

βξκ

= Dµ

(

ξλAλαβ)

− ξλ(

DµAλαβ − ∂λAµα

β)

− ΓµκλAλα

βξκ

= Dµ

(

ξλAλαβ)

− ξλFµλαβ . (18.46)

Under local Lorentz transformations, the vierbein field has already its simplest pos-sible form,

δLhαµ = ωα

βhαµ, (18.47)

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18.4 Matter with Integer Spin 393

while Aµαβ acquires the typical additive term of a gauge field

δLAµαβ = Dµωα

β. (18.48)

Using these covariant transformation rules, the variations of the action (18.6),(18.29) become

δLA =∫

d4x√−g

Θβαωα

βhβµ − 1

2Σα

β,µDµωα

β

, (18.49)

δEA =∫

d4x√−g

Θµα(

−Dλξµhα

λ + (Aλαµ − 2Sλα

µ) ξλ)

− 1

2Σα

β,µ[

(

ξλAλαβ)

− ξλFµλαβ]

. (18.50)

A partial integration of (18.49) [using (15.35), (15.39)] then gives directly the diver-gence of the spin current (18.13). A partial integration of (18.50) leads to

D∗λΘµλ +

(

Aµαβ − 2Sµα

β)

Θβα +

1

2D∗νΣ

αβ +

1

2Σα

β,νFνµα

β = 0 (18.51)

which, after inserting (18.13), reduces correctly to the covariant, conservation lawfor the canonical energy-momentum tensor (18.41).

18.4 Matter with Integer Spin

If matter fields only carried integer spin it would ne be necessary to introduce thehαµ, Aµα

β fields. Then there would only be invariance under Einstein transforma-tions from symmetry considerations. The law of angular momentum conservationrequires the use of equation of motion. The action may be written in terms of gµνand Kµν

λ with the aid of Γµνλ = Γµν

λ +Kµνλ and Einstein invariance amounts to

δEA =∫

d4x

δAδgµν

SµνλδEgµν +

δAδKµν

λ

gµν

δEKµνλ

= −1

2

d4x√−g

T µν(

ξλ∂λgµν + ∂µξλgλν + ∂νξ

λgµλ)

+Σνκ,µ(

ξλ∂λKµνκ + ∂µξ

λKλνκ + ∂νξ

λKµλκ − ∂λξ

κKµνλ)

, (18.52)

where we have used the definitions (15.14) and (15.16) and inserted the transfor-mation laws under general coordinate transformations. We have omitted the super-scripts m since the equations in this section apply just as well to the gravitational

actionf

A, if we use the definitions (15.15) and (15.17). The further calculations aresimplified by defining the symmetrized canonical energy-momentum tensor by

δAδgµν

Γµνλ=const.≡ −1

2

√−g(

Θµν + Θνµ

)

. (18.53)

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394 18 Covariant Conservation Law

It is easy to see that this definition agrees with (17.134) (by differentiating withrespect to hα

µ at fixed Aµαβ and changing the index α to v) if there are no spin−1/2

fields. It may also be verified by forming

δAδgµν(x)

Sµνλ=const.=

δAδgµν(x)

Γµνλ=const.

+∫

dyδA

δΓστλ(x)

gµν=const.

δΓστλ(y)

δgµν(x)

Sµνλ=const., (18.54)

so that one obtains the standard Belinfante relation (17.147) between Tµν and Θµν ,now derived from geometric arguments for spaces with curvature and torsion:

T µν = Θµν − 1

2∂λ(Σ

µν,λ − Σνλ,µ + Σλµ,ν) (18.55)

For pure gravity, (18.53) is in accord with (17.138) which states that Θµν is theEinstein tensor [recall (17.138)] up to a factor −κ

−κΘµν = Gµν = Rµν −1

2gµνR,

as can be seen from (15.28) and the Belinfante relation (17.147) again coincides with(15.58).

Thus we can evaluate the consequences of Einstein invariance by using Θ and Σand considering, instead of (18.52), the variation

0 = δEA =∫

d4x

δAδgµν

ΓµνλδEgµν +

δAδΓµν

λ

gµν

δEΓµνλ

= −1

2

d4x√−g

Θµν(

ξλ∂λgµν + ∂µξλgλν + ∂νξ

λgµλ)

−Σνκ,µ(

ξλ∂λΓµνκ + ∂µξ

λΓλνκ + ∂νξ

λΓµλκ − ∂λξ

κΓµνλ + ∂µ∂νξ

κ)

. (18.56)

It is again useful to bring the variations δEgµν , δEΓµνλ into covariant form. We

rewrite the Einstein variation of the metric as

δEgµν = Dµξν + Dνξµ = Dµξν +Dνξµ +[

Kµνλ + (µ↔ ν)

]

ξλ

= Dµξν +Dνξµ + 2[

Sλµν + (µ ↔ ν)]

ξλ, (18.57)

and the variation of the connection

δEΓµνκ = DµDνξ

κ − 2Dµ

(

Sνλκξλ

)

+Rλµνκξλ. (18.58)

Inserting this into (18.54) gives

δEA =∫

d4x√−g

(Θνµ + Θµν)(

Dνξµ + 2Sλµν)

ξλ

+Σνκ,µ[

DµDνξκ − 2Dµ

(

Sνλκξλ

)

+Rλµνκξλ

]

.

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18.5 Relations between Conservation Laws and Bianchi Identities 395

By partially integrating the Σ term and using the spin divergence law (18.13), weobtain immediately

δEA = 2∫

d4x√−g

−DµΘλµ − 2Sµλ

νΘνµ +

1

2Σν

κ,µRλµν

κ

ξλ, (18.59)

leading directly to the covariant conservation law

D∗µΘλµ + 2Sµλ

νΘνµ − 1

2Σν

κ,µRλµν

κ = 0. (18.60)

This is not in manifest agreement with (18.41) since the last term is Σνκ,µRλµν

κ,

while we had Σβγ,µFλµβ

γ in (18.41), which is the same due to Eq. (17.74).When expressing the energy-momentum tensor and the spin-current density in

terms of the Einstein and Palatini tensors Gµν = Rµν − gµνR and (1/2)Sνκ,µ =

Sνκµ + gνµSκ− δκ

µSν , the two covariant conservations laws (18.13) and (18.60) of apure gravitational field take the form [recall (15.55), (17.138)]

1

2D∗µS

λκ,µ = G[λ,κ], (18.61)

D∗µGλµ + 2Sνλ

κGκν − 1

2Sνκ

,µRλµνκ = 0, (18.62)

18.5 Relations between Conservation Laws and BianchiIdentities

It is a fact that for the gravitational field, by itself, both covariant laws are au-tomatically satisfied irrespective of the presence of matter due to the fundamentalidentity (12.108) and the Bianchi identity (12.120). To see this we apply (15.40) to(15.46) and obtain

1

2D∗λSνµ

,λ = D∗λ(Sνµλ + Sν

λSµ − SµλSν)

= DλSνµλ +DνSµ −DµSν

= DλSνµλ +DνSµ −DµSν + 2SλSνµ

λ. (18.63)

Now we take (12.108) and contract the subscript ν with the superscript κ to obtain

Rµλ − Rλµ =2(

DκSµλκ +DµSλκ

κ +DλSκµκ)

− 4(

SκµρSλρ

κ + SµλρSκρ

κ + SλκρSκµρ

)

= 2(

DκSµλκ +DµSλ −DλSµ

)

+ 4SρSµλρ = D∗λSνµ

,λ. (18.64)

Since Rµλ differs from Gµλ only by the symmetric tensor gµλ/2, the same equationholds for Gµλ −Gλµ, so that we find

D∗λSνµ,λ = Gνµ −Gµν (18.65)

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396 18 Covariant Conservation Law

in agreement with (18.61).Similarly, using (12.120) and permuting the indices we have

DτRσνµτ +DσRντµ

τ +DνRτσµτ = 2Sτσ

λRνλµτ + 2Sσν

λRτλµτ + 2Sντ

λRσλµτ . (18.66)

Contracting ν and µ, this becomes

2DτRστ −DσR = 2Dτ Gσ

τ = −2Sτσλ + 2Sσ

µλRλµ + 2SµτλRσλµ

τ

= −4SτσλRλ

τ + 2SµτλRσλµ

τ (18.67)

or

D∗µGσµ − 2Sµ

(

Rλµ − 1

2δλµR)

+ Sτσλ(

Gλτ +

1

2δλτR)

− SµτλRσλµ

τ = 0, (18.68)

in agreement with (18.62).Within the defect interpretation of curvature and torsion, we observed before

that the fundamental identities are nonlinear generalizations of the conservationlaws of defect densities. From what we have just learned, the same equation can beobtained as conservation laws of energy-momentum and angular momentum froman Einstein action.

The two laws follow from the invariance of the Einstein action under generalcoordinate transformations, which are local translations, and under local Lorentztransformations, respectively.

These transformations correspond to elastic deformations (translational and ro-tational) of the world crystal and the invariance of the action expresses the fact thatelastic deformations do not change the defect structure.

It is important to realize that due to the intimate relationship between theconservation laws and the fundamental identities for the gravitational fields, theyremain valid in the presence of any matter distribution. Then, by the field equations(17.127), (17.148), the spin density and energy-momentum tensor of the matter fieldshave to satisfy the same divergence laws by themselves. Indeed, it can easily be seenthat this is a direct consequence of the Einstein invariance of the matter action inan arbitrary but fixed affine space, i.e., in a space whose geometry is specified fromthe outset rather than being determined by the matter fields via the field equations.

18.6 Particle Trajectories from Energy-MomentumConservation

The classical equations of motion for a point particle has the consequence that its

energy-momentum tensorm

T λµ = 0 in Eq. (15.21) satisfies the covariant conservation

law (18.41) all by itself. Otherwise the Einstein equations (18.53) would not be

satisfied. Since the symmetric energy-momentum tensorsf

T λµ = 0 and

m

T λµ = 0 of

gravitational field and matter are proportional to each other, they must separatelysatisfy the covariant conservation law.

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18.6 Particle Trajectories from Energy-Momentum Conservation 397

Consider first a particle without spin in a space without torsion. Starting pointis the covariant conservation law (18.41) which reads now

DκTλκ(x) = 0. (18.69)

Expressing the covariant derivative in terms of the Riemann connection, and thisfurther in terms of derivatives of the metric using the identity

1√−g∂ν√−g =

1

2gλκ∂νgλκ = Γνλ

λ, (18.70)

Eq. (18.69) becomes

∂ν [√−gT µν(x)] +

√−g Γνλµ(x)T λν(x) = 0. (18.71)

This must hold for the energy-momentum tensor of the particle trajectories (15.21).Inserting this gives

m∫

dτ [qµ(τ)qν(τ)∂νδ(4)(x− q(τ)) + Γνλ

µ(q)qν(τ)qλ(τ) δ(4)(x− q(τ))] = 0. (18.72)

The first term in the integrand can also be written as −qµ(τ)∂τδ(4)(x − q(τ)), sothat a partial integration leads to

m∫

dτ [qµ(τ) + Γνλµ(q)qν(τ)qλ(τ)] δ(4)(x− q(τ)) = 0. (18.73)

Integrating this over a this tube around the trajectory qµ(τ), we obtain the equation(11.25) for the geodesic trajectory [6].

The same result may be derived from the following consideration. Accordingto Eq. (18.59), the variation of the action under Einstein transformations of thecoordinates δEx

µ = −ξµ is in Riemannian space

δEA = −2∫

d4x√−gDµTλ

µξλ. (18.74)

Due to Einstein’s equation (15.62) this holds separately for field and matter parts.If matter consists of point particles only, we obtain:

δEA = −∫

dτδm

Aδqµ(τ)

ξµ(q(τ)). (18.75)

This vanishes along the geodesic trajectory implying that the energy-momentumtensor is covariantly conserved.

Let us now allow for torsion in the curved spacetime, where the covariant con-servation law (18.69) becomes (18.60):

D∗µΘλµ + 2Sµλ

νΘνµ − 1

2Σν

κ,µRλµν

κ = 0. (18.76)

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398 18 Covariant Conservation Law

Recalling the Belinfante relation (18.55) and the definition of D∗µ in Eq. (15.40), weobtain for scalar particles with Σν

κ,µ = 0:

DµTλµ = 0, (18.77)

This coincides with the conservation law (18.69) in Riemannian space and leads oncemore the geodesic trajectories (11.25).

How can we remove the discrepancy with respect to the autoparallel trajec-tory found by the multivalued mapping procedure in Eq. (14.7)? Apparently, thevariational procedure of the metric which has led to the Einstein equation (15.62)must be modified to account for the torsion. This question is further discussed inSubsection 20.2.1.

Notes and References

[1] R. Utiyama, Phys. Rev. 101, 1597 (1956).

[2] T.W.B. Kibble, J. Math. Phys. 2, 212 (1961).

[3] F.W. Hehl, P. von der Heyde, , G.D. Kerlick and J.M. Nester, Rev. Mod. Phys.48, 393 (1976).

[4] F.W. Hehl, J.D. McCrea, E.W. Mielke, and Y. Ne’eman, Phys. Rep. 258, 1(1995).

[5] H. Kleinert, Gauge Fields in Condensed Matter , Vol. II Stresses and De-fects , World Scientific, Singapore, 1989, pp. 744-1443 (http://www.phy-sik.fu-berlin.de/~kleinert/b2).

[6] F.W. Hehl, Phys. Lett. A 36, 225 (1971).

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Gravitation cannot be held responsible for people falling in love

Albert Einstein (1879 - 1955)

19Gravitation of Spinning Matter as a Gauge Theory

The alert reader will have noticed by now that the theory of gravity of spinningmatter, when formulated in terms of fields hα

µ, Aµαµ introduced in Eqs. (17.17)

and (18.32), is really a gauge theory of local Lorentz transformations. Gauge prop-erties have become apparent before in when we observed in Eq. (11.106) that theconnection Γµν

λ transforms like a nonabelian gauge field under general coordinatetransformations. But at that early stage, we could not have really spoken about agauge theory since the connection Γµν

λ was not an independent field of the system.When introducing spinning particles, the metric gµν(x) as a fundamental field wasreplaced by the vierbein field hα

µ(x) which transforms like a gauge field under trans-lations. The Dirac theory in curved space, in which the covariant derivative containsthe spin connection Aαβ

γ of Eq. (17.66) with only the objects of anholonomity and

no torsion tensor Kµνλ0, is a bona fide gauge theory of local Einstein and Lorentz

transformations. If the space has also torsion, the spin connection Aαβγ becomes a

gauge field which is completely independent of the gauge field hαµ. Let us study the

properties of such a theory in more detail.

19.1 Local Lorentz Transformations

Recall that under infinitesimal Lorentz transformations a vector field behaves like[see Eq. (17.101)]

δLvα(x) = ωαβvβ(x), δLv

α(x) = ωαβvβ(x), (19.1)

where the physical coordinates xµ remain unchanged since only the local directionsare transformed. Due to the antisymmetry of the matrix ω this can also be writtenas

δLvα(x) = −vβ(x)ωβα. (19.2)

This shows that the transformation law (I.3.108b) coincides precisely with (18.4) forthe special case of the local Lorentz group.

δLAµβγ = ωβ

β′

Aµβ′

γ + ωγγ′Aµβγ′ + ∂µωβ

γ. (19.3)

399

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400 19 Gravitation of Spinning Matter as a Gauge Theory

Observe that the spacetime variables xµ are not transformed so that the Lorentzgroup plays the same role as an internal symmetry group. there is, however, a certainsimilarity with external gauge symmetries discussed in Section 3.5, Part I. This isbecause hα

µ can couple Lorentz and Einstein indices, just as in (I.3.135), thus givingrise to more invariants. For instance, there is no need of forming (Fµνα

β)2 in orderto get an invariant action. There also exists an invariant expression linear in thefield strength,

Af = − 1

d4x√−g hαµhβνFνµαβ. (19.4)

In fact, from (17.74) this is just the Einstein-Cartan action (15.8).

For completeness, let us see once more how the spin current and energy-momentum tensor follow from this action with independent fields hα

µ, Aµαβ. First

we calculate the spin current of the field. By definition,

1

2

√−gf

Σαβ,µ(x) = − δ

f

AδAµ

αβ(x)(19.5)

=1

δ

δAµαβ(x)

d4x√−g hα′µ′hβ′

ν′(

∂ν′Aµ′α′

β′−∂µ′Aν′α′

β′−Aν′α′

γAµ′γβ′

+Aµ′α′

γAν′γβ′)

=− 1

∂ν√−g

[

hαµhβ

ν−(α↔β)]

+√−g

[

(Aνα′αhα′µhβ

ν+Aνβ′βhαµhβ

′ν)−(α↔β)]

.

We may write this in terms of the partially covariant derivatives (18.10), (18.11) as

−κf

Σαβ,µ =

L

[

hαµhβ

ν − (α ↔ β)]

+ Γνσσ[

hαµhβ

ν − (α↔ β)]

.

Applying the chain rule of differentiation this becomes

−κf

Σ,µ

αβ=(

L

Dβ hαµ − hα

µ L

Dν hβν + hα

µΓβσσ)

− (α↔ β). (19.6)

We now observe that, due to the identity Dµhαν ≡ 0, the connection can be rewritten

as

Γµνλ = hαλ

L

Dµ hαν = −hανL

Dµ hαλ. (19.7)

This relation is complementary to the relation Γµβγ = hγν

Γ

Dµ hβν of Eq. (17.140).

Using (19.7), the spin current of the field becomes

−κf

Σαβ,µ = 2

(

Sαβµ + hα

µSβ − hβµSα

)

= Sαβµ, (19.8)

in agreement with (15.46), (15.55).

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19.2 Local Translations 401

We now calculate the functional derivative of the action with respect to hαµ. It

shows directly that the canonical energy-momentum tensor of the gravitational fieldcoincides with the Einstein tensor,

√−gΘµα =

(

δAf/δhαµ)

=√−g

(

hδνFµνδα − hβµFβδ

δα)

hαν√−g

(

Rµν − gµνR)

=√−g Gµ

α. (19.9)

The use of the field hαµ has made it possible to retrieve the Einstein tensor without

projecting out the symmetric part ot it, as in the previous formulas, (15.28) and(18.53).

19.2 Local Translations

In the literature one often finds the statement that the vierbein field may be con-sidered as a gauge field of local translations. In fact, Einstein’s transformations

x′ = x− ξ(x) (19.10)

can be considered as local translations and the vierbein field does ensure that thetheory is invariant under these, just as any bona fide gauge field is supposed to. Thecovariant derivative

Dα ≡ hαµ∂µ +

i

2Aαβ

γΣβγ (19.11)

may be viewed as a combination of hαµ times the translational “functional matrix”

∂µ and (i/2)Aαβγ times the Lorentz matrix Σβ

γ. This viewpoint becomes most trans-parent by considering the expression in (17.82), the commutator of two covariantderivatives with respect to the dislocation coordinates,

[

Dα, Dβ

]

ψ =i

2Fαβγ

δΣγδψ + i2Sαβ

γiDγψ. (19.12)

Since the factor of Fαβγδ is the curl of the gauge field of Lorentz transformations, the

factor 2Sαβγ of Dγψ may be considered as the curl of the gauge field of translations.

Indeed, if we write 2Sαβγ in the form

2Sαβγ = −hγν

[

hαµ L

Dµ hβν − (α↔ β)

]

= hαµhβ

ν[

L

Dµ hγν − (µ↔ ν)

]

, (19.13)

we arrive at the standard form of a curl, and the present formulation of gravity ofspinning matter can be considered as a gauge theory of both local Lorentz transfor-mations and local translations.

If the space has no torsion, then Aµαβ is completely composed of derivatives of

vierbein fields [recall Eq. (17.63)]

Aαβγ = −hγλhαµhβν

h

Γµνλ. (19.14)

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402 19 Gravitation of Spinning Matter as a Gauge Theory

Inserting this into (19.13) we verify that this is equivalent to a vanishing torsion.In recent years, this aspect of gravitational theory has received increasing atten-

tion, due to the shift in emphasis from geometric principles to gauge principles.

Notes and References

For more details on general relativity, the reader may consultL.D. Landau and E.M. Lifshitz, Classical field Theory Addison-Wesley Reading,Mass., 1958),C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation , Freeman and Co., NewYork, 1973,E. Schmutzer, Relativistische Physik , Akad. Verlagsgesellschaft, Geest und Portig,Leipzig, 1968,S. Weinberg, Gravitation and Cosmology , J. Wiley and Sons, New York, 1972.

The mathematics of metric-affine spaces is discussed inJ.A. Schouten, Ricci Calculus , Springer, Berlin, 1954,whose notation we use.F.W. Hehl, Abh. Braunschweig. Wiss. Ges. 18 (1966) 98.

The gauge aspects of gravity are discussed inD.W. Sciama, in Recent Developments in General Relativity , Pergamon Press, Ox-ford, 1962, p. 4,5,T.W.B. Kibble, J. Math. Phys. 2 (1961) 212,R. Utiyama, Phys. Rev. 101 (1956) 1597,F.W. Hehl and B.K. Datta, J. Math. Phys., 12 (1971) 1334,F.W. Hehl, P.v.d. Heyde, G.D. Kerlick, and J.M. Nester, Rev. Mod. Phys. 48

(1976) 393.F.W. Hehl, in Spin, Torsion, Rotation, and Supergravity , ed. by P.G. Bergmannand V. De Sabbata, Plenum Press, New York, 1980, p. 5;H. Kleinert, Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects, WorldScientific, Singapore, 1989,

(http://www.physik.fu-berlin.de/~kleinert/b1/contents2.html)

The symmetric energy-momentum tensor was first constructed byF.J. Belinfante, Physica 6 (1939) 887,L. Rosenfeld, Mem. Acad. Roy. Belg. U. Sci. 18 (1938) 2.

H. Kleinert, MULTIVALUED FIELDS

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To be, or not to be: that is the question

William Shakespeare (1564 - 1616)

20Evanescent Properties of Torsion in Gravity

What additional physics is brought about by torsion? If the field action is of theEinstein-Cartan type (19.4), the consequences turn out to be practically unobserv-able. This remains true if higher powers of the curvature tensor are added. Thefield equation (17.129) determines the torsion by the field equation:

Sµνλ =κ

2

(

m

Σµν,λ +1

2gνλ

m

Σµκ,κ − 1

2gνλ

m

Σµκ,κ)

. (20.1)

Let us discuss the effect of this equation for fields od various spins.

20.1 Local Four-Fermion Interaction due to Torsion

A non-trivial effect of torsion can be derived for Dirac fields. The spin density ofmatter is, from (17.124),

m

Σαβ,γ= −i 1

2ψ[γγ,Σαβ]+ψ, (20.2)

with Σαβ = (i/4)[γα, γβ]. This can be written as

m

Σαβ,γ=1

2ψγ[αγβγγ]ψ =

1

2εαβγλψγ

λγ5ψ (20.3)

with γ5 ≡ (1/4!)εαβγδγαγβγγγδ, where the brackets around the subscripts denote

their complete antisymmetrization. Due to antisymmetry, the Palatini tensor di-

vided by 2, the torsion, and the contortion tensor are all equal to (κ/2)m

Σαβ,γ

1

2Sαβ,γ = Sαβγ = Kαβ,γ =

κ

2

m

Σαβ,γ . (20.4)

In Eq. (11.141) we have expressed the curvature tensor in terms of the Riemanncurvature tensor plus the contortion. Two contractions give the corresponding de-composition of the scalar curvature

R = R + DµKννµ − DνKµ

νµ +(

KµµρKνρ

ν −KνµρKµρ

ν)

. (20.5)

403

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404 20 Evanescent Properties of Torsion in Gravity

In the gravitational Einstein-Cartan action, R is integrated over the total invariantvolume of the universe. The terms DµKν

νµ and DνKµνµ produce irrelevant surface

terms and can be ignored. The action can therefore be separated into a Hilbert-Einstein action

f

A = − 1

d4x√−gR, (20.6)

plus a field torsion action

f

A S =∫

d4x√−g

f

LS, (20.7)

with a Lagrangian density

f

LS = − 1

(

KµµρKρ

ν −KνµρKµρ

ν)

. (20.8)

This can be rearranged to

f

LS =1

2κSµν,λK

λνµ, (20.9)

where Sµν,λ is the Palatini tensor (20.31). As a cross check we differentiate this with

respect to Kλνµ and obtain

∂f

LS

∂Kλνµ =1

2κSµν,λ. (20.10)

in accordance with (15.17) and (15.55).We now add to (20.9) the matter-torsion interaction Lagrangian density ex-

tracted from the Dirac action (17.15):

m

LS =1

2

m

Σµν,λ Kλν (20.11)

Extremizing the combined torsion Lagrangian L =f

L S+m

L S, we recover once more(20.4). Inserting this back into the total Lagrange density gives, at the extremum,the effective torsion Lagrangian

Leff =κ

4

m

Σµνλ Kλνµ =

κ

8

m

Σµνλ

m

Σµνλ

, (20.12)

or explicitly, with (20.3) (see [1])

Leff =3κ

16ψγµγ5ψψγ

µγ5ψ. (20.13)

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20.2 Scalar Fields 405

Unfortunately, this interaction is too weak to be detectable by present-day experi-ments, and probably also for many generations to come. The interaction (20.13) willinterfere with the weak interactions of nuclear β-decay which have the Lagrangian

L = − G√2

[

pγλ(

1 − gAgVγ5

)

n

]

[

eγλ (1 − γ5) ν]

+ c.c. , (20.14)

with the coupling constant

G = (1.14730 ± 0.000641) × 10−5GeV−2, (20.15)

and the ratio

gA/gV = 1.255 ± 0.006. (20.16)

In the unifies theory of weak and electromagnetic interactions, the coupling constant(20.15) is given by

G ≈ e2

m2W

4πα

m2W

(20.17)

wheremW = 80.423 ± 0.039GeV (20.18)

is the mass of the charged vector mesonsW±. This shows that the torsion interaction(20.13) is smaller than the weak interaction by the immense factor [recall mP fromEq. (12.47)]

m2W

m2P

≈ 4.34 × 10−35. (20.19)

Thus any hope for a detection in the foreseeable future is an illusion.An additional problem is that a four-fermion interaction such as (20.13) is not

renormalizable, so that it cannot possibly be a fundamental interaction, but at besta phenomenological approximation to some more fundamental theory.

20.2 Scalar Fields

In Eq. (15.61) we have already noted that, as a consequence of their avanishing spindensity, scalar fields do not give rise to torsion. This result contradicts our findingin Section 14.1.2 that classical particle trajectories are coupled to torsion, whichmakes them autoparallel rather than geodesic. Let us study this problem in moredetail by deriving particle trajectories once more from field-theoretic equations.

20.2.1 Possible Cure

How can we remove the discrepancy with respect to the previous derivation of anautoparallel trajectory in Eq. (14.7) , where it was found as a consequence of the

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406 20 Evanescent Properties of Torsion in Gravity

multivalued mapping procedure? Apparently, the variational procedure of the metricwhich has led to the Einstein equation (15.62) must be modified to account for thetorsion.

One cure is clearly if spin-one-half particles are the only sources of torsion, whichis compatible with the present knowledge of the composition od fundamental par-ticles. The the torsion is completely antisymmetric so that it decouples from theclassical equation of motion. (14.7).

Another solution may ultimatel emerge from the following consideration. In the

presence of torsion, the particle trajectory does not satisfy δm

A/δqµ(τ) = 0 but,according to (14.42),

δm

Aδqµ(τ)

=∂L

∂q µ− d

∂L

∂qµ= −2S λ

µν qν ∂L

∂qλ. (20.20)

For the Lagrangian in the action (11.11), reparametrized with the the proper timeτ = s/c, the right-hand side becomes

2S λµν q

ν ∂L

∂qλ= −m 2Sµνλq

ν(τ)qλ(τ). (20.21)

The autoparallel equation of motion would obviously be obtained if the energy-momentum tensor of a free spinless point particles would satisfy the covariant con-servation law

D∗νTµν(x) = 0, (20.22)

rather than (18.77).In order to see how such a conservation law could be obtained recall its deriva-

tion: We calculated in Eq. (18.56) the change of the total action under Einsteintransformations (11.80). For a point particle, this may be written explicitly as

δEA =∫

d4xδ

f

Aδgµν(x)

δEgµν(x) +∫

dτδm

Aδqµ(τ)

δEqµ(τ). (20.23)

Now the last term does not vanish on autoparallel trajectories. However, if wechange δq(τ) into the nonholonomic -δq(τ) defined in Eq. (14.19) with the property(14.35), then the second term vanishes.

The vanishing of the first term in (20.23) for arbitrary infinitesimal δEgµν(x) ofEq. (18.19) has produced the covariant conservation law (18.76) leading to autopar-allel trajectories. It is interesting to realize that if we were to replace the Einsteintransformation (18.19) in (20.23) by a transformation defined by

-δEgµν(x) = Dµξν(x) +Dνξµ(x) = Dµξν(x) + Dνξµ(x) − 4Sλµνξλ(x), (20.24)

which looks like (11.80), but with the Riemann covariant derivative replaced by thefull covariant derivative Dµ, the variation would contain an extra term,

-δEgµν(x) = δEgµν(x) − 4Sλµνξλ(x). (20.25)

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20.2 Scalar Fields 407

The change of the field part of the action would become

δEA = −1

2

d4x√−gT µν(x) -δEgµν = −

d4x√−gT µν(x)Dνξµ(x). (20.26)

Integrals over invariant expressions containing the covariant derivative Dµ can beintegrated by parts according to a rule (15.39). If we neglect the surface terms wefind

δEA =∫

d4x√−g D∗νT µν(x)ξµ(x), (20.27)

where D∗ν = Dν + 2Sνλλ. From this Einstein transformation we would thus find

the covariant conservation law of the energy-momentum tensor of a spinless pointparticles in a space with curvature and torsion D∗νT

µν(x) = 0, which is precisely thelaw (20.22) corresponding to autoparallel trajectories.

The question arises whether the new conservation law (20.22) allows for theconstruction of an extension of Einstein’s field equation

Gµν = κT µν (20.28)

to spaces with torsion, where Gµν is the Einstein tensor formed from the Ricci tensorRµν ≡ Rλµν

λ in Riemann spacetime (11.140). The minimal extension of (20.28) tospacetimes with torsion replaces the left-hand side by the Einstein-Cartan tensorGµν ≡ Rµν − 1

2gµνRτ

τ and the rigth-hand side by the canonical energy-momentumtensor Θµν , and becomes

Gµν = κΘµν (20.29)

The Einstein-Cartan tensor Gµν satisfies a Bianchi identity

D∗νGµν + 2Sλµ

κGκλ − 1

2Sλκ

;νRµνλκ = 0, (20.30)

where Sλκ;ν is the Palatini tensor (15.46),

Sλκ;ν ≡ 2(Sλκ

ν + δλνSκτ

τ − δκνSλτ

τ ). (20.31)

It is then concluded that the energy-momentum tensor satisfies the conservation law

D∗νΘµν + 2Sλµ

κΘκλ − 1

2κSλκ

;νRµνλκ = 0. (20.32)

For standard field theories of matter, this is indeed true if the Palatini tensor satisfiesthe second Einstein-Cartan field equation

Sλκ;ν = κΣλκ;ν , (20.33)

where Σλκ;ν is the canonical spin density of the matter fields. A spinless pointparticle contributes only to the first two terms in (20.32).

What tensor will stand on the left-hand side of the field equation (20.29) if theenergy-momentum tensor satisfies the conservation law (20.22) instead of (18.77)?

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408 20 Evanescent Properties of Torsion in Gravity

At present, we can give an answer [3] only for the case of a pure gradient torsionwhich has the general form [4]

Sµνλ =

1

2(δµ

λ∂νσ − δνλ∂µσ). (20.34)

Then we may simply replace (20.29) by

eσGµν = κT µν . (20.35)

Note that for gradient torsion, Gµν is symmetric as can be deduced from the fun-damental identity (which expresses merely the fact that the Einstein-Cartan tensorRµνλ

κ is the covariant curl of the affine connection)

D∗λSµν;λ = Gµν −Gνµ. (20.36)

Indeed, inserting (20.34) into (20.31), we find the Palatini tensor

Sλµ;κ ≡ −2[δλ

κ∂µσ − (λ↔ µ)]. (20.37)

This has a vanishing covariant derivative

D∗λSµν;λ = −2[D∗µ∂νσ −D∗ν∂µσ] = 2[Sµν

λ∂λσ − 2Sµλλ∂νσ + 2Sνλ

λ∂µσ], (20.38)

since the terms on the right-hand side cancel after using (20.34) and Sµλλ ≡ Sµ =

−3∂µσ/2. Now we insert (20.34) into the Bianchi identity (20.30), with the result

D∗νGλν + ∂λσGκ

κ − ∂νσGλν + 2∂νσRλ

ν = 0. (20.39)

Inserting here Rλκ = Gλκ − 12gλκGν

ν , this becomes

D∗νGλν + ∂νσGλ

ν = 0. (20.40)

Thus we find the Bianchi identity

D∗ν(eσGλ

ν) = 0. (20.41)

This makes the left-hand side of the new field equation (20.35) compatible with theautoparallel covariant conservation law (20.22).

If torsion exists in space and if it is not of the gradient type, there is, so far,no way of reconciling the classical result with field theory. The field theory is notable to render autoparallel trajectories for scalar particles. This has been the animportant problem of gravity with torsion.

There is, however, a simple resolution of this problem. If torsion has fermionssuch as quarks and leptons as the only sources, the torsion tensor is completelyantisymmetric. In this case all autoparallels are automatically geodesics and theconflict disappers.

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20.2 Scalar Fields 409

20.2.2 Electromagnetism

The most disturbing problem with torsion is that it is impossible to couple electro-magentism to it without destroying gauge invariance. As mentioned in the Preface,this was first observed by Schrodinger in the 1930s, causing him to derive upperbounds for the photon mass from experimental observations. The present upperbound is

mγ < 3 × 10−27eV. (20.42)

In order to be invariant under the usual electromagnetic gauge transformations

Aµ → Aµ + ∂µΛ, (20.43)

the electromagnetic action

Aem = −1

4

d4x√−gFµνF µν , (20.44)

must contain the same field strengths

Fµν = ∂µAν − ∂νAµ, (20.45)

in spaces with curvature and torsion as in flat space. Gauge invariance permitsreplacing the derivatives ∂µ in Fµν by the Riemann covariant derivatives Dν , sincethe Christoffel symbols drop out in front of the antisymmetric tensor Fµν :

DµAν − DνAµ = ∂µAν − ΓµνλAλ − ∂νAµ + Γνµ

λAλ = ∂µAν − ∂νAµ. (20.46)

Thus the field strength (20.45) is gauge invariant under electromagnetic, Einstein,and local Lorentz transformations.

For the full covariant derivative Dµ, however, the replacement introduces anadditional term

DµAν−DνAµ = ∂µAν−ΓµνλAλ−∂νAµ+Γνµ

λAλ = ∂µAν−∂νAµ−2SµνλAλ, (20.47)

which destroys the gauge invariance. Inserting this covariant curl into the action(20.44) would lead to massive photons in any region with nonvanishing torsion.

As far as physical observations are concerned, the problem is really academic.We have seen before in the discussion of fermions, that torsion gives only forcesat extremely short distances of the order of Planck length. Since no conceivableexperiment can invade into such a short distance, the mass of the photons wouldremain undetectable. Moreover, at such short distances from matter, the propertiesof photons would much stronger be modified by electromagnetic dispersion andabsorption than by any conceivable gravitational torsion field. Only if torsion couldpropagate, for which we see no possible mechanism, a non-gauge-invariant couplingwould have fatal consequences. In fact, the experimental upper bound (20.42) is incomplete agreement with the absence of torsion in the cosmos.

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410 20 Evanescent Properties of Torsion in Gravity

20.3 Compatibility Problems of Gravity with Torsion and

Electroweak Interactions

Apart from the problem to find autoparallel classical trajectories for scalar particlesand the violation of gauge invariance in the field theory of gravity with torsion,there are other difficulties. These arise on purely theoretical grounds and would beobservable only of torsion could propagate, which it does not in Einstein-Cartantheory or any conceivable extension of it.

As we just saw, the electromagnetic field cannot couple minimally to torsionsince this would destroy gauge invariance [5].

Massive vector bosons, on the other hand, such as the ρ-meson, whose wavefunction has a large amplitude in a state of a quark and an antiquark in an s-wavespin triplet channel, should certainly couple to torsion via their quark content.

By analogy with photons, the fundamental action describing electroweak pro-cesses should contain no minimally coupled torsion in the gradient terms of the barevector bosons W and Z. However, these particles acquire a mass via the Meissner-Higgs effect which makes them essentially composite particles, their fields being amixture of the original massless vector fields and the Higgs fields. By analogy withthe massive ρ-vector field, we could expect that also the massive electroweak vectorfields couple to torsion, and the question arises how the Meissner-Higgs effect iscapable of generating such a coupling.

20.3.1 Solution for Gradient Torsion

In general, we do not know the answer to the last problem. We can only give ananswer if the torsion is of the gradient type

Sµνλ(x) =

1

2

[

δ λµ ∂νσ(x) − δ λ

ν ∂µσ(x)]

. (20.48)

It is immediately obvious that a minimally-coupled scalar Higgs field with anaction

A[φ] =∫

d4x√−g

(

1

2gµν |∇µφ∇νφ| −

m2

2|φ|2 − λ

4|φ2|2

)

(20.49)

cannot equip a previously uncoupled massless vector field with a torsion coupling.For simplicity, we consider only a simple Ginzburg-Landau-type theory with a com-plex field to avoid inessential complications. As usual, g = detgµν denotes thedeterminant of the metric gµν(x), and ∇µ is the electromagnetic covariant deriva-tives ∇µ = ∂µ − ieVµ. The square mass is negative, so that the Higgs field has anonzero expectation value with |φ|2 = −m2/λ. From the derivative term, the vectorfield acquires a mass term e2|φ|2V µVµ/2, leading to the a free part of the vectorboson action

A[V ] =∫

d4x√−g

(

−1

4FµνF

µν − e2m2

2λVνV

µ

)

, (20.50)

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20.3 Compatibility Problems of Gravity with Torsion and Electroweak Interactions 411

where Fµν the covariant curl Fµν ≡ ∂µVν − ∂νVµ. Of course, the covariant curl ofthe nonabelian electroweak vector bosons would also have self-interactions, whichcan however be ignored in the present discussion since we are only interested in thefree-particle propagation.

Since the Meissner-Higgs effect creates the mass of the vector bosons by mixingthe uncoupled bare vector boson with the scalar Higgs field, it is obvious that themassive vector bosons can couple to torsion only if the scalar Higgs field has such acoupling. Indeed, it has recently been emphasized [6, 7] that, contrary to commonbelief [8], trajectories of scalar particles should be experience a torsion force. Thisconclusion was reached by a careful reinvestigation of the geometric properties of thevariational procedure of the action. Taking into account the fact that in the presenceof torsion parallelograms exhibit a closure failure, the variational procedure requireda modification of this procedure [6, 9, 10] which led to the conclusion that scalarparticles should move along autoparallel trajectories rather than geodesic ones asderived from a minimally coupled scalar field action [8]. The modification of thevariational procedure was suggested to us by the close analogy of spaces with torsionwith crystals containing defects [11].

20.3.2 New Scalar Product

In the textbook [6] it has been pointed out that there exists a consistent Schrodingerformulation for a particle in a space with torsion if this has the restricted gradientform (20.48) or if it is completely antisymmetric. The Schrodinger equation is drivenby the Laplace operator gµνDµDν , where Dµ is the covariant derivative involving the

full affine connection Γµνλ, including torsion. It differs from the Laplace-Beltrami

operator in torsion-free spaces ∆ ≡√

|g|−1∂µ

|g|gµν∂ν by a term −2Sνλλ∂ν =

−3(∂νσ)∂ν . This operator, however, is hermitian only in a scalar product whichcontains a factor e−3σ [12]. In the case of totally antisymmetric torsion, the twoLaplace operators are equal and the original scalar product ensures hermiticity andthus unitarity of time evolution. Such a torsion drops also out from the classicalequation of motion, so that autoparallel and geodesic trajectories coincide. For thisreason we shall continue the discussion only for gradient torsion.

The gradient torsion has the advantage that it can be incorporated into theclassical action of a scalar point particle in such a way that the modification of thevariational procedure found in [9, 10] becomes superfluous. The modified actionreads for a massive particle [13]

A[x] = −mc∫

dτ eσ(x)√

gµν(x))xµxν = −mc

ds eσ(x(s)), (20.51)

where τ is an arbitrary parameter and s the proper time. From the Euler-Lagrangeequation we find that for τ = s, the Lagrangian under the integral is a constant ofmotion, whose value is, moreover, fixed by the mass shell constraint

L = eσ(x)√

gµν(x))xµxν ≡ 1, τ = s. (20.52)

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412 20 Evanescent Properties of Torsion in Gravity

The necessity of a factor e−3σ(x) in the scalar product discovered in [6] became thebasis of a series of studies in general relativity [14, 15]. In the latter work, the actionof a relativistic free scalar field φ was found to be

A[φ] =∫

d4x√−ge−3σ

(

1

2gµν |∇µφ∇νφ| −

m2

2|φ|2e−2σ

)

. (20.53)

The associated Euler-Lagrange equation is

DµDµφ+m2e−2σ(x)φ = 0, (20.54)

whose eikonal approximation φ(x) ≈ eiE(x) yields the following equation for thephase E(x) [15]:

e2σ(x)gµν(x)[∂µE(x)][∂νE(x)] = m2. (20.55)

Since ∂µE is the momentum of the particle, the replacement ∂µE → mxµ shows thatthe eikonal equation (20.55) guarantees the constancy of the Lagrangian (20.52),thus describing autoparallel trajectories.

20.3.3 Self-Interacting Higgs Field

Apart from the factor e−3σ(x) accompanying the volume integral, the σ-field couplesto the scalar field like a dilaton, the power of e−σ being determined by the dimensionof the associated term. If we therefore add to the free-field action (20.53) a quarticself-interaction to have a Meissner-Higgs effect, this self-interaction will not carry anextra factor e−σ, so that the proper Higgs action in the presence of gradient torsionreads

A[φ] =∫

d4x√−ge−3σ

(

1

2gµν |∇µφ∇νφ| −

m2

2|φ|2e−2σ − λ

4|φ2|2

)

(20.56)

If m2 is negative, and the torsion depends only weakly on spacetime, the Higgs fieldhas a smooth vacuum expectation value

|φ|2 = −m2

λe−2σ. (20.57)

The smoothness of the torsion field is required over a length scale of the Comptonwavelength of the Higgs particle, i.e. over a distance of the order 1/20GeV ≈ 10−15

cm. For a torsion field of gravitational origin, this smoothness will certainly beguaranteed. From the gradient term in (20.56) we then extract in the gauge φ =realthe mass term of the vector bosons

d4x√−ge−3σ 1

2m2V e−2σ(x)V µVµ (20.58)

where

m2V = −e

2

λm2, m2 < 0. (20.59)

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Notes and References 413

Taking the physical scalar product in the presence of torsion into account, we obtainfor the massive vector bosons the free-field action

A[V ] =∫

d4x√−ge−3σ

(

−1

4FµνF

µν +m2V e−2σ(x)VνV

µ)

. (20.60)

The appearance of the factor e−2σ in the mass term guarantees again the sameautoparallel trajectories in the eikonal approximation as for spinless particles in theaction (20.53).

Note that the scalar product factor e−3σ(x) implies a coupling to torsion also forthe massless vector bosons which is fully compatible with gauge invariance. Due tothe symmetry between Z-boson and photon, this factor must be present also in theelectromagnetic action.

Notes and References

[1] F.W. Hehl and B.K. Datta, J. Math. Phys. 12, 1334 (1971).

[2] F.W. Hehl, Phys. Lett. A 36, 225 (1971).

[3] H. Kleinert and A. Pelster, Acta Phys. Polon. B 29 , 1015 (1998).

[4] S. Hojman, M. Rosenbaum, M.P. Ryan, Phys. Rev. D 19, 430 (1979).

[5] R. Utiyama, Phys. Rev. 101, 1597 (1956);T.W.B. Kibble, J. Math. Phys. 2, 212 (1961);F.W. Hehl, P. von der Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys.48, 393 (1976);F.W. Hehl, J.D. McCrea, E.W. Mielke and Y. Ne’eman, Phys. Rep. 258, 1(1995).

[6] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets , 4th ed., World Scientific, Singapore 2006(kl/b5), where kl is short for the www address http://www.physik.fu-ber-lin.de/~kleinert.

[7] H. Kleinert, Nonholonomic Mapping Principle for Classical and Quantum Me-chanics in Spaces with Curvature and Torsion, Gen. Rel. Grav. 32, 769 (2000)(kl/258).Short version presented as a lecture at the Workshop on Gauge Theories ofGravitation, Jadwisin, Poland, 4-10 September 1997, Acta Phys. Pol. B 29,1033 (1998) (gr-qc/9801003).

[8] F.W. Hehl, Phys. Lett. A 36, 225 (1971). See also Section 7 of [7].

[9] P. Fiziev and H. Kleinert, New Action Principle for Classical Particle Trajecto-ries In Spaces with Torsion, Europhys. Lett. 35, 241 (1996) (hep-th/9503074).

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414 20 Evanescent Properties of Torsion in Gravity

[10] H. Kleinert and A. Pelster, Autoparallels From a New Action Principle, Gen.Rel. Grav. 31, 1439 (1999) (gr-qc/9605028).

[11] Our notation of field-theoretic and geometric quantities is the same as inthe textbook H. Kleinert, Gauge Fields in Condensed Matter , Vol. IIStresses and Defects , World Scientific, Singapore 1989, pp. 744-1443(kl/b1/contents1.html).

[12] See Section 11.4 in Ref. [6]. Note that the normalization of the σ-field is nor-malized differently from the present one by a factor 2/3. There we introducedδ via the relation Sµν

ν = ∂µσ,whereas here Sµνν = (3/2)∂µσ.

[13] H. Kleinert and A. Pelster, Novel Geometric Gauge-Invariance of Autopar-allels , Lectures presented at Workshop Gauge Theories of Gravitation, Jad-wisin, Poland, 4-10 September 1997, Acta Phys. Pol. B 29, 1015 (1998) (gr-qc/9801030).

[14] A. Saa, Mod. Phys. Lett. A8, 2565, (1993); ibid. 971, (1994); Class. Quant.Grav. 12, L85, (1995); J. Geom. and Phys. 15, 102, (1995); Gen. Rel. andGrav. 29, 205, (1997).

[15] P. Fiziev, Spinless matter in Transposed-Equi-Affine theory of gravity , Gen.Rel. Grav. 30, 1341 (1998) (gr-qc/9712004). See also Gravitation Theory withPropagating Torsion, (gr-qc/9808006).

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Man is equally incapable of seeing the nothingness

from which he emerges and the infinity in which he is engulfed

Blaise Pascal (1623 - 1662)

21

Emerging Gravity

The natural length scale of gravitational physics is the Planck length lP ≈ 1.616 ×10−33 cm formed in Eq. (12.46) from combinations of Newton’s gravitational con-stant GN, the light velocity c, and Planck’s constant h. It is the Compton wave-length lP ≡ h/mPc associated with the Planck mass mP ≈ 2.177 × 10−5g =1.22 × 1022MeV/c2. The Planck length is an extremely small quantity whichpresently lies beyond any experimental resolution, and will probably be so in the nottoo distant future. Particle accelerators are presently able to probe distances whichare still 10 orders of magnitude larger than lP. Considering the fast growing costs ofaccelerators with energy, it is unimaginable, that they will get close to the Plancklength for many generations to come. This length may therefore be considered as theshortest length accessible to experimental physics. Thus it makes no physical senseto produce theories which predict properties of the universe at smaller length scales.Since the times of Galileo Galilei, such theories fall into the realm of philosophy ofeven religion. The history of science shows us that nature has always surprised uswith new discoveries as observations invaded into shorter and shorter distances. Sofar, all theories in the past which claimed for a while to be theories of everythinghave been falsified by such discoveries.

The history of theoretical physics is full of such examples for such exotic theories.The presently most popular example is string theory. Its main strength lies inmaking predictions for the trans-Planckian regime down to zero length. In theexperimentally accessible energy range, these theories require spacetime dimensionsand predict particles which are not found in nature. In particular, the assumptioin ofa string representing fundamental particles makes only sense if there are overtones,which any string must have. In the string model these overtones lie all in theinaccessible Planck regime. It is thus unclear how one can believe any of theirpredictions in this regime.

One of the most important features of string theories is that they predict thevalidity of Lorentz invariance at all energies in the trans-Planckian regime. Inthis chapter we would like to point out that if one is willing to spend time withspeculations, an entirely different scenario is possible.

415

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416 21 Emerging Gravity

21.1 World Crystal

Let us suppose, just for the fun of it, that we live in a world crystal with a latticeconstant of the order of the Planck length [1]. Up to now we would have been unableto notice this. And this would remain so for a long time to come. None of the present-day relativistic physical laws would be observably violated. The gravitational forcesand their geometric description would arise from variants of the plastic forces in thisworld crystal . The observed curvature of spacetime would be just a signal of thepresence of disclinations in the crystal. Matter would be sources of disclinations [2].

For simplicity, we shall present such a construction only for a system withouttorsion. If the world crystal is distorted by an infinitesimal displacement field

xµ → x′µ = xµ + uµ(x), (21.1)

it has a strain energy

A =µ

4

d4x (∂µuν + ∂νuµ)2, (21.2)

where µ is some elastic constant. We assume the second possible elastic constant,the Poisson ratio, to be zero. If the distortions are partly plastic, the world crystalcontains defects defined by Volterra surfaces, where crystalline sections have beencut out. The displacement field is multivalued, and the action (21.2) is the analogof the magnetic action (4.84) in the presence of a current loop. In order to do fieldtheory with this action, we have to make the displacement field single-valued withthe help of δ-functions which describe the jumps across the Volterra surfaces, bycomplete analogy with the magnetic energy (4.85):

A = µ∫

d4x (uµν − upµν)2, (21.3)

where uµν ≡ (∂µuν + ∂νuµ)/2 is the elastic strain tensor, and upµν the plastic straintensor [compare Eq. (9.71)] describing the Volterra surfaces via δ-functions on thesesurfaces. As explained in Section 9.11, the plastic strain tensor is a gauge field ofplastic deformations. The energy density is invariant under the single-valued defectgauge transformations [the continuum limit of (9.85)]

uµνp → uµν

p + (∂µλν + ∂νλµ)/2, uµ → uµ + λµ. (21.4)

Physically, they express the fact that defects are not affected by elastic distortionsof the crystal. Only multivalued gauge functions λµ change the defect content inupµν .

We now rewrite the action (21.5) in a canonical form [the analog of (4.86)] byintroducing an auxiliary symmetric stress tensor field σµν as

A =∫

d3x

[

1

4µσµνσ

µν + iσµν(uµν − upµν)

]

. (21.5)

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21.1 World Crystal 417

After a partial integration and extremization in uµ, the second term in the actionyields the equation

∂νσµν = 0. (21.6)

This may be guaranteed identically, as a Bianchi identity, by an ansatz

σµν = εµκλσεν

κλ′σ′∂λ∂λ′χσσ′ . (21.7)

The field χσσ′ plays the role of an elastic gauge field. It is the analog of the vectorpotential A(x) in Eq. (4.88). Inserting (21.7) into (21.5), we obtain the analog of(4.89):

A =∫

d4x

1

[

εµκλσενκλ′σ′∂λ∂λ′χσσ′

]2+ iενκλσεµκλ

′σ′∂λ∂λ′χσσ′upµν

. (21.8)

A further partial integration brings this to the form

A =∫

d4x

1

[

εµκλσενκλ′σ′∂λ∂λ′χσσ′

]2+ iχσσ′

[

εσκλνεσ′κλ′µ∂λ∂λ′u

pµν

]

, (21.9)

which is the analog of the double-gauge theory (4.90). The action is now double-gauge theory invariant under the defect gauge transformation (21.4), and understress gauge transformations

χστ → χστ + ∂σΛσ′ + ∂τΛσ. (21.10)

The action can further be rewritten as

A =∫

d4x

1

4µσµνσ

µν + iχµνηµν

, (21.11)

where ηµν is the four-dimensional extension of the defect density ηij in Eq. (12.37)[the analog of the magnetic current (4.91)]:

ηµν = εµκλσενκ

λ′τ∂λ∂λ′upστ . (21.12)

This is invariant under defect gauge transformations (21.4), and satisfies the con-servation law

∂νηµν = 0. (21.13)

We may now replace upσσ′ by half the metric field gµν in (12.23) and, recallingEq. (12.34), we recognize the tensor ηµν as the Einstein tensor associated with themetric tensor gµν .

Let us eliminate the stress gauge field from the action (21.11). For this we use theidentity (1A.23) for the product of two Levi-Civita tensors, and rewrite the stressfield (21.7) as

σµν = εµκλσεν

κλ′τ∂λ∂λ′ χστ

= −(∂2χµν + ∂µ∂νχλλ − ∂µ∂λχµ

λ − ∂ν∂λχµλ) + ηµν(∂

2χλλ − ∂λ∂κχ

λκ).(21.14)

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418 21 Emerging Gravity

Introducing the field φµν ≡ χµ

ν− 12δµνχλ

λ, and going to the Hilbert gauge ∂µφµν = 0,

the stress tensor reduces to

σµν = −∂2φµν , (21.15)

and the action of an arbitrary distribution of defects becomes

A =∫

d4x

1

4µ∂2φµν∂2φµν + iφµ

ν(ηµν − 12δµνη

λλ)

. (21.16)

Extremization with respect to the field φµν yields the interaction of an arbitrarydistribution of defects [the analog of (4.92)]:

A = µ∫

d4x (ηµν − 12δµνη

λλ)

1

(∂2)2 (ηµν − 1

2δµνηλλ). (21.17)

This is not the Einstein action for a Riemann spacetime. It would be so if thederivatives ∂2 in (21.31) were replaced by ∂. Then the Green function of (∂2)2

would be replaced by the Green function of −∂2. An index rearrangement wouldlead to the interaction

A = µ∫

d4x (ηµν − 12δµνη

λλ)

1

−∂2 ηµν . (21.18)

The defect tensor ηµν is composed of the plastic gauge fields upµν in the same way asthe stress tensor is in terms of the stress gauge field in Eq. (21.14), i.e.:

ηµν = εµκλσεν

κλ′τ∂λ∂λ′ upστ .

= −(∂2upµν + ∂µ∂νupλλ − ∂µ∂λu

pµλ − ∂ν∂λu

pµλ) + ηµν(∂

2upλλ − ∂λ∂κu

pλκ). (21.19)

If we introduce the auxiliary field wpµν ≡ upµ

ν− 12δµνupλ

λ, and chose the Hilbert gauge∂µwpµν = 0, the defect density reduces to

ηµν = −∂2wpµν , ηµν − 1

2δµνηλλ = −∂2upµν . (21.20)

and the interaction (21.18) of an arbitrary distribution of defects would become

A = µ∫

d4xupµν(x)ηµν(x). (21.21)

This coincides with the linearized Einstein action

A = − 1

d4x√−gR (21.22)

where κ is the gravitational constant. Indeed, in the linear approximation gµν =

δµν + hµ

ν with |hµν | 1, where the Christoffel symbols can be approximated by

Γµνλ ≈ 1

2

(

∂µhνλ + ∂νhµλ − ∂λhµν)

, (21.23)

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21.1 World Crystal 419

and the Riemann curvature tensor becomes

Rµνλκ ≈1

2

[

∂µ∂λhνκ − ∂ν∂κhµλ − (µ ↔ ν)]

, (21.24)

as can be seen directly from Eq. (11.147). This gives the Ricci tensor

Rµκ ≈ 1

2(∂µ∂λhλκ + ∂κ∂λhλµ − ∂µ∂κh− ∂2hµκ), (21.25)

where h ≡ hλλ is the trace of the tensor hµν . The ensuing scalar curvature is

R ≈ −(∂2h− ∂µ∂νhµν), (21.26)

so that the Einstein tensor becomes

Gµκ = Rµκ −1

2gµκR (21.27)

≈ −1

2(∂2hµκ + ∂µ∂κh− ∂µ∂λh

λκ − ∂κ∂λh

λµ) +

1

2ηµκ(∂

2h− ∂ν∂λhνλ).

This can be written as a four-dimensional version of a double curl

Gµκ =1

2εµδ

νλεκδστ∂ν∂σhλτ , (21.28)

as can be verified using the identity (1A.23).Thus the Einstein-Hilbert action has the linear approximation

A ≈ 1

d4xhµνGµν . (21.29)

Recalling the previously established identifications of plastic field and defect densitywith metric and Einstein tensor, respectively, the interaction between defects (21.21)is indeed the linearized version of the Einstein-Hilbert action (21.22), if we identifythe constant µ with 1/4κ.

The world crystal with the elastic energy (21.5) does not lead to this action. Itmust be modified to do so. A first modification is to introduce two more derivativesand assign to the crystal the higher-gradient elastic energy

A′ = µ∫

d4x [∂(uµν − upµν)]2. (21.30)

This removes one power of −∂2 from the denominator in the interaction (21.17).I order to obtain the correct contractions in (21.18), we must replace the action

(21.31) by

A =∫

d4x

[

− 1

(

φµν∂2φµν − 12φµ

µ∂2φµν)

+ iφνν(ηµν − 1

2δµνη

λλ)

]

. (21.31)

This, in turn, follows from an interaction energy

A = µ∫

d4x

[∂(uµν − upµν)]2 − 1

2 [∂(uµµ − uµ

pν)]2

. (21.32)

Thus we have shown that defects in the world crystal create a Riemannian space-time with a euclidean action of the Einstein type.

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420 21 Emerging Gravity

21.2 Gravity Emerging from Fluctuations of Matter andin Closed Friedmann Universe

In 1967 Sacharov put forward an interesting idea [3, 4] that the geometry does notpossess a dynamics of its own, but that the stiffness of spacetime could be entirelydue to the vacuum fluctuations of the fundamental fields in the universe (scalar,vector, tensor, spinor). These give rise to an Einstein action proportional to R,but also a cosmological term without R, and to all possible higher powers of Rµνλκ

contracted to scalars such as R2, RµνRµν , RµνλκR

µνλκ, R3, . . . . The lowest threecoefficients diverge in the ultraviolet, but if all fluctuating field stem from in arenormalizable quantum field theory, all infinities can be subtracted to leave a finitevalue to be fixed by experiment.

The cosmological term without R changes the gravitational action (15.8) to

f

A= − 1

d4x√−g(R + 2λ). (21.33)

The constant λ is the so-called cosmologial constant . It changes theenergy-momentum tensor of the gravitational field from −(1/κ)Gµν to

−(1/κ)(

Gµν − λgµν)

.When calculating the effect of fluctuations for any of the fundamental fields one

finds a contribution to the cosmological constant corresponding to an action density

Λ ≡ λ

κ=

λ c3

8πGN

(21.34)

which is of the order of ±h/l4P, where lP is the Planck length. For bosons, the signis positive, for fermions negative, reflecting the filling of all negative-energy statesin the vacuum.

A constant of this size is much larger than the present experimental estimate.In the literature one usually finds estimates for the dimensionless quantity

Ωλ0 ≡λ c2

3H20

(21.35)

where H0 is the Hubble constant which parametrizes the expansion velocity of theuniverse as a function of the distance r from us by Hubble’s law :

v = H0r. (21.36)

The inverse of H0 is roughly equal to the lifetime of the universe

H−10 ≈ 14 × 109 years. (21.37)

Present fits to distant supernovae and other cosmological data yield the estimate [5]

Ωλ0 ≡ 0.68 ± 0.10. (21.38)

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Notes and References 421

As a result, the experimental number for Λ is

Λ = Ωλ0

3H20

c2l2P8π

≈ 10−122 h

l4P. (21.39)

Such a small prefactor in front of the “natural” action density h/l4P can only arisefrom an almost perfect cancellation of the contributions of boson and fermion fields.This cancellation is the main reason why some people postulate the exostence ofa broken supersymmetry in the universe, in which every boson has a fermioniccounterpart. So far, the known particle spectra show no trace of such a symmetry.Thus there is need to explain it by some other not yet understood mechanism.

A mechanical model for Sacharov’s idea of emerging gravity would be an infinitelythin plastic bag filled with water. The bag represents the geometry which does nothave any dynamics of its own. All its movements are controlled by the dynamics ofthe water contained in it.

Sacharov’s idea is very appealing. Unfortunately, the calculation of the emerg-ing gravitational action requires the knowledge of what are all elementary fields innature. It is questionable whether this will ever be available. Moreover, it is unclearwhat role is played by the vacuum fluctuations of composite particles, such as themany elements in the periodic system. Do gold and lead contribute to the vacuumenergy?

Notes and References

[1] H. Kleinert, Gravity as Theory of Defects in a Crystal with Only Second-Gradient Elasticity , Ann. d. Physik, 44, 117(1987) (kl/172).

[2] For an attempt to explain why no torsion is found in the world crystal seeH. Kleinert and J. Zaanen, World Nematic Crystal Model of Gravity Explainingthe Absence of Torsion, Phys. Lett. A 324, 361 (2004) (gr-qc/0307033).

[3] A.D. Sacharov, DAN SSSR 177, 70, (1967). Reprinted inA.D. Sacharov, Gen. Rel. Grav. 32, 365, (2000).

[4] A cosmological model based on Sacharov’s idea is discussed inH. Kleinert and H.-J. Schmidt, Cosmology with Curvature-Saturated Grav-

itational Lagrangian R/√

1 + l4, Gen. Rel. Grav. 34, 1295 (2002) (gr-qc/0006074).

[5] For cosmological data see the internet pagehttp://super.colorado.edu/ michaele/Lambda/links.html.

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422 21 Emerging Gravity

H. Kleinert, MULTIVALUED FIELDS

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Index

acceleration, 1covariant, 209

action, 38Einstein-Cartan, 277Einstein-Hilbert, 277, 292local, 38, 41principle, 38

adjoint representation, 9affine connection, 213

for Dirac fields, 240for vector fields, 245

algebraLie, 9, 11

associativity, 9of charges, 90

Ampere law, 100angle

Euler, 27Frank, 123solid, 102, 131spherical, 102, 131

angularmomentum, 61

four-dimensional, 27velocity, 97

anholonomic objects, 247associativity, in Lie Algebra, 9autoparallel trajectories, 213auxiliary

magnetic field, 110nonholonomic variation, 214, 215stress field, 289

axial gauge, 45, 46

Baker-Campbell-Hausdorff formula, 8Balian, R., x, 144Barut, A.O., 99Belinfante energy-momentum tensor, 85,

86, 94, 99, 255, 257, 265Belinfante, F., 99Bessel-Hagen, E., 99

Bianchi identity, 94, 101Bilby, B.A., 144Biot-Savart energy, 111Birell, N.D., 257Bollmann, W., 144branching defect lines, 138Bullough, R., 144Burgers vector, 116Burgers, J.M., 144

canonicalcommutation rules, 67

fields, 89energy-momentum tensor, 74–76, 78–

80, 82, 85, 86momentum, 62, 66quantization, 67

Cartan, E., ixcenter-of-mass

theorem, 64, 79chain rule, curved space, 169charge

algebra, 90density, 69Noether, 57, 78

Cheng, K.S., 257closed-path variations in action principle,

214, 215closure failure, 217commutation rules

canonical, 67fields, 89

Comptonwavelength, 288

Compton wavelength, 52, 177, 236, 285conformal tensor

Weyl, 200connection

affine, 213spin

for Dirac fields, 240

423

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424 Index

for vector fields, 245

conservation law, 56

current, 53, 69

global, 69

local, 69, 72

conserved

current, 72

quantity, 57

constant

cosmological, 293

fine-structure, 22

group structure, 9

Hubble, 293

of motion, 57, 60

Planck, 288

continuous media, 123

contortion tensor, 162, 165, 176, 225, 240

definition, 161

Nye, 176, 232

contravariant, 15, 155

convention, summation a la Einstein, 2

coordinate

generalized, 38–40

transformation, 59, 61, 65, 73, 76

multivalued, 147

cosmological constant, 293

Coulomb

gauge, 45, 46

law, 45

covariant, 15

acceleration, 209

curl, definition, 163

derivative, 157

crystal

defects, 113

melting, 111

world, ix, 171, 176, 289

curl, covariant, 163

current

conservation, 53, 69, 72

Euler-Lagrange type equation, 59

loop in gauge field representation,109

Noether, 69

probability, 53

curvature, 179

scalarsphere, 202

scalar, definition, 165tensor, definition, 164

cuts, Volterra, 122, 124, 129

cutting procedure, Volterra, 134

Datta, B.K., 257, 275, 286

Davies, P.C.W., 257de Donder, T., 203

decompositionWeyl, 200

DeDonder, T., 194, 203

defect, 180density, 134

incompatibility, 139

disclintion, 116dislocation, 114, 116grain boundary, 116

in crystals, 113interdependence, 122

interstitial, 113lines, 122lines, branching, 138

point, 113vacancy, 113

definition

contortion, 161covariant curl, 163covariant derivative, 157

curvature tensor, 164spin, 33torsion, 161

densitycharge, 69

defect, 134incompatibility, 139

disclination, 131, 134

dislocation, 131, 134Lagrangian, 41, 42, 50, 223

derivative

covariant, 157functional, 50Lagrange, 220

DeWitt, B.S., 257Dirac

field, 238

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425

connection, 240matrices, 25

string, 106theory of magnetic monopoles, 106

disclination, 116

density, 131, 134edge, 177interdependence, 122

dislocation, 114, 116density, 131, 134edge, 177

interdependence, 122displacement

field, 125field, integrability, 129

distributions (generalized functions)

and Stokes theorem, 111divergence theorem

Gauss, 54, 109

double-gauge theory, 111, 290

Eddington, A.S., 203edge

disclination, 177dislocation, 177

Einstein

-Hilbert action, 277, 292summation convention, 2tensor, 227

definition, 165Einstein tensor, 227Einstein, A., ix, 3

Einstein.-Cartan action, 277electric

field, 40, 41electromagnetic

field

Euler-Lagrange equation, 42forces, 106

emerging gravity, 293

Endrias, S., ixenergy

-momentum tensorBelinfante, 85, 86, 94, 99, 255, 257canonical, 74–76, 78–80, 82, 85, 86

symmetric, 83, 85, 86, 94, 99, 255,257

Biot-Savart, 111rest, 20

energy-momentum tensorBelinfante, 265

Eotvos, R., 146

equationEuler-Lagrange

electromagnetic field, 42

point-particle, 39, 47Klein-Gordon, 44, 53Lorentz, 48

of motion, 39Heisenberg, 67

equivalence principle, 146, 147, 212

new, 212, 212Eshelby, J.D., 144

Euler angles, 27Euler-Lagrange equation, 57

current conservation, 59

electromagnetic field, 42point-particle, 39

relativistic, 47

Ewing, A., 144expansion, Lie, 12, 28extremal

principle, 38trajectory, 39

failure of closure, 217field

Dirac, 238displacement, 125electric, 40, 41

gauge, 100minimal coupling, 106

of current loop, 109magnetic, 40, 41

auxiliary, 110

multivalued, 125scalar, 233vector, 154

vierbein, 241reciprocal, 241

fine-structure constant, 22

Fitzgerald, G.F., 3fixing gauge, 44

Flugge, S., 144

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426 Index

Fock, V., 194, 257force

electromagnetic, 106Lorentz, 21, 48tidal, 147

formulaBaker-Campbell-Hausdorff, 8Lie expansion, 28

four-dimensional

angular momentum, 27

momentum, 18velocity, 18

-momentum, 18

-vectorof Dirac matrices, 25

-velocity, 18frame

inertial, 2, 17, 147, 151, 181, 191, 242

Lorentz, 3, 4of reference

absolute, 1

Frank vector, 120, 123Frank, F.C., 144Frenkel, J., 144

Friedel, J.P., 144functional

derivative, 50matrix, 274

Galileigroup, 2transformation, 68

gaugeaxial, 45, 46

Coulomb, 46field, 100

minimal coupling, 106

of current loop, 109fixing, 44Lorentz, 44

radiation, 45theory, 272

double, 111, 290

transformationrestricted, 44

second kind, 44

Gauss theorem, 54, 109generalized

coordinates, 38–40

momenta, 40, 42generator

Lorentz transformation, 7

rotation, 6geodesic trajectories, 150

global

conservation law, 69symmetry, 62

gradient representation of magnetic field,108, 111

grain boundary, 116gravity

emerging, 293

induced, 293group

Galilei, 2

rank, 9representation, 68

adjoint, 9

symmetry, 56

Hagen, C., 112

Hamiltonian, 39, 60, 67operator, 67

Hammond, R.T., x

Hehl, F.W., 257, 275, 286Heisenberg equation of motion, 67

Henderson, B., 144

Hessian matrix, 220Heyde, P.v.d., 275

Hubble

constant, 293

identity

Bianchi, 94, 101Jacobi, 9

Ricci, 249

incompatibility, 139induced gravity, 293

inertial

frame, 2, 17, 147, 151, 181, 191, 242mass, 146

integrability condition, 44, 103, 129

integral theorem

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427

Gauss, 54, 109interdependence, dislocations and discli-

nations, 122internal symmetry, 87interstitials, 113invariance

rotational, 61translational, 61

Iwanenko, D., 257

Jackiw, R., 112Jackson, J.D., 99Jacobi identity, 9Jaseja, T.S., 3Jaxan, A., 3

Kerlick, G.D., 275Kibble, T.W.B., x, 275Klein-Gordon

equation, 53equations, 44

Kleinert, A., ixKleinert, H., 37, 99, 112, 144, 145, 203,

211, 221, 294Kleman, M., 144Kondo, K., xKroner, E., x, 144Kroupa, F., 144

Lagrange derivative, 220Lagrangian, 38

density, 41, 42, 50, 223Lanczos, C., 194, 203Landau, L.D., 99, 275Laplace-Beltrami operator, 195, 234law

Ampere, 100conservation, 56

local, 72Coulomb, 45

Legendre transformation, 39, 60length

Planck, 177, 224Levi-Civita, 6Levi-Civita tensor, 34, 167Lie

algebra, 9, 11associativity, 9

expansion formula, 12, 28Lifshitz, E., 99

Lifshitz, E.M., 275lines

defect, 122

world, viii, 106, 225local

action, 38, 41

conservation law, 69, 72symmetry

transformation, 58, 70

transformation, 58of coordinates, 108

U(1) transformations, 107

Lodge, O., 3Lorentz

equation, 48force, 21, 48frame, 3, 4

gauge, 44transformation

generator, 7

transformations, 68Lorentz, H.A., 3

magneticfield, 40, 41

auxiliary, 110monopole, 100

Dirac theory, 106

massinertial, 146shell condition, 51

matrixDirac, 25

Dirac, four-vector, 25functiona, 274Hessian, 220

Pauli, 25media, continuous, 123melting of crystals, 111

Michelson, A.A., 3minimal

coupling

gauge field, 106Misner, C.W., 275

momenta

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428 Index

generalized, 42momentum

angular, 61four-dimensional, 27

canonical, 62, 66generalized, 40relativistic four-vector, 18

monopoleDirac theory, 106magnetic, 100

Morley, E.W., 3motion, equation of, 39Mugge, O., 144multivalued

coordinate transformation, 147field, 125

Mura, T., 144Murray, J., 3

Nabarro, F.R.N., 144natural units, 52Nester, J.M., 275new equivalence principle, 212Noether

charge, 57, 78current, 69

Noether, E., 99nonholonomic

variation, 214, 215auxiliary, 214, 215

Nye contortion tensor, 176, 232

object of anholonomity, 247operator

Hamiltonian, 67Laplace-Beltrami, 195, 234tensor, 11vector, 11, 26

orbitalangular momentum, 80transformation, 81

Orowan, E., 144

Palatini tensor, 229, 230, 255, 266, 270,271, 276, 277, 280, 281

pathautoparallel, 160closed, in action principle, 214, 215

geodesic, 150Pauli matrices, 25

Pelster, A., 221

Pi, S.-Y., 112Planck

constant, 288

length, 177, 224Poincare, J.H., 3

point defect, 113

point particleEuler-Lagrange equation, 39

relativistic, 47

Polanyi, M., 144precession, spin, 95

from coupling to orbit, 97

Thomas, 95, 96, 97precession, spin-orbit, 97

Price, P.B., 144

principleaction, 38

equivalence, 146, 147, 212

new, 212, 212extremal, 38

probability current, 53

process, Volterra, 134proper time, 147

quantization, canonical, 67

radiation gauge, 45

rank of group, 9

relativisticmomentum, four-vector, 18

velocity, four-vector, 18

Renner, J., 146representation, 68

adjoint, 9

rest energy, 20restricted gauge transformation, 44

Ricci

identity, 249tensor

definition, 165

Rosenfeld, L., 99Rosenhain, W., 144

rotation

generator, 6

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429

symmetry, 2, 61

scalarcurvature

sphere, 202curvature, definition, 165field, 233

Schmutzer, E., 275Schouten, J.A., 275

Schrodinger, E., ix, 257Schwarz integrability condition, 103Sciama, D.W., ix, 275

second-kind gauge transformation, 44Shabanov, S.V., 211, 221Smith, E., 144

soft symmetry breaking, 89solid angle, 102, 131

spherecurvature scalar, 202surface, 203

spin-orbit precession, 97connection

for Dirac fields, 240for vector fields, 245

definition, 33precession, 95rotation, 81

Stokes theorem, 102, 104, 129, 130, 132,136

for distributions, 111stress field, 289

stringDirac, 106theory, 288

structure constants, 9summation convention, Einstein, 2

surfacesphere, 203Volterra, 122, 124, 129

symmetric energy-momentum tensor, 83,85, 86, 94, 99, 255, 257

symmetrybreaking

soft, 89global, 62

group, 56

internal, 87transformation, 56, 75

spacetime-dependent, 70variation, 56, 57, 66, 73

Taylor, G.I., 144

tensorconformal, Weyl, 200contortion, 162, 165, 176, 225

definition, 161Nye, 176, 232

curvature, definition, 164Einstein, 227

definition, 165

energy-momentumBelinfante, 85, 86, 94, 99, 255, 257canonical, 74–76, 78–80, 82, 85, 86

symmetric, 83, 85, 86, 94, 99, 255,257

Levi-Civita, 34, 167

operator, 11Palatini, 229, 230, 255, 266, 270, 271,

276, 277, 280, 281Ricci

definition, 165torsion, definition, 161

theorem

center-of-mass, 64, 79divergence, Gauss, 54, 109Stokes, 102, 104, 129, 130, 132, 136

for distributions, 111Weingarten, 126

theorygauge, 272

double, 111, 290

string, 288Thomas precession, 95, 96, 97

Thorne, K.S., 275tidal force, 147time

proper, 147translation, 59

torsion tensor, 233, 237, 239, 242, 247,253, 255

definition, 161Toupin, R., 144

Townes, C.H., 3

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430 Index

trajectoriesautoparallel, 213

extremal, 39

geodesic, 150transformation

coordinate, 59, 61, 65, 73, 76

local, 108Galilei, 68

Legendre, 60

local U(1), 107Lorentz, 68

generator, 7

orbital, 81spin, 81

symmetry, 56, 75

spacetime-dependent, 70time, 59

Volterra

trivial, 137translation

symmetry, 2, 61

time, 59trivial Volterra transformation, 137

Truesdell, C., 144

U(1) local transformations, 107

unit tensorantisymmetric Levi-Civita, 34, 167

units, natural, 52

Utiyama, R., ix, 275

vacancy, 113variation

auxiliary nonholonomic, 214, 215

in action principle, 214, 215nonholonomic, 214, 215

symmetry, 56, 57, 66, 73

vectorBurgers, 116

contravariant, 15

covariant, 15field, 154

connection, 245

Frank, 120, 123operator, 11, 26

vector field

auxiliary, 110

velocityangular, 97four-vector, 18space, 207

vierbein field, 241reciprocal, 241

Volterracutting procedure, 122, 124, 129, 134surface, 122, 124, 129transformation

trivial, 137

wavelengthCompton, 52, 177, 236, 285, 288

Weinberg, S., 204, 275Weingarten theorem, 126Weyl

conformal tensor, 200decomposition, 200

Weyl, H., 203, 257Wheeler, J.A., 275Williams, L.P., 3world

crystal, ix, 171, 176, 289line, viii, 106, 225

Zaanen, J., 294

H. Kleinert, MULTIVALUED FIELDS


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