Representation and QM

Quantum Theory, Groups and Representations:

An Introduction

(under construction)

Peter WoitDepartment of Mathematics, Columbia University

[email protected]

October 14, 2014

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Contents

Preface xi

1 Introduction and Overview 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic principles of quantum mechanics . . . . . . . . . . . . . . . 2

1.2.1 Fundamental axioms of quantum mechanics . . . . . . . . 31.2.2 Principles of measurement theory . . . . . . . . . . . . . . 4

1.3 Unitary group representations . . . . . . . . . . . . . . . . . . . . 51.4 Representations and quantum mechanics . . . . . . . . . . . . . . 71.5 Symmetry groups and their representations on function spaces . 81.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Group U(1) and its Representations 132.1 Some representation theory . . . . . . . . . . . . . . . . . . . . . 142.2 The group U(1) and its representations . . . . . . . . . . . . . . 162.3 The charge operator . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Conservation of charge . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Two-state Systems and SU(2) 233.1 The two-state quantum system . . . . . . . . . . . . . . . . . . . 24

3.1.1 The Pauli matrices: observables of the two-state quantumsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Exponentials of Pauli matrices: unitary transformationsof the two-state system . . . . . . . . . . . . . . . . . . . 26

3.2 Commutation relations for Pauli matrices . . . . . . . . . . . . . 293.3 Dynamics of a two-state system . . . . . . . . . . . . . . . . . . . 313.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Linear Algebra Review, Unitary and Orthogonal Groups 334.1 Vector spaces and linear maps . . . . . . . . . . . . . . . . . . . . 334.2 Dual vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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4.4 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Orthogonal and unitary transformations . . . . . . . . . . . . . . 40

4.6.1 Orthogonal groups . . . . . . . . . . . . . . . . . . . . . . 41

4.6.2 Unitary groups . . . . . . . . . . . . . . . . . . . . . . . . 42

4.7 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 43

4.8 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Lie Algebras and Lie Algebra Representations 45

5.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Lie algebras of the orthogonal and unitary groups . . . . . . . . . 48

5.2.1 Lie algebra of the orthogonal group . . . . . . . . . . . . . 49

5.2.2 Lie algebra of the unitary group . . . . . . . . . . . . . . 50

5.3 Lie algebra representations . . . . . . . . . . . . . . . . . . . . . 50

5.4 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


6 The Rotation and Spin Groups in 3 and 4 Dimensions 57

6.1 The rotation group in three dimensions . . . . . . . . . . . . . . 57

6.2 Spin groups in three and four dimensions . . . . . . . . . . . . . 60

6.2.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6.2.2 Rotations and spin groups in four dimensions . . . . . . . 62

6.2.3 Rotations and spin groups in three dimensions . . . . . . 62

6.2.4 The spin group and SU(2) . . . . . . . . . . . . . . . . . 66

6.3 A summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69


7 Rotations and the Spin 12 Particle in a Magnetic Field 71

7.1 The spinor representation . . . . . . . . . . . . . . . . . . . . . . 71

7.2 The spin 1/2 particle in a magnetic field . . . . . . . . . . . . . . 72

7.3 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . 76

7.4 The Bloch sphere and complex projective space . . . . . . . . . . 77


8 Representations of SU(2) and SO(3) 83

8.1 Representations of SU(2): classification . . . . . . . . . . . . . . 84

8.1.1 Weight decomposition . . . . . . . . . . . . . . . . . . . . 84

8.1.2 Lie algebra representations: raising and lowering operators 86

8.2 Representations of SU(2): construction . . . . . . . . . . . . . . 90

8.3 Representations of SO(3) and spherical harmonics . . . . . . . . 93

8.4 The Casimir operator . . . . . . . . . . . . . . . . . . . . . . . . 99


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9 Tensor Products, Entanglement, and Addition of Spin 103

9.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.2 Composite quantum systems and tensor products . . . . . . . . . 105

9.3 Indecomposable vectors and entanglement . . . . . . . . . . . . . 106

9.4 Tensor products of representations . . . . . . . . . . . . . . . . . 107

9.4.1 Tensor products of SU(2) representations . . . . . . . . . 108

9.4.2 Characters of representations . . . . . . . . . . . . . . . . 108

9.4.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . 110

9.5 Bilinear forms and tensor products . . . . . . . . . . . . . . . . . 111


10 Energy, Momentum and Translation Groups 115

10.1 Energy, momentum and space-time translations . . . . . . . . . . 116

10.2 Periodic boundary conditions and the group U(1) . . . . . . . . . 121

10.3 The group R and the Fourier transform . . . . . . . . . . . . . . 124

10.3.1 Delta functions . . . . . . . . . . . . . . . . . . . . . . . . 126


11 The Heisenberg group and the Schrodinger Representation 131

11.1 The position operator and the Heisenberg Lie algebra . . . . . . 132

11.1.1 Position space representation . . . . . . . . . . . . . . . . 132

11.1.2 Momentum space representation . . . . . . . . . . . . . . 133

11.1.3 Physical interpretation . . . . . . . . . . . . . . . . . . . . 134

11.2 The Heisenberg Lie algebra . . . . . . . . . . . . . . . . . . . . . 135

11.3 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . 136

11.4 The Schrodinger representation . . . . . . . . . . . . . . . . . . . 137


12 The Poisson Bracket and Symplectic Geometry 141

12.1 Classical mechanics and the Poisson bracket . . . . . . . . . . . . 141

12.2 Poisson brackets and vector fields . . . . . . . . . . . . . . . . . . 143

12.3 Symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 148

12.4 Phase space and the Heisenberg Lie algebra . . . . . . . . . . . . 150

12.5 The Poisson bracket on the dual of a Lie algebra . . . . . . . . . 151


13 The Symplectic Group and the Moment Map 153

13.1 The symplectic group for d = 1 . . . . . . . . . . . . . . . . . . . 153

13.2 The Lie algebra of quadratic polynomials for d = 1 . . . . . . . . 157

13.3 The symplectic group for arbitary d . . . . . . . . . . . . . . . . 158

13.4 The moment map . . . . . . . . . . . . . . . . . . . . . . . . . . . 161


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14 Quantization 16514.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . 16514.2 The Groenewold-van Hove no-go theorem . . . . . . . . . . . . . 16714.3 Canonical quantization in d dimensions . . . . . . . . . . . . . . 16814.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 169

15 Semi-direct Products and their Representations 17115.1 An example: the Euclidean group . . . . . . . . . . . . . . . . . . 17115.2 Semi-direct products N oK and their representations, the case

of N commutative . . . . . . . . . . . . . . . . . . . . . . . . . . 17315.3 The Jacobi group and general semi-direct products . . . . . . . . 17715.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 178

16 Symmetries and Intertwining Operators 17916.1 Intertwining operators and the metaplectic representation . . . . 18016.2 Constructing the intertwining operators . . . . . . . . . . . . . . 18116.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

16.3.1 The SO(2) action on the d = 1 phase space . . . . . . . . 18316.3.2 The SO(2) action by rotations of the plane for d = 2 . . . 184


17 The Quantum Free Particle as a Representation of the Eu-clidean Group 18717.1 Representations of E(2) . . . . . . . . . . . . . . . . . . . . . . . 18717.2 The quantum particle and E(3) representations . . . . . . . . . . 19217.3 Other representations of E(3) . . . . . . . . . . . . . . . . . . . . 19417.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 195

18 Central Potentials and the Hydrogen Atom 19718.1 Quantum particle in a central potential . . . . . . . . . . . . . . 19718.2 so(4) symmetry and the Coulomb potential . . . . . . . . . . . . 20118.3 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . 20518.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 206

19 The Harmonic Oscillator 20719.1 The harmonic oscillator with one degree of freedom . . . . . . . . 20819.2 Creation and annihilation operators . . . . . . . . . . . . . . . . 21019.3 The Bargmann-Fock representation . . . . . . . . . . . . . . . . . 21319.4 Multiple degrees of freedom . . . . . . . . . . . . . . . . . . . . . 21519.5 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 217

20 The Harmonic Oscillator as a Representation of the HeisenbergGroup 21920.1 Complex structures and quantization . . . . . . . . . . . . . . . . 22020.2 The Bargmann transform . . . . . . . . . . . . . . . . . . . . . . 22420.3 Coherent states and the Heisenberg group action . . . . . . . . . 225

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21 The Harmonic Oscillator and the Metaplectic Representation,d = 1 229

21.1 The metaplectic representation for d = 1 . . . . . . . . . . . . . . 229

21.2 Normal-ordering and the choice of complex structure . . . . . . . 233


22 The Harmonic Oscillator as a Representation of U(d) 239

22.1 The metaplectic representation for d degrees of freedom . . . . . 240

22.2 Examples in d = 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . 244

22.2.1 Two degrees of freedom and SU(2) . . . . . . . . . . . . . 244

22.2.2 Three degrees of freedom and SO(3) . . . . . . . . . . . . 247


23 The Fermionic Oscillator 249

23.1 Canonical commutation relations and the bosonic oscillator . . . 249

23.2 Canonical anti-commutation relations and the fermionic oscillator 250

23.3 Multiple degrees of freedom . . . . . . . . . . . . . . . . . . . . . 252


24 Weyl and Clifford Algebras 255

24.1 The Complex Weyl and Clifford algebras . . . . . . . . . . . . . . 255

24.1.1 One degree of freedom, bosonic case . . . . . . . . . . . . 255

24.1.2 One degree of freedom, fermionic case . . . . . . . . . . . 256

24.1.3 Multiple degrees of freedom . . . . . . . . . . . . . . . . . 258

24.2 Real Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . 259


25 Clifford Algebras and Geometry 263

25.1 Non-degenerate bilinear forms . . . . . . . . . . . . . . . . . . . . 263

25.2 Clifford algebras and geometry . . . . . . . . . . . . . . . . . . . 265

25.2.1 Rotations as iterated orthogonal reflections . . . . . . . . 267

25.2.2 The Lie algebra of the rotation group and quadratic ele-ments of the Clifford algebra . . . . . . . . . . . . . . . . 268


26 Anti-commuting Variables and Pseudo-classical Mechanics 271

26.1 The Grassmann algebra of polynomials on anti-commuting gen-erators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

26.2 Pseudo-classical mechanics and the fermionic Poisson bracket . . 274

26.3 Examples of pseudo-classical mechanics . . . . . . . . . . . . . . 277

26.3.1 The classical spin degree of freedom . . . . . . . . . . . . 277

26.3.2 The classical fermionic oscillator . . . . . . . . . . . . . . 278


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27 Fermionic Quantization and Spinors 281

27.1 Quantization of pseudo-classical systems . . . . . . . . . . . . . . 281

27.2 The Schrodinger representation for fermions: ghosts . . . . . . . 284

27.3 Spinors and the Bargmann-Fock construction . . . . . . . . . . . 285

27.4 Parallels between bosonic and fermionic . . . . . . . . . . . . . . 290


28 Supersymmetry, Some Simple Examples 291

28.1 The supersymmetric oscillator . . . . . . . . . . . . . . . . . . . . 291

28.2 Supersymmetric quantum mechanics with a superpotential . . . . 294

28.3 Supersymmetric quantum mechanics and differential forms . . . . 296


29 The Dirac Operator 299

29.1 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . 299

29.2 The Pauli operator and free spin 12 particles in d = 3 . . . . . . . 300


30 Lagrangian Methods and the Path Integral 305

30.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . 305

30.2 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310


31 Quantization of Infinite-dimensional Phase Spaces 315

31.1 Inequivalent irreducible representations . . . . . . . . . . . . . . 316

31.2 The anomaly and the Schwinger term . . . . . . . . . . . . . . . 317

31.3 Higher order operators and renormalization . . . . . . . . . . . . 319


32 Multi-particle Systems and Non-relativistic Quantum Fields 321

32.1 Multi-particle quantum systems as quanta of a harmonic oscillator322

32.1.1 Bosons and the quantum harmonic oscillator . . . . . . . 322

32.1.2 Fermions and the fermionic oscillator . . . . . . . . . . . . 324

32.2 Solutions to the free particle Schrodinger equation . . . . . . . . 324

32.2.1 Box normalization . . . . . . . . . . . . . . . . . . . . . . 325

32.2.2 Continuum normalization . . . . . . . . . . . . . . . . . . 328

32.3 Quantum field operators . . . . . . . . . . . . . . . . . . . . . . . 329


33 Field Quantization and Dynamics for Non-relativistic QuantumFields 335

33.1 Quantization of classical fields . . . . . . . . . . . . . . . . . . . . 335

33.2 Dynamics of the free quantum field . . . . . . . . . . . . . . . . . 338


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34 Symmetries and Non-relativistic Quantum Fields 343

34.1 Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 343

34.1.1 U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . 344

34.1.2 U(m) symmetry . . . . . . . . . . . . . . . . . . . . . . . 346

34.2 Spatial symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 348

34.3 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351


35 Minkowski Space and the Lorentz Group 353

35.1 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

35.2 The Lorentz group and its Lie algebra . . . . . . . . . . . . . . . 357

35.3 Spin and the Lorentz group . . . . . . . . . . . . . . . . . . . . . 359


36 Representations of the Lorentz Group 363

36.1 Representations of the Lorentz group . . . . . . . . . . . . . . . . 363

36.2 Dirac γ matrices and Cliff(3, 1) . . . . . . . . . . . . . . . . . . . 368


37 The Poincare Group and its Representations 373

37.1 The Poincare group and its Lie algebra . . . . . . . . . . . . . . . 373

37.2 Representations of the Poincare group . . . . . . . . . . . . . . . 375

37.2.1 Positive energy time-like orbits . . . . . . . . . . . . . . . 377

37.2.2 Negative energy time-like orbits . . . . . . . . . . . . . . . 378

37.2.3 Space-like orbits . . . . . . . . . . . . . . . . . . . . . . . 378

37.2.4 The zero orbit . . . . . . . . . . . . . . . . . . . . . . . . 378

37.2.5 Positive energy null orbits . . . . . . . . . . . . . . . . . . 379

37.2.6 Negative energy null orbits . . . . . . . . . . . . . . . . . 379


38 The Klein-Gordon Equation and Scalar Quantum Fields 381

38.1 The Klein-Gordon equation and its solutions . . . . . . . . . . . 382

38.2 Classical relativistic scalar field theory . . . . . . . . . . . . . . . 385

38.3 The complex structure on the space of Klein-Gordon solutions . 387

38.4 Quantization of the real scalar field . . . . . . . . . . . . . . . . . 389

38.5 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

38.6 Fermionic scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . 392


39 Symmetries and Relativistic Scalar Quantum Fields 393

39.1 Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 393

39.1.1 SO(m) symmetry and real scalar fields . . . . . . . . . . . 394

39.1.2 U(1) symmetry and complex scalar fields . . . . . . . . . 396

39.2 Poincare symmetry and scalar fields . . . . . . . . . . . . . . . . 399


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40 U(1) Gauge Symmetry and Coupling to the ElectromagneticField 40140.1 U(1) gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . 40140.2 Electric and magnetic fields . . . . . . . . . . . . . . . . . . . . . 40340.3 The Pauli-Schrodinger equation in an electromagnetic field . . . 40440.4 Non-abelian gauge symmetry . . . . . . . . . . . . . . . . . . . . 40440.5 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 404

41 Quantization of the Electromagnetic Field: the Photon 40541.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . 40541.2 Hamiltonian formalism for electromagnetic fields . . . . . . . . . 40641.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40641.4 Field operators for the vector potential . . . . . . . . . . . . . . . 40641.5 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 406

42 The Dirac Equation and Spin-1/2 Fields 40742.1 The Dirac and Weyl Equations . . . . . . . . . . . . . . . . . . . 40742.2 Quantum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41042.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41042.4 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41042.5 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 410

43 An Introduction to the Standard Model 41143.1 Non-Abelian gauge fields . . . . . . . . . . . . . . . . . . . . . . . 41143.2 Fundamental fermions . . . . . . . . . . . . . . . . . . . . . . . . 41143.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . 41143.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 411

44 Further Topics 413

A Conventions 415

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Preface

This document began as course notes prepared for a class taught at Columbiaduring the 2012-13 academic year. The intent was to cover the basics of quantummechanics, up to and including basic material on relativistic quantum fieldtheory, from a point of view emphasizing the role of unitary representations ofLie groups in the foundations of the subject. It has been significantly rewrittenand extended during the past year and the intent is to continue this processbased upon experience teaching the same material during 2014-5. The currentstate of the document is that of a first draft of a book. As changes are made,the latest version will be available at

http://www.math.columbia.edu/~woit/QM/qmbook.pdf

Corrections, comments, criticism, and suggestions about improvements areencouraged, with the best way to contact me email to [email protected]

The approach to this material is simultaneously rather advanced, using cru-cially some fundamental mathematical structures normally only discussed ingraduate mathematics courses, while at the same time trying to do this in aselementary terms as possible. The Lie groups needed are relatively simple onesthat can be described purely in terms of small matrices. Much of the represen-tation theory will just use standard manipulations of such matrices. The onlyprerequisite for the course as taught was linear algebra and multi-variable cal-culus. My hope is that this level of presentation will simultaneously be useful tomathematics students trying to learn something about both quantum mechan-ics and representation theory, as well as to physics students who already haveseen some quantum mechanics, but would like to know more about the mathe-matics underlying the subject, especially that relevant to exploiting symmetryprinciples.

The topics covered often intentionally avoid overlap with the material ofstandard physics courses in quantum mechanics and quantum field theory, forwhich many excellent textbooks are available. This document is best read inconjunction with such a text. Some of the main differences with standard physicspresentations include:

• The role of Lie groups, Lie algebras, and their unitary representations issystematically emphasized, including not just the standard use of these toderive consequences for the theory of a “symmetry” generated by operatorscommuting with the Hamiltonian.

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http://www.math.columbia.edu/~woit/QM/qmbook.pdf

mailto:[email protected]

• Symplectic geometry and the role of the Lie algebra of functions on phasespace in Hamiltonian mechanics is emphasized, with quantization just thepassage to a unitary representation of (a subalgebra of) this Lie algebra.

• The role of the metaplectic representation and the subtleties of the pro-jective factor involved are described in detail.

• The parallel role of the Clifford algebra and spinor representation areextensively investigated.

• Some topics usually first encountered in the context of relativistic quan-tum field theory are instead first developed in simpler non-relativistic orfinite-dimensional contexts. Non-relativistic quantum field theory basedon the Schrodinger equation is described in detail before moving on tothe relativistic case. The topic of irreducible representations of space-time symmetry groups is first encountered with the case of the Euclideangroup, where the implications for the non-relativistic theory are explained.The analogous problem for the relativistic case, that of the irreducible rep-resentations of the Poincare group, is then worked out later on.

• The emphasis is on the Hamiltonian formalism and its representation-theoretical implications, with the Lagrangian formalism de-emphasized.In particular, the operators generating symmetry transformations are de-rived using the moment map for the action of such transformations onphase space, not by invoking Noether’s theorem for transformations thatleave invariant a Lagrangian.

• Care is taken to keep track of the distinction between vector spaces andtheir duals, as well as the distinction between real and complex vectorspaces, making clear exactly where complexification and the choice of acomplex structure enters the theory.

• A fully rigorous treatment of the subject is beyond the scope of what iscovered here, but an attempt is made to keep clear the difference betweenwhere a rigorous treatment could be pursued relatively straight-forwardly,and where there are serious problems of principle making a rigorous treat-ment very hard to achieve.

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Chapter 1

Introduction and Overview

1.1 Introduction

A famous quote from Richard Feynman goes “I think it is safe to say that no oneunderstands quantum mechanics.”[17]. In this book we’ll pursue one possibleroute to such an understanding, emphasizing the deep connections of quantummechanics to fundamental ideas and powerful techniques of modern mathemat-ics. The strangeness inherent in quantum theory that Feynman was referringto has two rather different sources. One of them is the inherent disjunction andincommensurability between the conceptual framework of the classical physicswhich governs our everyday experience of the physical world, and the very dif-ferent framework which governs physical reality at the atomic scale. Familiaritywith the powerful formalisms of classical mechanics and electromagnetism pro-vides deep understanding of the world at the distance scales familiar to us.Supplementing these with the more modern (but still “classical” in the senseof “not quantum”) subjects of special and general relativity extends our under-standing into other less accessible regimes, while still leaving atomic physics amystery.

Read in context though, Feynman was pointing to a second source of diffi-culty, contrasting the mathematical formalism of quantum mechanics with thatof the theory of general relativity, a supposedly equally hard to understandsubject. General relativity can be a difficult subject to master, but its math-ematical and conceptual structure involves a fairly straight-forward extensionof structures that characterize 19th century physics. The fundamental physicallaws (Einstein’s equations for general relativity) are expressed as partial differ-ential equations, a familiar if difficult mathematical subject. The state of thesystem is determined by the set of fields satisfying these equations, and observ-able quantities are functionals of these fields. The mathematics is just that ofthe usual calculus: differential equations and their real-valued solutions.

In quantum mechanics, the state of a system is best thought of as a differentsort of mathematical object: a vector in a complex vector space, the so-called

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state space. One can sometimes interpret this vector as a function, the wave-function, although this comes with the non-classical feature that wave-functionsare complex-valued. What’s truly completely different is the treatment of ob-servable quantities, which correspond to self-adjoint linear operators on the statespace. This has no parallel in classical physics, and violates our intuitions abouthow physics should work, with observables now often no longer commuting.

During the earliest days of quantum mechanics, the mathematician HermannWeyl quickly recognized that the mathematical structures being used were oneshe was quite familiar with from his work in the field of representation theory.From the point of view that takes representation theory as a fundamental struc-ture, the framework of quantum mechanics looks perfectly natural. Weyl soonwrote a book expounding such ideas [67], but this got a mixed reaction fromphysicists unhappy with the penetration of unfamiliar mathematical structuresinto their subject (with some of them characterizing the situation as the “Grup-penpest”, the group theory plague). One goal of this course will be to try andmake some of this mathematics as accessible as possible, boiling down Weyl’sexposition to its essentials while updating it in the light of many decades ofprogress and better understanding of the subject.

Weyl’s insight that quantum mechanics crucially involves understanding theLie groups that act on the phase space of a physical system and the unitary rep-resentations of these groups has been vindicated by later developments whichdramatically expanded the scope of these ideas. The use of representation the-ory to exploit the symmetries of a problem has become a powerful tool that hasfound uses in many areas of science, not just quantum mechanics. I hope thatreaders whose main interest is physics will learn to appreciate the mathematicalstructures that lie behind the calculations of standard textbooks, helping themunderstand how to effectively exploit them in other contexts. Those whose maininterest is mathematics will hopefully gain some understanding of fundamen-tal physics, at the same time as seeing some crucial examples of groups andrepresentations. These should provide a good grounding for appreciating moreabstract presentations of the subject that are part of the standard mathemat-ical curriculum. Anyone curious about the relation of fundamental physics tomathematics, and what Eugene Wigner described as “The Unreasonable Ef-fectiveness of Mathematics in the Natural Sciences”[68] should benefit from anexposure to this remarkable story at the intersection of the two subjects.

The following sections give an overview of the fundamental ideas behindmuch of the material to follow. In this sketchy and abstract form they willlikely seem rather mystifying to those meeting them for the first time. As wework through basic examples in coming chapters, a better understanding of theoverall picture described here should start to emerge.

1.2 Basic principles of quantum mechanics

We’ll divide the conventional list of basic principles of quantum mechanics intotwo parts, with the first covering the fundamental mathematics structures.

2

1.2.1 Fundamental axioms of quantum mechanics

In classical physics, the state of a system is given by a point in a “phase space”,which one can think of equivalently as the space of solutions of an equationof motion, or as (parametrizing solutions by initial value data) the space ofcoordinates and momenta. Observable quantities are just functions on this space(i.e. functions of the coordinates and momenta). There is one distinguishedobservable, the energy or Hamiltonian, and it determines how states evolve intime through Hamilton’s equations.

The basic structure of quantum mechanics is quite different, with the for-malism built on the following simple axioms:

Axiom (States). The state of a quantum mechanical system is given by a vectorin a complex vector space H with Hermitian inner product 〈·, ·〉.

We’ll review in chapter 4 some linear algebra, including the properties of in-ner products on complex vector spaces. H may be finite or infinite dimensional,with further restrictions required in the infinite-dimensional case (e.g. we maywant to require H to be a Hilbert space). Note two very important differenceswith classical mechanical states:

• The state space is always linear: a linear combination of states is also astate.

• The state space is a complex vector space: these linear combinations canand do crucially involve complex numbers, in an inescapable way. In theclassical case only real numbers appear, with complex numbers used onlyas an inessential calculational tool.

In this course we will sometimes use the notation introduced by Dirac forvectors in the state space H: such a vector with a label ψ is denoted

|ψ〉

Axiom (Observables). The observables of a quantum mechanical system aregiven by self-adjoint linear operators on H.

We’ll also review the notion of self-adjointness in our review of linear algebra.When H is infinite-dimensional, further restrictions will be needed on the classof linear operators to be used.

Axiom (Dynamics). There is a distinguished observable, the Hamiltonian H.Time evolution of states |ψ(t)〉 ∈ H is given by the Schrodinger equation

d

dt|ψ(t)〉 = − i

~H|ψ(t)〉

The Hamiltonian observable H will have a physical interpretation in termsof energy, and one may also want to specify some sort of positivity property onH, in order to assure the existence of a stable lowest energy state.

3

~ is a dimensional constant, the value of which depends on what units oneuses for time and for energy. It has the dimensions [energy] · [time] and itsexperimental values are

1.054571726(47)× 10−34Joule · seconds = 6.58211928(15)× 10−16eV · seconds

(eV is the unit of “electron-Volt”, the energy acquired by an electron movingthrough a one-Volt electric potential). The most natural units to use for quan-tum mechanical problems would be energy and time units chosen so that ~ = 1.For instance one could use seconds for time and measure energies in the verysmall units of 6.6 × 10−16 eV, or use eV for energies, and then the very smallunits of 6.6× 10−16 seconds for time. Schrodinger’s equation implies that if oneis looking at a system where the typical energy scale is an eV, one’s state-vectorwill be changing on the very short time scale of 6.6 × 10−16 seconds. Whenwe do computations, usually we will just set ~ = 1, implicitly going to a unitsystem natural for quantum mechanics. When we get our final result, we caninsert appropriate factors of ~ to allow one to get answers in more conventionalunit systems.

It is sometimes convenient however to carry along factors of ~, since thiscan help make clear which terms correspond to classical physics behavior, andwhich ones are purely quantum mechanical in nature. Typically classical physicscomes about in the limit where

(energy scale)(time scale)

~is large. This is true for the energy and time scales encountered in everydaylife, but it can also always be achieved by taking ~ → 0, and this is what willoften be referred to as the “classical limit”.

1.2.2 Principles of measurement theory

The above axioms characterize the mathematical structure of a quantum theory,but they don’t address the “measurement problem”. This is the question ofhow to apply this structure to a physical system interacting with some sortof macroscopic, human-scale experimental apparatus that “measures” what isgoing on. This is highly thorny issue, requiring in principle the study of twointeracting quantum systems (the one being measured, and the measurementapparatus) in an overall state that is not just the product of the two states,but is highly “entangled” (for the meaning of this term, see chapter 9). Since amacroscopic apparatus will involve something like 1023 degrees of freedom, thisquestion is extremely hard to analyze purely within the quantum mechanicalframework (for one thing, one would need to solve a Schrodinger equation in1023 variables).

Instead of trying to resolve in general this problem of how classical physicsbehavior emerges for macroscopic objects, one can adopt the following two prin-ciples as describing what will happen, and these allow one to make precise sta-tistical predictions using quantum theory:

4

Principle (Observables). States where the value of an observable can be char-acterized by a well-defined number are the states that are eigenvectors for thecorresponding self-adjoint operator. The value of the observable in such a statewill be a real number, the eigenvalue of the operator.

This principle identifies the states we have some hope of sensibly associatinga label to (the eigenvalue), a label which in some contexts corresponds to anobservable quantity characterizing states in classical mechanics. The observablesof most use will turn out to correspond to some group action on the physicalsystem (for instance the energy, momentum, angular momentum, or charge).

Principle (The Born rule). Given an observable O and two unit-norm states|ψ1〉 and |ψ2〉 that are eigenvectors of O with eigenvalues λ1 and λ2 (i.e. O|ψ1〉 =λ1|ψ1〉 and O|ψ2〉 = λ2|ψ2〉), the complex linear combination state

c1|ψ1〉+ c2|ψ2〉

may not have a well-defined value for the observable O. If one attempts tomeasure this observable, one will get either λ1 or λ2, with probabilities

|c21||c21|+ |c22|

and|c22|

|c21|+ |c22|respectively.

The Born rule is sometimes raised to the level of an axiom of the theory, butit is plausible to expect that, given a full understanding of how measurementswork, it can be derived from the more fundamental axioms of the previoussection. Such an understanding though of how classical behavior emerges inexperiments is a very challenging topic, with the notion of “decoherence” playingan important role. See the end of this chapter for some references that discussthe these issues in detail.

Note that the state c|ψ〉 will have the same eigenvalues and probabilities asthe state |ψ〉, for any complex number c. It is conventional to work with statesof norm fixed to the value 1, which fixes the amplitude of c, leaving a remainingambiguity which is a phase eiθ. By the above principles this phase will notcontribute to the calculated probabilities of measurements. We will howevernot at all take the point of view that this phase information can be ignored. Itplays an important role in the mathematical structure, and the relative phaseof two different states certainly does affect measurement probabilities.

1.3 Unitary group representations

The mathematical framework of quantum mechanics is closely related to whatmathematicians describe as the theory of “unitary group representations”. We

5

will be examining this notion in great detail and working through many examplesin coming chapters, but here’s a quick summary of the relevant definitions, aswell as an indication of the relationship to the quantum theory formalism.

Definition (Group). A group G is a set with an associative multiplication, suchthat the set contains an identity element, as well as the multiplicative inverseof each element.

Many different kinds of groups are of interest in mathematics, with an ex-ample of the sort that we will be interested in the group of all rotations abouta point in 3-dimensional space. Most of the groups we will consider are “matrixgroups”, i.e. subgroups of the group of n by n invertible matrices (with real orcomplex coefficients).

Definition (Representation). A (complex) representation (π, V ) of a group Gis a homomorphism

π : g ∈ G→ π(g) ∈ GL(V )

where GL(V ) is the group of invertible linear maps V → V , with V a complexvector space.

Saying the map π is a homomorphism means

π(g1)π(g2) = π(g1g2)

for all g1, g2 ∈ G. When V is finite dimensional and we have chosen a basis ofV , then we have an identification of linear maps and matrices

GL(V ) ' GL(n,C)

where GL(n,C) is the group of invertible n by n complex matrices. We willbegin by studying representations that are finite dimensional and will try tomake rigorous statements. Later on we will get to representations on functionspaces, which are infinite dimensional, and from then on will need to consider theserious analytical difficulties that arise when one tries to make mathematicallyprecise statements in the infinite-dimensional case.

One source of confusion is that representations (π, V ) are sometimes referredto by the map π, leaving implicit the vector space V that the matrices π(g) acton, but at other times referred to by specifying the vector space V , leavingimplicit the map π. One reason for this is that the map π may be the identitymap: often G is a matrix group, so a subgroup of GL(n,C), acting on V ' Cn

by the standard action of matrices on vectors. One should keep in mind thoughthat just specifying V is generally not enough to specify the representation,since it may not be the standard one. For example, it could very well carry thetrivial representation, where

π(g) = 1n

i.e. each element of G acts on V as the identity.It turns out that in mathematics the most interesting classes of complex

representations are “unitary”, i.e. preserving the notion of length given by

6

the standard Hermitian inner product in a complex vector space. In physicalapplications, the group representations under consideration typically correspondto physical symmetries, and will preserve lengths in H, since these correspondto probabilities of various observations. We have the definition

Definition (Unitary representation). A representation (π, V ) on a complex vec-tor space V with Hermitian inner product 〈·, ·〉 is a unitary representation if itpreserves the inner product, i.e.

〈π(g)v1, π(g)v2〉 = 〈v1, v2〉

for all g ∈ G and v1, v2 ∈ V .

For a unitary representation, the matrices π(g) take values in a subgroupU(n) ⊂ GL(n,C). In our review of linear algebra we will see that U(n) can becharacterized as the group of n by n complex matrices U such that

U−1 = U†

where U† is the conjugate-transpose of U . Note that we’ll be using the notation“†” to mean the “adjoint” or conjugate-transpose matrix. This notation is prettyuniversal in physics, whereas mathematicians prefer to use “∗” instead of “†”.

1.4 Representations and quantum mechanics

The fundamental relationship between quantum mechanics and representationtheory is that whenever we have a physical quantum system with a group Gacting on it, the space of states H will carry a unitary representation of G (atleast up to a phase factor). For physicists working with quantum mechanics,this implies that representation theory provides information about quantummechanical state spaces. For mathematicians studying representation theory,this means that physics is a very fruitful source of unitary representations tostudy: any physical system with a symmetry group G will provide one.

For a representation π and group elements g that are close to the identity,one can use exponentiation to write π(g) ∈ GL(n,C) as

π(g) = eA

where A is also a matrix, close to the zero matrix.We will study this situation in much more detail and work extensively with

examples, showing in particular that if π(g) is unitary (i.e. in the subgroupU(n) ⊂ GL(n,C)), then A will be skew-adjoint:

A† = −A

where A† is the conjugate-transpose matrix. Defining B = iA, we find that Bis self-adjoint

B† = B

7

We thus see that, at least in the case of finite-dimensional H, the unitaryrepresentation π of G on H coming from a symmetry G of our physical sys-tem gives us not just unitary matrices π(g), but also corresponding self-adjointoperators B on H. Symmetries thus give us quantum mechanical observables,with the fact that these are self-adjoint linear operators corresponding to thefact that symmetries are realized as unitary representations on the state space.

In the following chapters we’ll see many examples of this phenomenon. Afundamental example that we will study in detail is that of time-translationsymmetry. Here the group G = R and we get a unitary representation ofR on the space of states H. The corresponding self-adjoint operator is theHamiltonian operator H. This unitary representation gives the dynamics of thetheory, with the Schrodinger equation just the statement that i

~H∆t is the skew-adjoint operator that gets exponentiated to give the unitary transformation thatmoves states ψ(t) ahead in time by an amount ∆t.

1.5 Symmetry groups and their representationson function spaces

It is conventional to refer to the groups that appear in this subject as “symmetrygroups”, which emphasizes the phenomenon of invariance of properties of objectsunder sets of transformations that form a group. This is a bit misleading though,since we are interested in not just invariance, but the more general phenomenonof groups acting on sets, according to the following definition:

Definition (Group action on a set). An action of a group G on a set M isgiven by a map

(g, x) ∈ G×M → g · x ∈M

such thatg1 · (g2 · x) = (g1g2) · x

ande · x = x

where e is the identity element of G

A good example to keep in mind is that of 3-dimensional space M = R3 withthe standard inner product. This comes with an action of the group G = R3 onX = M by translations, and of the group G′ = O(3) of 3-dimensional orthogonaltransformations (by rotations about the origin). Note that order matters: wewill often be interested in non-commutative groups like G′ where g1g2 6= g2g1

for some group elements g1, g2

A fundamental principle of modern mathematics is that the way to under-stand a space X, given as some set of points, is to look at Fun(M), the set offunctions on this space. This “linearizes” the problem, since the function spaceis a vector space, no matter what the geometrical structure of the original setis. If our original set has a finite number of elements, the function space will be

8

a finite dimensional vector space. In general though it will be infinite dimen-sional and we will need to further specify the space of functions (i.e. continuousfunctions, differentiable functions, functions with finite integral, etc.).

Given a group action of G on M , taking complex functions on M providesa representation (π, Fun(M)) of G, with π defined on functions f by

(π(g)f)(x) = f(g−1 · x)

Note the inverse that is needed to get the group homomorphism property towork since one has

(π(g1)π(g2)f)(x) = (π(g2)f)(g−11 · x)

= f(g−12 · (g−1

1 · x))

= f((g−12 g−1

1 ) · x)

= f((g1g2)−1 · x)

= π(g1g2)f(x)

This calculation would not work out properly for non-commutative G if onedefined (π(g)f)(x) = f(g · x).

One way to construct quantum mechanical state spaces H is as “wave-functions”, meaning complex-valued functions on space-time. The above showsthat given any group action on space-time, we get a representation π on thestate space H of such wave-functions.

Note that only in the case of M a finite set of points will we get a finite-dimensional representation this way, since only then will Fun(M) be a finite-dimensional vector space (C# of points in M). A good example to consider tounderstand this construction is the following:

• Take M to be a set of 3 elements x1, x2, x3. So Fun(M) = C3. Forf ∈ Fun(M), f is a vector in C3, with components (f(x1), f(x2), f(x3)).

• Take G = S3, the group of permutations of 3 elements. This group has3! = 6 elements.

• Take G to act on M by permuting the 3 elements.

(g, xi)→ g · xi

where i = 1, 2, 3 gives the three elements of M .

• Find the representation matrices π(g) for the representation of G onFun(M) as above

(π(g)f)(xi) = f(g−1 · xi)

This construction gives six 3 by 3 complex matrices, which under multiplicationof matrices satisfy the same relations as the elements of the group under groupmultiplication. In this particular case, all the entries of the matrix will be 0 or1, but that is special to the permutation representation.

9

The discussion here has been just a quick sketch of some of the ideas behindthe material we will cover in later chapters. These ideas will be examined inmuch greater detail, beginning with the next two chapters, where they willappear very concretely when we discuss the simplest possible quantum systems,those with one and two-complex dimensional state spaces.

1.6 For further reading

We will be approaching the subject of quantum theory from a different direc-tion than the conventional one, starting with the role of symmetry and with thesimplest possible finite-dimensional quantum systems, systems which are purelyquantum mechanical, with no classical analog. This means that the early dis-cussion one finds in most physics textbooks is rather different than the onehere. They will generally include the same fundamental principles describedhere, but often begin with the theory of motion of a quantized particle, tryingto motivate it from classical mechanics. The state space is then a space of wave-functions, which is infinite-dimensional and necessarily brings some analyticaldifficulties. Quantum mechanics is inherently a quite different conceptual struc-ture than classical mechanics. The relationship of the two subjects is rathercomplicated, but it is clear that quantum mechanics cannot be derived fromclassical mechanics, so attempts to motivate it that way are of necessity un-convincing, although they correspond to the very interesting historical story ofhow the subject evolved. We will come to the topic of the quantized motion ofa particle only in chapter 10, at which point it should become much easier tofollow the standard books.

There are many good physics quantum mechanics textbooks available, aimedat a wide variety of backgrounds, and a reader of this book should look for oneat an appropriate level to supplement the discussions here. One example wouldbe [53], which is not really an introductory text, but it includes the physicist’sversion of many of the standard calculations we will also be considering. Someuseful textbooks on the subject aimed at mathematicians are [13], [28], [29], [36],and [59]. The first few chapters of [20] provide an excellent while very concisesummary of both basic physics and quantum mechanics. One important topicwe won’t discuss is that of the application of the representation theory of finitegroups in quantum mechanics. For this as well as a discussion that overlapsquite a bit with the point of view of this course while emphasizing differentareas, see [55].

For the difficult issue of how measurements work and how classical physicsemerges from quantum theory, an important part of the story is the notion of“decoherence”. Good places to read about this are Wojciech Zurek’s updatedversion of his 1991 Physics Today article [73], as well as his more recent workon “quantum Darwinism” [74]. There is an excellent book on the subject bySchlosshauer [48] and for the details of what happens in real experimental setups,see the book by Haroche and Raimond [30]. For a review of how classicalphysics emerges from quantum written from the mathematical point of view,

10

see Landsman [33]. Finally, to get an idea of the wide variety of points of viewavailable on the topic of the “interpretation” of quantum mechanics, there’s avolume of interviews [49] with experts on the topic.

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12

Chapter 2

The Group U(1) and itsRepresentations

The simplest example of a Lie group is the group of rotations of the plane,with elements parametrized by a single number, the angle of rotation θ. It isuseful to identify such group elements with unit vectors in the complex plane,given by eiθ. The group is then denoted U(1), since such complex numberscan be thought of as 1 by 1 unitary matrices . We will see in this chapter howthe general picture described in chapter 1 works out in this simple case. Statespaces will be unitary representations of the group U(1), and we will see that anysuch representation decomposes into a sum of one-dimensional representations.These one-dimensional representations will be characterized by an integer q, andsuch integers are the eigenvalues of a self-adjoint operator we will call Q, whichis an observable of the quantum theory.

One motivation for the notation Q is that this is the conventional physicsnotation for electric charge, and this is one of the places where a U(1) groupoccurs in physics. Examples of U(1) groups acting on physical systems include:

• Quantum particles can be described by a complex-valued “wave-function”,and U(1) acts on such wavefunctions by phase transformations of thevalue of the function. This phenomenon can be used to understand howparticles interact with electromagnetic fields, and in this case the physicalinterpretation of the eigenvalues of the Q operator will be the electriccharge of the particle. We will discuss this in detail in chapter 40.

• If one chooses a particular direction in three-dimensional space, then thegroup of rotations about that axis can be identified with the group U(1).The eigenvalues of Q will have a physical interpretation as the quantumversion of angular momentum in the chosen direction. The fact that sucheigenvalues are not continuous, but integral, shows that quantum angularmomentum has quite different behavior than classical angular momentum.

• When we study the harmonic oscillator we will find that it has a U(1) sym-

13

metry (rotations in the position-momentum plane), and that the Hamil-tonian operator is a multiple of the operator Q for this case. This im-plies that the eigenvalues of the Hamiltonian (which give the energy ofthe system) will be integers times some fixed value. When one describesmulti-particle systems in terms of quantum fields one finds a harmonicoscillator for each momentum mode, and then the Q for that mode countsthe number of particles with that momentum.

We will sometimes refer to the operator Q as a “charge” operator, assigninga much more general meaning to the term than that of the specific example ofelectric charge. U(1) representations are also ubiquitous in mathematics, whereoften the integral eigenvalues of the Q operator will be called “weights”.

In a very real sense, the reason for the “quantum” in “quantum mechanics”is precisely because of the role of U(1) symmetries. Such symmetries implyobservables that characterize states by an integer eigenvalue of an operator Q,and it is this “quantization” of observables that motivates the name of thesubject.

2.1 Some representation theory

Recall the definition of a group representation:

Definition (Representation). A (complex) representation (π, V ) of a group Gon a complex vector space V (with a chosen basis identifying V ' Cn) is ahomomorphism

π : G→ GL(n,C)

This is just a set of n by n matrices, one for each group element, satisfyingthe multiplication rules of the group elements. n is called the dimension of therepresentation.

The groups G we are interested in will be examples of what mathematicianscall “Lie groups”. For those familiar with differential geometry, such groupsare examples of smooth manifolds. This means one can define derivatives offunctions on G and more generally the derivative of maps between Lie groups.We will assume that our representations are given by differentiable maps π.Some difficult general theory shows that considering the more general case ofcontinuous maps gives nothing new since the homomorphism property of thesemaps is highly constraining. In any case, our goal in this course will be to studyquite explicitly certain specific groups and representations which are central inquantum mechanics, and these representations will always be easily seen to bedifferentiable.

Given two representations one can form their direct sum:

Definition (Direct sum representation). Given representations π1 and π2 ofdimensions n1 and n2, one can define another representation, of dimensionn1 + n2 called the direct sum of the two representations, denoted by π1 ⊕ π2.

14

This representation is given by the homomorphism

(π1 ⊕ π2) : g ∈ G→(π1(g) 0

0 π2(g)

)In other words, one just takes as representation matrices block-diagonal

matrices with π1 and π2 giving the blocks.To understand the representations of a group G, one proceeds by first iden-

tifying the irreducible ones, those that cannot be decomposed into two repre-sentations of lower dimension:

Definition (Irreducible representation). A representation π is called irreducibleif it is not of the form π1⊕π2, for π1 and π2 representations of dimension greaterthan zero.

This criterion is not so easy to check, and the decomposition of an arbitraryreducible representation into irreducible components can be a very non-trivialproblem. Recall that one gets explicit matrices for the π(g) of a representation(π, V ) only when a basis for V is chosen. To see if the representation is reducible,one can’t just look to see if the π(g) are all in block-diagonal form. One needsto find out whether there is some basis for V with respect to which they areall in such form, something very non-obvious from just looking at the matricesthemselves.

Digression. Another approach to this would be to check to see if the represen-tation has no proper non-trivial sub-representations (subspaces of V preservedby the π(g)). This is not necessarily equivalent to our definition of irreducibil-ity (which is often called “indecomposability”), since a sub-representation mayhave no complement that is also a sub-representation. A simple example ofthis occurs for the action of upper triangular matrices on column vectors. Suchrepresentations are however non-unitary. In the unitary case indecomposabilityand irreducibility are equivalent. In these notes unless otherwise specified, oneshould assume that all representations are unitary, so the distinction betweenirreducibility and indecomposability will generally not arise.

The following theorem provides a criterion for determining if a representationis irreducible or not:

Theorem (Schur’s lemma). If a complex representation (π, V ) is irreducible,then the only linear maps M : V → V commuting with all the π(g) are λ1,multiplication by a scalar λ ∈ C.

Proof. Since we are working over the field C (this doesn’t work for R), we canalways solve the eigenvalue equation

det(M − λ1) = 0

to find the eigenvalues λ of M . The eigenspaces

Vλ = v ∈ V : Mv = λv

15

are non-zero vector subspaces of V and can also be described as ker(M − λ1),the kernel of the operator M−λ1. Since this operator and all the π(g) commute,we have

v ∈ ker(M − λ1) =⇒ π(g)v ∈ ker(M − λ1)

so ker(M − λ1) ⊂ V is a representation of G. If V is irreducible, we musthave either ker(M − λ1) = V or ker(M − λ1) = 0. Since λ is an eigenvalue,ker(M − λ1) 6= 0, so ker(M − λ1) = V and thus M = λ1 as a linear operatoron V .

More concretely Schur’s lemma says that for an irreducible representation, ifa matrix M commutes with all the representation matrices π(g), then M mustbe a scalar multiple of the unit matrix.

Note that the proof crucially uses the fact that one can solve the eigenvalueequation. This will only be true in general if one works with C and thus withcomplex representations. For the theory of representations on real vector spaces,Schur’s lemma is no longer true.

An important corollary of Schur’s lemma is the following characterization ofirreducible representations of G when G is commutative.

Theorem. If G is commutative, all of its irreducible representations are one-dimensional.

Proof. For G commutative, g ∈ G, any representation will satisfy

π(g)π(h) = π(h)π(g)

for all h ∈ G. If π is irreducible, Schur’s lemma implies that, since they commutewith all the π(g), the matrices π(h) are all scalar matrices, i.e. π(h) = λh1 forsome λh ∈ C. π is then irreducible exactly when it is the one-dimensionalrepresentation given by π(h) = λh.

2.2 The group U(1) and its representations

One can think of the group U(1) as the unit circle, with the multiplication ruleon its points given by addition of angles. More explicitly:

Definition (The group U(1)). The elements of the group U(1) are points onthe unit circle, which can be labeled by the unit complex number eiθ, for θ ∈ R.Note that θ and θ+N2π label the same group element for N ∈ Z. Multiplicationof group elements is just complex multiplication, which by the properties of theexponential satisfies

eiθ1eiθ2 = ei(θ1+θ2)

so in terms of angles the group law is just addition (mod 2π).

16

By our theorem from the last section, since U(1) is a commutative group,all irreducible representations will be one-dimensional. Such an irreducible rep-resentation will be given by a map

π : U(1)→ GL(1,C)

but an invertible 1 by 1 matrix is just an invertible complex number, and we willdenote the group of these as C∗. We will always assume that our representationsare given by differentiable maps, since we will often want to study them in termsof their derivatives. A differentiable map π that is a representation of U(1) mustsatisfy homomorphism and periodicity properties which can be used to show:

Theorem 2.1. All irreducible representations of the group U(1) are unitary,and given by

πk : θ ∈ U(1)→ πk(θ) = eikθ ∈ U(1) ⊂ GL(1,C) ' C∗

for k ∈ Z.

Proof. The given πk satisfy the homomorphism property

πk(θ1 + θ2) = πk(θ1)πk(θ2)

and periodicity property

πk(2π) = πk(0) = 1

We just need to show that any differentiable map

f : U(1)→ C∗

satisfying the homomorphism and periodicity properties is of this form. Com-puting the derivative f ′(θ) = df

dθ we find

f ′(θ) = lim∆θ→0

f(θ + ∆θ)− f(θ)

∆θ

= f(θ) lim∆θ→0

(f(∆θ)− 1)

∆θ(using the homomorphism property)

= f(θ)f ′(0)

Denoting the constant f ′(0) by C, the only solutions to this differential equationsatisfying f(0) = 1 are

f(θ) = eCθ

Requiring periodicity we find

f(2π) = eC2π = f(0) = 1

which implies C = ik for k ∈ Z, and f = πk for some integer k.

17

The representations we have found are all unitary, with πk taking values notjust in C∗, but in U(1) ⊂ C∗. One can check that the complex numbers eikθ

satisfy the condition to be a unitary 1 by 1 matrix, since

(eikθ)−1 = e−ikθ = eikθ

These representations are restrictions to the unit circle U(1) of the irre-ducible representations of the group C∗, which are given by

πk : z ∈ C∗ → πk(z) = zk ∈ C∗

Such representations are not unitary, but they have an extremely simple form,so it sometimes is convenient to work with them, later restricting to the unitcircle, where the representation is unitary.

Digression (Fourier analysis of periodic functions). We’ll discuss Fourier anal-ysis more seriously in chapter 10 when we come to the case of the translationgroups and of state-spaces that are spaces of “wave-functions” on space-time.For now though, it might be worth pointing out an important example of a rep-resentation of U(1): the space Fun(S1) of complex-valued functions on the circleS1. We will evade discussion here of the very non-trivial analysis involved, bynot specifying what class of functions we are talking about (e.g. continuous,integrable, differentiable, etc.). Periodic functions can be studied by rescalingthe period to 2π, thus looking at complex-valued functions of a real variable φsatisfying

f(φ+N2π) = f(φ)

for integer N , which we can think of as functions on a circle, parametrized byangle φ. We have an action of the group U(1) on the circle by rotation, withthe group element eiθ acting as:

φ→ φ+ θ

where φ is the angle parametrizing the circle S1.In chapter 1 we saw that given an action of a group on a space X, we can

“linearize” and get a representation (π, Fun(X)) of the group on the functionson the space, by taking

(π(g)f)(x) = f(g−1 · x)

for f ∈ Fun(X), x ∈ X. Here X = S1, the action is the rotation action and wefind

(π(θ)f)(φ) = f(φ− θ)since the inverse of a rotation by θ is a rotation by −θ.

This representation (π, Fun(S1)) is infinite-dimensional, but one can stillask how it decomposes into the one-dimensional irreducible representations (πk,C)of U(1). What we learn from the subject of Fourier analysis is that each (πk,C)occurs exactly once in the decomposition of Fun(S1) into irreducibles, i.e.

(π, Fun(S1)) =⊕

k∈Z(πk,C)

18

where we have matched the sin of not specifying the class of functions in Fun(S1)on the left-hand side with the sin of not explaining how to handle the infinite

direct sum⊕

on the right-hand side. What can be specified precisely is howthe irreducible sub-representation (πk,C) sits inside Fun(S1). It is the set offunctions f satisfying

(π(θ)f)(φ) = f(φ− θ) = eikθf(φ)

so explicitly given by the one-complex dimensional space of functions propor-tional to e−ikφ.

One part of the relevance of representation theory to Fourier analysis isthat the representation theory of U(1) picks out a distinguished basis of theinfinite-dimensional space of periodic functions by using the decomposition ofthe function space into irreducible representations. One can then effectivelystudy functions by expanding them in terms of their components in this specialbasis, writing an f ∈ Fun(S1) as

f(φ) =∑k∈Z

ckeikφ

for some complex coefficients ck, with analytical difficulties then appearing asquestions about the convergence of this series.

2.3 The charge operator

Recall from chapter 1 the claim of a general principle that, since the state spaceH is a unitary representation of a Lie group, we get an associated self-adjointoperator on H. We’ll now illustrate this for the simple case of G = U(1). For Hirreducible, the representation is one-dimensional, of the form (πq,C) for someq ∈ Z, and the self-adjoint operator will just be multiplication by the integer q.In general, we have

H = Hq1 ⊕Hq2 ⊕ · · · ⊕ Hqnfor some set of integers q1, q2, . . . , qn (n is the dimension of H, the qi may notbe distinct) and can define:

Definition. The charge operator Q for the U(1) representation (π,H) is theself-adjoint linear operator on H that acts by multiplication by qj on the irre-ducible representation Hqj . Taking basis elements in Hqj it acts on H as thematrix

q1 0 · · · 00 q2 · · · 0· · · · · ·0 0 · · · qn

Q is our first example of a quantum mechanical observable, a self-adjoint

operator on H. States in the subspaces Hqj will be eigenvectors for Q and will

19

have a well-defined numerical value for this observable, the integer qj . A generalstate will be a linear superposition of state vectors from different Hqj and therewill not be a well-defined numerical value for the observable Q on such a state.

From the action of Q onH, one can recover the representation, i.e. the actionof the symmetry group U(1) on H, by multiplying by i and exponentiating, toget

π(θ) = eiQθ =

eiq1θ 0 · · · 0

0 eiq2θ · · · 0· · · · · ·0 0 · · · eiqnθ

∈ U(n) ⊂ GL(n,C)

The standard physics terminology is that “Q generates the U(1) symmetrytransformations”.

The general abstract mathematical point of view (which we will discuss inmore detail later) is that the representation π is a map between manifolds,from the Lie group U(1) to the Lie group GL(n,C) that takes the identity ofU(1) to the identity of GL(n,C). As such it has a differential π′, which is amap from the tangent space at the identity of U(1) (which here is iR) to thetangent space at the identity of GL(n,C) (which is the space M(n,C) of n byn complex matrices). The tangent space at the identity of a Lie group is calleda “Lie algebra”. We will later study these in detail in many examples to come,including their role in specifying a representation.

Here the relation between the differential of π and the operator Q is

π′ : iθ ∈ iR→ π′(iθ) = iQθ

One can sketch the situation like this:

20

The right-hand side of the picture is supposed to somehow representGL(n,C),which is the 2n2 dimensional real vector space of n by n complex matrices, mi-nus the locus of matrices with zero determinant, which are those that can’t beinverted. It has a distinguished point, the identity. The derivative π′ of therepresentation map π is the linear operator iQ.

In this very simple example, this abstract picture is over-kill and likely con-fusing. We will see the same picture though occurring in many other examplesin later chapters, examples where the abstract technology is increasingly useful.Keep in mind that, just like in this U(1) case, the maps π will just be exponen-tial maps in the examples we care about, with very concrete incarnations givenby exponentiating matrices.

2.4 Conservation of charge

The way we have defined observable operators in terms a group representationon H, the action of these operators has nothing to do with the dynamics. Ifwe start at time t = 0 in a state in Hqj , with definite numerical value qj forthe observable, there is no reason that time evolution should preserve this.Recall from one of our basic axioms that time evolution of states is given by theSchrodinger equation

d

dt|ψ(t)〉 = −iH|ψ(t)〉

(we have set ~ = 1). We will later more carefully study the relation of thisequation to the symmetry of time translation (basically the Hamiltonian op-erator H generates an action of the group R of time translations, just as theoperator Q generates an action of the group U(1)). For now though, note thatfor time-independent Hamiltonian operators H, the solution to this equation isgiven by exponentiating H, with

|ψ(t)〉 = U(t)|ψ(0)〉

for

U(t) = e−itH

The commutator of two operators O1, O2 is defined by

[O1, O2] := O1O2 −O2O1

and such operators are said to commute if [O1, O2] = 0. If the Hamiltonianoperator H and the charge operator Q commute then Q will also commute withall powers of H

[Hk, Q] = 0

and thus with the exponential of H, so

[U(t), Q] = 0

21

This conditionU(t)Q = QU(t)

implies that if a state has a well-defined value qj for the observable Q at timet = 0, it will continue to have the same value at any other time t, since

Q|ψ(t)〉 = QU(t)|ψ(0)〉 = U(t)Q|ψ(0)〉 = U(t)qj |ψ(0)〉 = qj |ψ(t)〉

This will be a general phenomenon: if an observable commutes with the Hamil-tonian observable, we get a conservation law. This conservation law says thatif we start in a state with a well-defined numerical value for the observable (aneigenvector for the observable operator), we will remain in such a state, withthe value not changing, i.e. “conserved”.

2.5 Summary

To summarize the situation for G = U(1), we have found

• Irreducible representations π are one-dimensional and characterized bytheir derivative π′ at the identity. If G = R, π′ could be any complexnumber. If G = U(1), periodicity requires that π′ must be iq, q ∈ Z, soirreducible representations are labeled by an integer.

• An arbitrary representation π of U(1) is of the form

π(eiθ) = eiθQ

where Q is a matrix with eigenvalues a set of integers qj . For a quan-tum system, Q is the self-adjoint observable corresponding to the U(1)symmetry of the system, and is said to be a “generator” of the symmetry.

• If [Q,H] = 0, the qj will be “conserved quantities”, numbers that char-acterize the quantum states, and do not change as the states evolve intime.


I’ve had trouble finding another source that covers the material here. Mostquantum mechanics books consider it somehow too trivial to mention, startingtheir discussion of symmetries with more complicated examples.

22

Chapter 3

Two-state Systems andSU(2)

The simplest truly non-trivial quantum systems have state spaces that are in-herently two-complex dimensional. This provides a great deal more structurethan that seen in chapter 2, which could be analyzed by breaking up the spaceof states into one-dimensional subspaces of given charge. We’ll study these two-state systems in this section, encountering for the first time the implications ofworking with representations of non-commutative groups. Since they give thesimplest non-trivial realization of many quantum phenomena, such systems arethe fundamental objects of quantum information theory (the “qubit”) and thefocus of attempts to build a quantum computer (which would be built out ofmultiple copies of this sort of fundamental object). Many different possible two-state quantum systems could potentially be used as the physical implementationof a qubit.

One of the simplest possibilities to take would be the idealized situationof a single electron, somehow fixed so that its spatial motion could be ignored,leaving its quantum state described just by its so-called “spin degree of freedom”,which takes values in H = C2. The term “spin” is supposed to call to mindthe angular momentum of an object spinning about about some axis, but suchclassical physics has nothing to do with the qubit, which is a purely quantumsystem.

In this chapter we will analyze what happens for general quantum systemswith H = C2 by first finding the possible observables. Exponentiating thesewill give the group U(2) of unitary 2 by 2 matrices acting on H = C2. Thisis a specific representation of U(2), the “defining” representation. By restrict-ing to the subgroup SU(2) ⊂ U(2) of elements of determinant one, we get arepresentation of SU(2) on C2 often called the “spin 1/2” representation.

Later on, in chapter 8, we will find all the irreducible representations ofSU(2). These are labeled by a natural number

N = 0, 1, 2, 3, . . .

23

and have dimension N + 1. The corresponding quantum systes are said to have“spin N/2”. The case N = 0 is the trivial representation on C and the caseN = 1 is the case of this chapter. In the limit N → ∞ one can make contactwith classical notions of spinning objects and angular momentum, but the spin1/2 case is at the other limit, where the behavior is purely quantum-mechanical.

3.1 The two-state quantum system

3.1.1 The Pauli matrices: observables of the two-statequantum system

For a quantum system with two-dimensional state space H = C2, observablesare self-adjoint linear operators on C2. With respect to a chosen basis of C2,these are 2 by 2 complex matricesM satisfying the conditionM = M† (M† is theconjugate transpose of M). Any such matrix will be a (real) linear combinationof four matrices:

M = c01 + c1σ1 + c2σ2 + c3σ3

with cj ∈ R and the standard choice of basis elements given by

1 =

(1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)The σj are called the “Pauli matrices” and are a pretty universal choice of basisin this subject. This choice of basis is a convention, with one aspect of thisconvention that of taking the basis element in the 3-direction to be diagonal.In common physical situations and conventions, the third direction is the dis-tinguished “up-down” direction in space, so often chosen when a distinguisheddirection in R3 is needed.

Recall that the basic principle of how measurements are supposed to workin quantum theory says that the only states that have well-defined values forthese four observables are the eigenvectors for these matrices. The first matrixgives a trivial observable (the identity on every state), whereas the last one, σ3,has the two eigenvectors

σ3

(10

)=

(10

)and

σ3

(01

)= −

(01

)with eigenvalues +1 and −1. In quantum information theory, where this isthe qubit system, these two eigenstates are labeled |0〉 and |1〉 because of theanalogy with a classical bit of information. Later on when we get to the theory ofspin, we will see that 1

2σ3 is the observable corresponding to the SO(2) = U(1)symmetry group of rotations about the third spatial axis, and the eigenvalues

24

− 12 ,+

12 of this operator will be used to label the two eigenstates

|+ 1

2〉 =

(10

)and | − 1

2〉 =

(01

)The two eigenstates |+ 1

2 〉 and | − 12 〉 provide a basis for C2, so an arbitrary

vector in H can be written as

|ψ〉 = α|+ 1

2〉+ β| − 1

2〉

for α, β ∈ C. Only if α or β is 0 does the observable σ3 correspond to a well-defined number that characterizes the state and can be measured. This will beeither 1

2 (if β = 0 so the state is an eigenvector |+ 12 〉), or − 1

2 (if α = 0 so thestate is an eigenvector | − 1

2 〉).An easy to check fact is that |+ 1

2 〉 and | − 12 〉 are NOT eigenvectors for the

operators σ1 and σ2. One can also check that no pair of the three σj commute,which implies that one cannot find vectors that are simultaneous eigenvectors formore than one σj . This non-commutativity of the operators is responsible for thecharacteristic classically paradoxical property of quantum observables: one canfind states with a well defined number for the measured value of one observableσj , but such states will not have a well-defined number for the measured valueof the other two non-commuting observables. The physical description of thisphenomenon in the realization of this system as a spin 1

2 particle is that if oneprepares states with a well-defined spin component in the j-direction, the twoother components of the spin can’t be assigned a numerical value in such astate. Any attempt to prepare states that simultaneously have specific chosennumerical values for the 3 observables corresponding to the σj is doomed. So isany attempt to simultaneously measure such values: if one measures the valuefor a particular observable σj , then going on to measure one of the other twowill ensure that the first measurement is no longer valid (repeating it will notnecessarily give the same thing). There are many subtleties in the theory ofmeasurement for quantum systems, but this simple two-state example alreadyshows some of the main features of how the behavior of observables is quitedifferent than in classical physics.

The choice we have made for the σj corresponds to a choice of basis for Hsuch that the basis vectors are eigenvectors of σ3. σ1 and σ2 take these basisvectors to non-trivial linear combinations of basis vectors. It turns out thatthere are two specific linear combinations of σ1 and σ2 that do something verysimple to the basis vectors, since

(σ1 + iσ2) =

(0 20 0

)and (σ1 − iσ2) =

(0 02 0

)we have

(σ1 + iσ2)

(01

)= 2

(10

)(σ1 + iσ2)

(10

)=

(00

)25

and

(σ1 − iσ2)

(10

)= 2

(01

)(σ1 − iσ2)

(01

)=

(00

)(σ1 + iσ2) is called a “raising operator”: on eigenvectors of σ3 it either

increases the eigenvalue by 2, or annihilates the vector. (σ1 − iσ2) is calleda “lowering operator”: on eigenvectors of σ3 it either decreases the eigenvalueby 2, or annihilates the vector. Note that these linear combinations are notself-adjoint, (σ1 + iσ2) is the adjoint of (σ1 − iσ2) and vice-versa.

3.1.2 Exponentials of Pauli matrices: unitary transforma-tions of the two-state system

We saw in chapter 2 that in the U(1) case, knowing the observable operator Q onH determined the representation of U(1), with the representation matrices foundby exponentiating iθQ. Here we will find the representation corresponding tothe two-state system observables by exponentiating the observables in a similarway.

Taking the the identity matrix first, multiplication by iθ and exponentiationgives the diagonal unitary matrix

eiθ1 =

(eiθ 00 eiθ

)This is just exactly the case studied in chapter 2, for a U(1) group acting onH = C2, with

Q =

(1 00 1

)This matrix commutes with any other 2 by 2 matrix, so we can treat its actionon H independently of the action of the σj .

Turning to the other three basis elements of the space of observables, thePauli matrices, it turns out that since all the σj satisfy σ2

j = 1, their exponentialsalso take a simple form.

eiθσj = 1 + iθσj +1

2(iθ)2σ2

j +1

3!(iθ)3σ3

j + · · ·

= 1 + iθσj −1

2θ21− i 1

3!θ3σj + · · ·

= (1− 1

2!θ2 + · · · )1 + i(θ − 1

3!θ3 + · · · )σj

= (cos θ)1 + iσj(sin θ) (3.1)

As θ goes from θ = 0 to θ = 2π, this exponential traces out a circle in thespace of unitary 2 by 2 matrices, starting and ending at the unit matrix. Thiscircle is a group, isomorphic to U(1). So, we have found three different U(1)

26

subgroups inside the unitary 2 by 2 matrices, but only one of them (the casej = 3) will act diagonally on H, with the U(1) representation determined by

Q =

(1 00 −1

)For the other two cases j = 1 and j = 2, by a change of basis one could puteither one in the same diagonal form, but doing this for one value of j makesthe other two no longer diagonal. All three values of j need to be treatedsimultaneously, and one needs to consider not just the U(1)s but the group onegets by exponentiating general linear combinations of Pauli matrices.

To compute such exponentials, one can check that these matrices satisfy thefollowing relations, useful in general for doing calculations with them instead ofmultiplying out explicitly the 2 by 2 matrices:

[σj , σk]+ = σjσk + σkσj = 2δjk1

Here [·, ·]+ is the anti-commutator. This relation says that all σj satisfy σ2j = 1

and distinct σj anti-commute (e.g. σjσk = −σkσj for j 6= k).Notice that the anti-commutation relations imply that, if we take a vector

v = (v1, v2, v3) ∈ R3 and define a 2 by 2 matrix by

v · σ = v1σ1 + v2σ2 + v3σ3 =

(v3 v1 − iv2

v1 + iv2 −v3

)then taking powers of this matrix we find

(v · σ)2 = (v21 + v2

2 + v23)1 = |v|21

If v is a unit vector, we have

(v · σ)n =

1 n even

(v · σ) n odd

Replacing σj by v · σ, the same calculation as for equation 3.1 gives (for va unit vector)

eiθv·σ = (cos θ)1 + i(sin θ)v · σ

Notice that one can easily compute the inverse of this matrix:

(eiθv·σ)−1 = (cos θ)1− i(sin θ)v · σ

since

((cos θ)1 + i(sin θ)v · σ)((cos θ)1− i(sin θ)v · σ) = (cos2 θ + sin2 θ)1 = 1

We’ll review linear algebra and the notion of a unitary matrix in chapter 4, butone form of the condition for a matrix M to be unitary is

M† = M−1

27

so the self-adjointness of the σj implies unitarity of eiθv·σ since

(eiθv·σ)† = ((cos θ)1 + i(sin θ)v · σ)†

= ((cos θ)1− i(sin θ)v · σ†)= ((cos θ)1− i(sin θ)v · σ)

= (eiθv·σ)−1

One can also easily compute the determinant of eiθv·σ, finding

det(eiθv·σ) = det((cos θ)1 + i(sin θ)v · σ)

= det

(cos θ + i sin θv3 i sin θ(v1 − iv2)i sin θ(v1 + iv2) cos θ − i sin θv3

)= cos2 θ + sin2 θ(v2

1 + v22 + v2

3)

= 1

So, we see that by exponentiating i times linear combinations of the self-adjoint Pauli matrices (which all have trace zero), we get unitary matrices ofdeterminant one. These are invertible, and form the group named SU(2), thegroup of unitary 2 by 2 matrices of determinant one. If we exponentiated notjust iθv · σ, but i(φ1 + θv · σ) for some real constant φ (such matrices will nothave trace zero unless φ = 0), we would get a unitary matrix with determinantei2φ. The group of unitary 2 by 2 matrices with arbitrary determinant is calledU(2). It contains as subgroups SU(2) as well as the U(1) described at thebeginning of this section. U(2) is slightly different than the product of thesetwo subgroups, since the group element(

−1 00 −1

)is in both subgroups. In our review of linear algebra to come we will encounterthe generalization to SU(n) and U(n), groups of unitary n by n complex ma-trices.

To get some more insight into the structure of the group SU(2), consider anarbitrary 2 by 2 complex matrix (

α βγ δ

)Unitarity implies that the rows are orthonormal. One can see this explicitlyfrom the condition that the matrix times its conjugate-transpose is the identity(

α βγ δ

)(α γ

β δ

)=

(1 00 1

)Orthogonality of the two rows gives the relation

γα+ δβ = 0 =⇒ δ = −γαβ

28

The condition that the first row has length one gives

αα+ ββ = |α|2 + |β|2 = 1

Using these two relations and computing the determinant (which has to be 1)gives

αδ − βγ = −ααγβ− βγ = −γ

β(αα+ ββ) = −γ

β= 1

so one must haveγ = −β, δ = α

and an SU(2) matrix will have the form(α β

−β α

)where (α, β) ∈ C2 and

|α|2 + |β|2 = 1

So, the elements of SU(2) are parametrized by two complex numbers, withthe sum of their length-squareds equal to one. Identifying C2 = R4, these arejust vectors of length one in R4. Just as U(1) could be identified as a space withthe unit circle S1 in C = R2, SU(2) can be identified with the unit three-sphereS3 in R4.

3.2 Commutation relations for Pauli matrices

An important set of relations satisfied by Pauli matrices are their commutationrelations:

[σj , σk] = σjσk − σkσj = 2i

3∑l=1

εjklσl

where εjkl satisfies ε123 = 1, is antisymmetric under permutation of two of itssubscripts, and vanishes if two of the subscripts take the same value. Moreexplicitly, this says:

[σ1, σ2] = 2iσ3, [σ2, σ3] = 2iσ1, [σ3, σ1] = 2iσ2

One can easily check these relations by explicitly computing with the matrices.Putting together the anticommutation and commutation relations, one gets aformula for the product of two Pauli matrices:

σjσk = δjk1 + i

3∑l=1

εjklσl

While physicists prefer to work with self-adjoint Pauli matrices and theirreal eigenvalues, one can work instead with the following skew-adjoint matrices

Xj = −iσj2

29

which satisfy the slightly simpler commutation relations

[Xj , Xk] =

3∑l=1

εjklXl

or more explicitly

[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

If these commutators were zero, the SU(2) elements one gets by exponentiat-ing linear combinations of the Xj would be commuting group elements. Thenon-triviality of the commutators reflects the non-commutativity of the group.Group elements U ∈ SU(2) near the identity satisfy

U ' 1 + ε1X1 + ε2X2 + ε3X2

for εj small and real, just as group elements z ∈ U(1) near the identity satisfy

z ' 1 + iε

One can think of the Xj and their commutation relations as an infinites-imal version of the full group and its group multiplication law, valid nearthe identity. In terms of the geometry of manifolds, recall that SU(2) is thespace S3. The Xj give a basis of the tangent space R3 to the identity ofSU(2), just as i gives a basis of the tangent space to the identity of U(1).

30

3.3 Dynamics of a two-state system

Recall that the time dependence of states in quantum mechanics is given by theSchrodinger equation

d

dt|ψ(t)〉 = −iH|ψ(t)〉

where H is a particular self-adjoint linear operator on H, the Hamiltonian op-erator. The most general such operator on C2 will be given by

H = h01 + h1σ1 + h2σ2 + h3σ3

for four real parameters h0, h1, h2, h3. The solution to the Schrodinger equationis just given by exponentiation:

|ψ(t)〉 = U(t)|ψ(0)〉

whereU(t) = e−itH

The h01 term in H just contributes an overall phase factor e−ih0t, with theremaining factor of U(t) an element of the group SU(2) rather than the largergroup U(2) of all 2 by 2 unitaries.

Using our earlier equation

eiθv·σ = (cos θ)1 + i(sin θ)v · σ

valid for a unit vector v, our U(t) is given by taking h = (h1, h2, h3), v = h|h|

and θ = −t|h|, so we find

U(t) =e−ih0t(cos(−t|h|)1 + i sin(−t|h|)h1σ1 + h2σ2 + h3σ3

|h|)

=e−ih0t(cos(t|h|)1− i sin(t|h|)h1σ1 + h2σ2 + h3σ3

|h|)

=

(e−ih0t(cos(t|h|)− i h3

|h| sin(t|h|)) −i sin(t|h|)h1−ih2

|h|−i sin(t|h|)h1+ih2

|h| e−ih0t(cos(t|h|) + i h3

|h| sin(t|h|))

)In the special case h = (0, 0, h3) we have

U(t) =

(e−it(h0+h3) 0

0 e−it(h0−h3)

)so if our initial state is

|ψ(0)〉 = α|+ 1

2〉+ β| − 1

2〉

for α, β ∈ C, at later times the state will be

|ψ(t)〉 = αe−it(h0+h3)|+ 1

2〉+ βe−it(h0−h3)| − 1

2〉

31

In this special case, one can see that the eigenvalues of the Hamiltonian areh0 ± h3.

In the physical realization of this system by a spin 1/2 particle (ignoring itsspatial motion), the Hamiltonian is given by

H =ge

4mc(B1σ1 +B2σ2 +B3σ3)

where the Bj are the components of the magnetic field, and the physical con-stants are the gyromagnetic ratio (g), the electric charge (e), the mass (m) andthe speed of light (c), so we have solved the problem of the time evolution ofsuch a system, setting hj = ge

4mcBj . For magnetic fields of size |B| in the 3-direction, we see that the two different states with well-defined energy (| + 1

2 〉and | − 1

2 〉) will have an energy difference between them of

ge

2mc|B|

This is known as the Zeeman effect and is readily visible in the spectra of atomssubjected to a magnetic field. We will consider this example in more detail inchapter 7, seeing how the group of rotations of R3 appears. Much later, inchapter 40, we will derive this Hamiltonian term from general principles of howelectromagnetic fields couple to such spin 1/2 particles.


Many quantum mechanics textbooks now begin with the two-state system, giv-ing a much more detailed treatment than the one given here, including muchmore about the physical interpretation of such systems (see for example [62]).Volume III of Feynman’s Lectures on Physics [16] is a quantum mechanics textwith much of the first half devoted to two-state systems. The field of “QuantumInformation Theory” gives a perspective on quantum theory that puts such sys-tems (in this context called the “qubit”) front and center. One possible referencefor this material is John Preskill’s notes on quantum computation [44].

32

Chapter 4

Linear Algebra Review,Unitary and OrthogonalGroups

A significant background in linear algebra will be assumed in later chapters,and we’ll need a range of specific facts from that subject. These will includesome aspects of linear algebra not emphasized in a typical linear algebra course,such as the role of the dual space and the consideration of various classes ofinvertible matrices as defining a group. For now our vector spaces will be finite-dimensional. Later on we will come to state spaces that are infinite dimensional,and will address the various issues that this raises at that time.

4.1 Vector spaces and linear maps

A vector space V over a field k is just a set such that one can consistently takelinear combinations of elements with coefficients in k. We will only be usingthe cases k = R and k = C, so such finite-dimensional V will just be Rn orCn. Choosing a basis (set of n linearly independent vectors) ej, an arbitraryvector v ∈ V can be written as

v = v1e1 + v2e2 + · · ·+ vnen

giving an explicit identification of V with n-tuples vj of real or complex numberswhich we will usually write as column vectors

v =

v1

v2

...vn

33

The choice of a basis ej also allows us to express the action of a linearoperator Ω on V

Ω : v ∈ V → Ωv ∈ Vas multiplication by an n by n matrix:

v1

v2

...vn

→

Ω11 Ω12 . . . Ω1n

Ω21 Ω22 . . . Ω2n

......

......

Ωn1 Ωn2 . . . Ωnn

v1

v2

...vn

The invertible linear operators on V form a group under composition, a

group we will sometimes denote GL(V ). Choosing a basis identifies this groupwith the group of invertible matrices, with group law matrix multiplication.For V n-dimensional, we will denote this group by GL(n,R) in the real case,GL(n,C) in the complex case.

Note that when working with vectors as linear combinations of basis vectors,we can use matrix notation to write a linear transformation as

v → Ωv =(e1 · · · en

)

Ω11 Ω12 . . . Ω1n

Ω21 Ω22 . . . Ω2n

......

......

Ωn1 Ωn2 . . . Ωnn

v1

v2

...vn

One sees from this that we can think of the transformed vector as we did

above in terms of transformed coefficients vj with respect to fixed basis vectors,but also could leave the vj unchanged and transform the basis vectors. At timeswe will want to use matrix notation to write formulas for how the basis vectorstransform in this way, and then will write

e1

e2

...en

→

Ω11 Ω21 . . . Ωn1

Ω12 Ω22 . . . Ωn2

......

......

Ω1n Ω2n . . . Ωnn

e1

e2

...en

Note that putting the basis vectors ej in a column vector like this causes thematrix for Ω to act on them by the transposed matrix. This is not a group actionsince in general the product of two transposed matrices is not the transpose ofthe product.

4.2 Dual vector spaces

To any vector space V one can associate a new vector space, its dual:

Definition (Dual vector space). Given a vector space V over a field k, the dualvector space V ∗ is the set of all linear maps V → k, i.e.

V ∗ = l : V → k such that l(αv + βw) = αl(v) + βl(w)

34

for α, β ∈ k, v, w ∈ V .

Given a linear transformation Ω acting on V , one can define:

Definition (Transpose transformation). The transpose of Ω is the linear trans-formation

ΩT : V ∗ → V ∗

that satisfies(ΩT l)(v) = l(Ωv)

for l ∈ V ∗, v ∈ V .

For any representation (π, V ) of a group G on V , one can define a corre-sponding representation on V ∗

Definition (Dual or contragredient representation). The dual or contragredientrepresentation on V ∗ is given by taking as linear operators

(πT )−1(g) : V ∗ → V ∗

These satisfy the homomorphism property since

(πT (g1))−1(πT (g2))−1 = (πT (g2)πT (g1))−1 = ((π(g1)π(g2))T )−1

For any choice of basis ej of V , one has a dual basis e∗j of V ∗ thatsatisfies

e∗j (ek) = δjk

Coordinates on V with respect to a basis are linear functions, and thus elementsof V ∗. One can identify the coordinate function vj with the dual basis vectore∗j since

e∗j (v1e1 + v2e2 + · · ·+ vnen) = vj

One can easily show that the elements of the matrix for Ω in the basis ejare given by

Ωjk = e∗j (Ωek)

and that the matrix for the transpose map (with respect to the dual basis) isjust the matrix transpose

(ΩT )jk = Ωkj

One can use matrix notation to write elements

l = l1e∗1 + l2e

∗2 + · · ·+ lne∗n ∈ V ∗

of V ∗ as row vectors (l1 l2 · · · ln

)of coordinates on V ∗. Then evaluation of l on a vector v given by matrixmultiplication

l(v) =(l1 l2 · · · ln

)v1

v2

...vn

= l1v1 + l2v2 + · · ·+ lnvn

35

4.3 Change of basis

Any invertible transformation A on V can be used to change the basis ej of Vto a new basis e′j by taking

ej → e′j = Aej

The matrix for a linear transformation Ω transforms under this change of basisas

Ωjk = e∗j (Ωek)→ (e′j)∗(Ωe′k) =(Aej)

∗(ΩAek)

=(AT )−1(e∗j )(ΩAek)

=e∗j (A−1ΩAek)

=(A−1ΩA)jk

In the second step we are using the fact that elements of the dual basis transformas the dual representation. One can check that this is what is needed to ensurethe relation

(e′j)∗(e′k) = δjk

The change of basis formula shows that if two matrices Ω1 and Ω2 are relatedby conjugation by a third matrix A

Ω2 = A−1Ω1A

then one can think of them as both representing the same linear transforma-tion, just with respect to two different choices of basis. Recall that a finite-dimensional representation is given by a set of matrices π(g), one for each groupelement. If two representations are related by

π2(g) = A−1π1(g)A

(for all g, A does not depend on g), then we can think of them as being thesame representation, with different choices of basis. In such a case the represen-tations π1 and π2 are called “equivalent”, and we will often implicitly identifyrepresentations that are equivalent.

4.4 Inner products

An inner product on a vector space V is an additional structure that providesa notion of length for vectors, of angle between vectors, and identifies V ∗ ' V .One has, in the real case:

Definition (Inner Product, real case). An inner product on a real vector spaceV is a map

〈·, ·〉 : V × V → R

that is linear in both variables and symmetric (〈v, w〉 = 〈w, v〉).

36

Our inner products will usually be positive-definite (〈v, v〉 ≥ 0 and 〈v, v〉 =0 =⇒ v = 0), with indefinite inner products only appearing in the con-text of special or general relativity, where an indefinite inner product on four-dimensional space-time is used.

In the complex case, one has

Definition (Inner Product, complex case). An Hermitian inner product on acomplex vector space V is a map

〈·, ·〉 : V × V → C

that is conjugate symmetric

〈v, w〉 = 〈w, v〉

as well as linear in the second variable, and antilinear in the first variable: forα ∈ C and u, v, w ∈ V

〈u+ v, w〉 = 〈u,w〉+ 〈v, w〉, 〈αu, v〉 = α〈u, v〉

An inner product gives a notion of length || · || for vectors, with

||v||2 = 〈v, v〉

Note that whether to specify anti-linearity in the first or second variable is amatter of convention. The choice we are making is universal among physicists,with the opposite choice common among mathematicians.

An inner product also provides an (anti-linear in the complex case) isomor-phism V ' V ∗ by the map

v ∈ V → lv ∈ V ∗

where lv is defined bylv(w) = 〈v, w〉

Physicists have a useful notation for elements of vector space and their duals,for the case when V is a complex vector space with an Hermitian inner product(such as the state space for a quantum theory). An element of such a vectorspace V is written as a “ket vector”

|v〉

where v is a label for a vector. An element of the dual vector space V ∗ is writtenas a “bra vector”

〈l|

Evaluating l ∈ V ∗ on v ∈ V gives an element of C, written

〈l|v〉

37

If Ω : V → V is a linear map

〈l|Ω|v〉 = 〈l|Ωv〉 = l(Ωv)

In the bra-ket notation, one denotes the dual vector lv by 〈v|. Note that inthe inner product the angle bracket notation means something different than inthe bra-ket notation. The similarity is intentional though, since in the bra-ketnotation one has

〈v|w〉 = 〈v, w〉Note that our convention of linearity in the second variable of the inner product,anti-linearity in the first, implies

|αv〉 = α|v〉, 〈αv| = α〈v|

for α ∈ C.For a choice of orthonormal basis ej, i.e. satisfying

〈ej , ek〉 = δjk

a useful notation is|j〉 = ej

Because of orthonormality, coefficients of vectors can be calculated as

vj = 〈ej , v〉

In bra-ket notation we havevj = 〈j|v〉

and

|v〉 =

n∑j=1

|j〉〈j|v〉

For corresponding elements of V ∗, one has (using anti-linearity)

〈v| =n∑j=1

vj〈j| =n∑j=1

〈v|j〉〈j|

With respect to the chosen orthonormal basis ej, one can represent vectorsv as column vectors and the operation of taking a vector |v〉 to a dual vector 〈v|corresponds to taking a column vector to the row vector that is its conjugate-transpose.

〈v| =(v1 v2 · · · vn

)Then one has

〈v|w〉 =(v1 v2 · · · vn

)w1

w2

...wn

= v1w1 + v2w2 + · · ·+ vnwn

38

If Ω is a linear operator Ω : V → V , then with respect to the chosen basis itbecomes a matrix with matrix elements

Ωkj = 〈k|Ωj〉

The decomposition of a vector v in terms of coefficients

|v〉 =

n∑j=1

|j〉〈j|v〉

can be interpreted as a matrix multiplication by the identity matrix

1 =

n∑j=1

|j〉〈j|

and this kind of expression is referred to by physicists as a “completeness rela-tion”, since it requires that the set of |j〉 be a basis with no missing elements.The operator

Pj = |j〉〈j|

is called the projection operator onto the j’th basis vector, it corresponds to thematrix that has 0s everywhere except in the jj component.

Digression. In this course, all our indices will be lower indices. One way tokeep straight the difference between vectors and dual vectors is to use upperindices for components of vectors, lower indices for components of dual vectors.This is quite useful in Riemannian geometry and general relativity, where theinner product is given by a metric that can vary from point to point, causingthe isomorphism between vectors and dual vectors to also vary. For quantummechanical state spaces, we will be using a single, standard, fixed inner product,so there will be a single isomorphism between vectors and dual vectors. Thebra-ket notation will take care of the notational distinction between vectors anddual vectors as necessary.

4.5 Adjoint operators

When V is a vector space with inner product, one can define the adjoint of Ωby

Definition (Adjoint Operator). The adjoint of a linear operator Ω : V → V isthe operator Ω† satisfying

〈Ωv, w〉 = 〈v,Ω†w〉

or, in bra-ket notation〈Ωv|w〉 = 〈v|Ω†w〉

for all v, w ∈ V .

39

Generalizing the fact that

〈αv| = α〈v|

for α ∈ C, one can write〈Ωv| = 〈v|Ω†

Note that mathematicians tend to favor Ω∗ as notation for the adjoint of Ω,as opposed to the physicist’s notation Ω† that we are using.

In terms of explicit matrices, since 〈Ωv| is the conjugate-transpose of |Ωv〉,the matrix for Ω† will be given by the conjugate transpose ΩT of the matrix forΩ:

Ω†jk = Ωkj

In the real case, the matrix for the adjoint is just the transpose matrix. Wewill say that a linear transformation is self-adjoint if Ω† = Ω, skew-adjoint ifΩ† = −Ω.

4.6 Orthogonal and unitary transformations

A special class of linear transformations will be invertible transformations thatpreserve the inner product, i.e. satisfying

〈Ωv,Ωw〉 = 〈Ωv|Ωw〉 = 〈v, w〉 = 〈v|w〉

for all v, w ∈ V . Such transformations take orthonormal bases to orthonormalbases, so they will appear in one role as change of basis transformations.

In terms of adjoints, this condition becomes

〈Ωv,Ωw〉 = 〈v,Ω†Ωw〉 = 〈v, w〉

soΩ†Ω = 1

or equivalentlyΩ† = Ω−1

In matrix notation this first condition becomesn∑k=1

(Ω†)jkΩkl =

n∑k=1

ΩkjΩkl = δjl

which says that the column vectors of the matrix for Ω are orthonormal vectors.Using instead the equivalent condition

ΩΩ† = 1

one finds that the row vectors of the matrix for Ω are also orthornormal.Since such linear transformations preserving the inner product can be com-

posed and are invertible, they form a group, and some of the basic examples ofLie groups are given by these groups for the cases of real and complex vectorspaces.

40

4.6.1 Orthogonal groups

We’ll begin with the real case, where these groups are called orthogonal groups:

Definition (Orthogonal group). The orthogonal group O(n) in n-dimensionsis the group of invertible transformations preserving an inner product on a realn-dimensional vector space V . This is isomorphic to the group of n by n realinvertible matrices Ω satisfying

Ω−1 = ΩT

The subgroup of O(n) of matrices with determinant 1 (equivalently, the subgrouppreserving orientation of orthonormal bases) is called SO(n).

Recall that for a representation π of a group G on V , one has a dual repre-sentation on V ∗ given by taking the transpose-inverse of π. If G is an orthogonalgroup, then π and its dual are the same matrices, with V identified by V ∗ bythe inner product.

Since the determinant of the transpose of a matrix is the same as the deter-minant of the matrix, we have

Ω−1Ω = 1 =⇒ det(Ω−1)det(Ω) = det(ΩT )det(Ω) = (det(Ω))2 = 1

sodet(Ω) = ±1

O(n) is a continuous Lie group, with two components: SO(n), the subgroup oforientation-preserving transformations, which include the identity, and a com-ponent of orientation-changing transformations.

The simplest non-trivial example is for n = 2, where all elements of SO(2)are given by matrices of the form(

cos θ − sin θsin θ cos θ

)These matrices give counter-clockwise rotations in R2 by an angle θ. The othercomponent of O(2) will be given by matrices of the form(

cos θ sin θsin θ − cos θ

)Note that the group SO(2) is isomorphic to the group U(1) by(


)⇔ eiθ

so the representation theory of SO(2) is just as for U(1), with irreducible com-plex representations one-dimensional and classified by an integer.

In chapter 6 we will consider in detail the case of SO(3), which is crucialfor physical applications because it is the group of rotations in the physicalthree-dimensional space.

41

4.6.2 Unitary groups

In the complex case, groups of invertible transformations preserving the Hermi-tian inner product are called unitary groups:

Definition (Unitary group). The unitary group U(n) in n-dimensions is thegroup of invertible transformations preserving an Hermitian inner product on acomplex n-dimensional vector space V . This is isomorphic to the group of n byn complex invertible matrices satisfying

Ω−1 = ΩT

= Ω†

The subgroup of U(n) of matrices with determinant 1 is called SU(n).

In the unitary case, the dual of a representation π has representation matricesthat are transpose-inverses of those for π, but

(π(g)T )−1 = π(g)

so the dual representation is given by conjugating all elements of the matrix.The same calculation as in the real case here gives

det(Ω−1)det(Ω) = det(Ω†)det(Ω) = det(Ω)det(Ω) = |det(Ω)|2 = 1

so det(Ω) is a complex number of modulus one. The map

Ω ∈ U(n)→ det(Ω) ∈ U(1)

is a group homomorphism.We have already seen the examples U(1), U(2) and SU(2). For general

values of n, the case of U(n) can be split into the study of its determinant,which lies in U(1) so is easy to deal with, and the subgroup SU(n), which is amuch more complicated story.

Digression. Note that is not quite true that the group U(n) is the productgroup SU(n) × U(1). If one tries to identify the U(1) as the subgroup of U(n)of elements of the form eiθ1, then matrices of the form

eimn 2π1

for m an integer will lie in both SU(n) and U(1), so U(n) is not a product ofthose two groups.

We saw at the end of section 3.1.2 that SU(2) can be identified with the three-sphere S3, since an arbitrary group element can be constructed by specifying onerow (or one column), which must be a vector of length one in C2. For the casen = 3, the same sort of construction starts by picking a row of length one in C3,which will be a point in S5. The second row must be orthornormal, and one canshow that the possibilities lie in a three-sphere S3. Once the first two rows arespecified, the third row is uniquely determined. So as a manifold, SU(3) is eight-dimensional, and one might think it could be identified with S5×S3. It turns out

42

that this is not the case, since the S3 varies in a topologically non-trivial wayas one varies the point in S5. As spaces, the SU(n) are topologically “twisted”products of odd-dimensional spheres, providing some of the basic examples ofquite non-trivial topological manifolds.

4.7 Eigenvalues and eigenvectors

We have seen that that the matrix for a linear transformation Ω of a vectorspace V changes by conjugation when we change our choice of basis of V . Toget basis-independent information about Ω, one considers the eigenvalues of thematrix. Complex matrices behave in a much simpler fashion than real matrices,since in the complex case the eigenvalue equation

det(λ1− Ω) = 0

can always be factored into linear factors, and solved for the eigenvalues λ. Foran arbitrary n by n complex matrix there will be n solutions (counting repeatedeigenvalues with multiplicity). One can always find a basis for which the matrixwill be in upper triangular form.

The case of self-adjoint matrices Ω is much more constrained, since transpo-sition relates matrix elements. One has:

Theorem (Spectral theorem for self-adjoint matrices). Given a self-adjointcomplex n by n matrix Ω, one can always find a unitary matrix U such that

UΩU−1 = D

where D is a diagonal matrix with entries Djj = λj , λj ∈ R.

Given Ω, one finds the eigenvalues λj by solving the eigenvalue equation. Onecan then go on to solve for the eigenvectors and use these to find U . For distincteigenvalues one finds that the corresponding eigenvectors are orthogonal.

This theorem is of crucial importance in quantum mechanics, where for Ω anobservable, the eigenvectors are the states in the state space with well-definednumerical values characterizing the state, and these numerical values are theeigenvalues. The theorem also tells us that given an observable, we can useit to choose distinguished orthonormal bases for the state space by picking abasis of eigenvectors, normalized to length one. This is a theorem about finite-dimensional vector spaces, but later on in the course we will see that somethingsimilar will be true even in the case of infinite-dimensional state spaces.

One can also diagonalize unitary matrices themselves by conjugation byanother unitary. The diagonal entries will all be complex numbers of unit length,so of the form eiλj , λj ∈ R.

For the simplest examples, consider the cases of the groups SU(2) and U(2).Any matrix in U(2) can be conjugated by a unitary matrix to the diagonalmatrix (

eiλ1 00 eiλ2

)43

which is the exponential of a corresponding diagonalized skew-adjoint matrix(iλ1 00 iλ2

)For matrices in the subgroup SU(2), one has λ2 = −λ1 = λ so in diagonal forman SU(2) matrix will be (

eiλ 00 e−iλ

)which is the exponential of a corresponding diagonalized skew-adjoint matrixthat has trace zero (

iλ 00 −iλ

)


Almost any of the more advanced linear algebra textbooks should cover thematerial of this chapter.

44

Chapter 5

Lie Algebras and LieAlgebra Representations

For a groupG we have defined unitary representations (π, V ) for finite-dimensionalvector spaces V of complex dimension n as homomorphisms

π : G→ U(n)

Recall that in the case of G = U(1), we could use the homomorphism propertyof π to determine π in terms of its derivative at the identity. This turns out tobe a general phenomenon for Lie groups G: we can study their representationsby considering the derivative of π at the identity, which we will call π′. Becauseof the homomorphism property, knowing π′ is often sufficient to characterizethe representation π it comes from. π′ is a linear map from the tangent spaceto G at the identity to the tangent space of U(n) at the identity. The tangentspace to G at the identity will carry some extra structure coming from the groupmultiplication, and this vector space with this structure will be called the Liealgebra of G.

The subject of differential geometry gives many equivalent ways of definingthe tangent space at a point of manifolds like G, but we do not want to enterhere into the subject of differential geometry in general. One of the standarddefinitions of the tangent space is as the space of tangent vectors, with tangentvectors defined as the possible velocity vectors of parametrized curves g(t) inthe group G.

More advanced treatments of Lie group theory develop this point of view(see for example [64]) which applies to arbitrary Lie groups, whether or notthey are groups of matrices. In our case though, since we are interested inspecific groups that are explicitly given as groups of matrices, we can give amore concrete definition, just using the exponential map on matrices. For amore detailed exposition of this subject, using the same concrete definition ofthe Lie algebra in terms of matrices, see Brian Hall’s book [26] or the abbreviatedon-line version [27].

45

Note that the material of this chapter is quite general, and may be hardto make sense of until one has some experience with basic examples. The nextchapter will discuss in detail the groups SU(2) and SO(3) and their Lie algebras,as well as giving some examples of their representations, and this may be helpfulin making sense of the general theory of this chapter.

5.1 Lie algebras

We’ll work with the following definition of a Lie algebra:

Definition (Lie algebra). For G a Lie group of n by n invertible matrices, theLie algebra of G (written Lie(G) or g) is the space of n by n matrices X suchthat etX ∈ G for t ∈ R.

Notice that while the group G determines the Lie algebra g, the Lie algebradoes not determine the group. For example, O(n) and SO(n) have the sametangent space at the identity, and thus the same Lie algebra, but elements inO(n) not in the component of the identity can’t be written in the form etX

(since then you could make a path of matrices connecting such an element tothe identity by shrinking t to zero). Note also that, for a given X, differentvalues of t may give the same group element, and this may happen in differentways for different groups sharing the same Lie algebra. For example, considerG = U(1) and G = (R,+), which both have the same Lie algebra g = R. In thefirst case an infinity of values of t give the same group element, in the second,only one does. In the next chapter we’ll see a more subtle example of this:SU(2) and SO(3) are different groups with the same Lie algebra.

We have G ⊂ GL(n,C), and X ∈ M(n,C), the space of n by n complexmatrices. For all t ∈ R, the exponential etX is an invertible matrix (with inversee−tX), so in GL(n,C). For each X, we thus have a path of elements of GL(n,C)going through the identity matrix at t = 0, with velocity vector

d

dtetX = XetX

which takes the value X at t = 0:

d

dt(etX)|t=0 = X

To calculate this derivative, just use the power series expansion for the expo-nential, and differentiate term-by-term.

For the case G = GL(n,C), we just have gl(n,C) = M(n,C), which is alinear space of the right dimension to be the tangent space to G at the identity,so this definition is consistent with our general motivation. For subgroups G ⊂GL(n,C) given by some condition (for example that of preserving an innerproduct), we will need to identify the corresponding condition on X ∈M(n,C)and check that this defines a linear space.

The existence of such a linear space g ⊂ M(n,C) will provide us with adistinguished representation, called the “adjoint representation”

46

Definition (Adjoint representation). The adjoint representation (Ad, g) is givenby the homomorphism

Ad : g ∈ G→ Ad(g) ∈ GL(g)

where Ad(g) acts on X ∈ g by

(Ad(g))(X) = gXg−1

To show that this is well-defined, one needs to check that gXg−1 ∈ g whenX ∈ g, but this can be shown using the identity

etgXg−1

= getXg−1

which implies that etgXg−1 ∈ G if etX ∈ G. To check this, just expand the

exponential and use

(gXg−1)k = (gXg−1)(gXg−1) · · · (gXg−1) = gXkg−1

It is also easy to check that this is a homomorphism, with

Ad(g1)Ad(g2) = Ad(g1g2)

A Lie algebra g is not just a real vector space, but comes with an extrastructure on the vector space

Definition (Lie bracket). The Lie bracket operation on g is the bilinear anti-symmetric map given by the commutator of matrices

[·, ·] : (X,Y ) ∈ g× g→ [X,Y ] = XY − Y X ∈ g

We need to check that this is well-defined, i.e. that it takes values in g.

Theorem. If X,Y ∈ g, [X,Y ] = XY − Y X ∈ g.

Proof. Since X ∈ g, we have etX ∈ G and we can act on Y ∈ g by the adjointrepresentation

Ad(etX)Y = etXY e−tX ∈ g

As t varies this gives us a parametrized curve in g. Its velocity vector will alsobe in g, so

d

dt(etXY e−tX) ∈ g

One has (by the product rule, which can easily be shown to apply in this case)

d

dt(etXY e−tX) = (

d

dt(etXY ))e−tX + etXY (

d

dte−tX)

= XetXY e−tX − etXY Xe−tX

Evaluating this at t = 0 givesXY − Y X

which is thus shown to be in g.

47

The relationd

dt(etXY e−tX)|t=0 = [X,Y ] (5.1)

used in this proof will be continually useful in relating Lie groups and Lie alge-bras.

To do calculations with a Lie algebra, one can just choose a basisX1, X2, . . . , Xn

for the vector space g, and use the fact that the Lie bracket can be written interms of this basis as

[Xj , Xk] =

n∑l=1

cjklXl

where cjkl is a set of constants known as the “structure constants” of the Liealgebra. For example, in the case of su(2), the Lie algebra of SU(2) one has abasis X1, X2, X3 satisfying

[Xj , Xk] =

3∑l=1

εjklXl

so the structure constants of su(2) are just the totally anti-symmetric εjkl.

5.2 Lie algebras of the orthogonal and unitarygroups

The groups we are most interested in are the groups of linear transformationspreserving an inner product: the orthogonal and unitary groups. We have seenthat these are subgroups of GL(n,R) or GL(n,C), consisting of those elementsΩ satisfying the condition

ΩΩ† = 1

In order to see what this condition becomes on the Lie algebra, write Ω = etX ,for some parameter t, and X a matrix in the Lie algebra. Since the transpose ofa product of matrices is the product (order-reversed) of the transposed matrices,i.e.

(XY )T = Y TXT

and the complex conjugate of a product of matrices is the product of the complexconjugates of the matrices, one has

(etX)† = etX†

The condition

ΩΩ† = 1

thus becomes

etX(etX)† = etXetX†

= 1

48

Taking the derivative of this equation gives

etXX†etX†

+XetXetX†

= 0

Evaluating this at t = 0 one finds

X +X† = 0

so the matrices we want to exponentiate are skew-adjoint, satisfying

X† = −X

Note that physicists often choose to define the Lie algebra in these casesas self-adjoint matrices, then multiplying by i before exponentiating to get agroup element. We will not use this definition, with one reason that we want tothink of the Lie algebra as a real vector space, so want to avoid an unnecessaryintroduction of complex numbers at this point.

5.2.1 Lie algebra of the orthogonal group

Recall that the orthogonal group O(n) is the subgroup of GL(n,R) of matricesΩ satisfying ΩT = Ω−1. We will restrict attention to the subgroup SO(n) ofmatrices with determinant 1 which is the component of the group containingthe identity, and thus elements that can be written as

Ω = etX

These give a path connecting Ω to the identity (taking esX , s ∈ [0, t]). Wesaw above that the condition ΩT = Ω−1 corresponds to skew-symmetry of thematrix X

XT = −XSo in the case of G = SO(n), we see that the Lie algebra so(n) is the space ofskew-symmetric (XT = −X) n by n real matrices, together with the bilinear,antisymmetric product given by the commutator:

(X,Y ) ∈ so(n)× so(n)→ [X,Y ] ∈ so(n)

The dimension of the space of such matrices will be

1 + 2 + · · ·+ (n− 1) =n2 − n

2

and a basis will be given by the matrices εjk, with j, k = 1, . . . , n, j < k definedas

(εjk)lm =

−1 if j = l, k = m

+1 if j = m, k = l

0 otherwise

(5.2)

In chapter 6 we will examine in detail the n = 3 case, where the Lie algebraso(3) is R3, realized as the space of anti-symmetric real 3 by 3 matrices.

49

5.2.2 Lie algebra of the unitary group

For the case of the group U(n), the group is connected and one can write allgroup elements as etX , where now X is a complex n by n matrix. The unitaritycondition implies that X is skew-adjoint (also called skew-Hermitian), satisfying

X† = −X

So the Lie algebra u(n) is the space of skew-adjoint n by n complex matrices,together with the bilinear, antisymmetric product given by the commutator:

(X,Y ) ∈ u(n)× u(n)→ [X,Y ] ∈ u(n)

Note that these matrices form a subspace of Cn2

of half the dimension,so of real dimension n2. u(n) is a real vector space of dimension n2, but itis NOT a space of real n by n matrices. It is the space of skew-Hermitianmatrices, which in general are complex. While the matrices are complex, onlyreal linear combinations of skew-Hermitian matrices are skew-Hermitian (recallthat multiplication by i changes a skew-Hermitian matrix into a Hermitianmatrix). Within this space of complex matrices, if one looks at the subspaceof real matrices one gets the sub-Lie algebra so(n) of anti-symmetric matrices(the Lie algebra of SO(n) ⊂ U(n)). If we take all complex linear combinationsof skew-Hermitian matrices, then we get M(n,C), all n by n complex matrices.

There is an identity relating the determinant and the trace of a matrix

det(eX) = etrace(X)

which can be proved by conjugating the matrix to upper-triangular form andusing the fact that the trace and the determinant of a matrix are conjugation-invariant. Since the determinant of an SU(n) matrix is 1, this shows that theLie algebra su(n) of SU(n) will consist of matrices that are not only skew-Hermitian, but also of trace zero. So in this case su(n) is again a real vectorspace, of dimension n2 − 1.

One can show that U(n) and u(n) matrices can be diagonalized by conju-gation by a unitary matrix to show that any U(n) matrix can be written as anexponential of something in the Lie algebra. The corresponding theorem is alsotrue for SO(n) but requires looking at diagonalization into 2 by 2 blocks. It isnot true for O(n) (you can’t reach the disconnected component of the identityby exponentiation). It also turns out to not be true for the groups GL(n,R)and GL(n,C) for n ≥ 2.

5.3 Lie algebra representations

We have defined a group representation as a homomorphism (a map of groupspreserving group multiplication)

π : G→ GL(n,C)

50

We can similarly define a Lie algebra representation as a map of Lie algebraspreserving the Lie bracket:

Definition (Lie algebra representation). A (complex) Lie algebra representation(φ, V ) of a Lie algebra g on an n-dimensional complex vector space V is givenby a linear map

φ : X ∈ g→ φ(X) ∈ gl(n,C) = M(n,C)

satisfying

φ([X,Y ]) = [φ(X), φ(Y )]

Such a representation is called unitary if its image is in u(n), i.e. it satisfies

φ(X)† = −φ(X)

More concretely, given a basis X1, X2, . . . , Xd of a Lie algebra g of dimensiond with structure constants cjkl, a representation is given by a choice of d complexn-dimensional matrices φ(Xj) satisfying the commutation relations

[φ(Xj), φ(Xk)] =

d∑l=1

cjklφ(Xl)

The representation is unitary when the matrices are skew-adjoint.

The notion of a Lie algebra is motivated by the fact that the homomorphismproperty causes the map π to be largely determined by its behavior infinitesi-mally near the identity, and thus by the derivative π′. One way to define thederivative of such a map is in terms of velocity vectors of paths, and this sort ofdefinition in this case associates to a representation π : G→ GL(n,C) a linearmap

π′ : g→M(n,C)

where

π′(X) =d

dt(π(etX))|t=0

51

In the case of U(1) we classified all irreducible representations (homomor-phisms U(1) → GL(1,C) = C∗) by looking at the derivative of the map atthe identity. For general Lie groups G, one can do something similar, show-ing that a representation π of G gives a representation of the Lie algebra (bytaking the derivative at the identity), and then trying to classify Lie algebrarepresentations.

Theorem. If π : G→ GL(n,C) is a group homomorphism, then

π′ : X ∈ g→ π′(X) =d

dt(π(etX))|t=0 ∈ gl(n,C) = M(n,C)

satisfies

1.

π(etX) = etπ′(X)

2. For g ∈ G

π′(gXg−1) = π(g)π′(X)(π(g))−1

3. π′ is a Lie algebra homomorphism:

π′([X,Y ]) = [π′(X), π′(Y )]

52

Proof. 1. We have

d

dtπ(etX) =

d

dsπ(e(t+s)X)|s=0

=d

dsπ(etXesX)|s=0

= π(etX)d

dsπ(esX)|s=0

= π(etX)π′(X)

So f(t) = π(etX) satisfies the differential equation ddtf = fπ′(X) with

initial condition f(0) = 1. This has the unique solution f(t) = etπ′(X)

2. We have

etπ′(gXg−1) = π(etgXg

−1

)

= π(getXg−1)

= π(g)π(etX)π(g)−1

= π(g)etπ′(X)π(g)−1

Differentiating with respect to t at t = 0 gives

π′(gXg−1) = π(g)π′(X)(π(g))−1

3. Recall that (5.1)

[X,Y ] =d

dt(etXY e−tX)|t=0

so

π′([X,Y ]) = π′(d

dt(etXY e−tX)|t=0)

=d

dtπ′(etXY e−tX)|t=0 (by linearity)

=d

dt(π(etX)π′(Y )π(e−tX))|t=0 (by 2.)

=d

dt(etπ

′(X)π′(Y )e−tπ′(X))|t=0 (by 1.)

= [π′(X), π′(Y )]

This theorem shows that we can study Lie group representations (π, V )by studying the corresponding Lie algebra representation (π′, V ). This willgenerally be much easier since the π′(X) are just linear maps. We will proceedin this manner in chapter 8 when we construct and classify all SU(2) and SO(3)representations, finding that the corresponding Lie algebra representations aremuch simpler to analyze.

53

For any Lie group G, we have seen that there is a distinguished representa-tion, the adjoint representation (Ad, g). The corresponding Lie algebra represen-tation is also called the adjoint representation, but written as (Ad′, g) = (ad, g).From the fact that

Ad(etX)(Y ) = etXY e−tX

we can differentiate with respect to t to get the Lie algebra representation

ad(X)(Y ) =d

dt(etXY e−tX)|t=0 = [X,Y ] (5.3)

From this we see that one can define

Definition (Adjoint Lie algebra representation). ad is the Lie algebra repre-sentation given by

X ∈ g→ ad(X)

where ad(X) is defined as the linear map from g to itself given by

Y → [X,Y ]

Note that this linear map ad(X), which one can write as [X, ·], can bethought of as the infinitesimal version of the conjugation action

(·)→ etX(·)e−tX

The Lie algebra homomorphism property of ad says that

ad([X,Y ]) = ad(X) ad(Y )− ad(Y ) ad(X)

where these are linear maps on g, with composition of linear maps, so operatingon Z ∈ g we have

ad([X,Y ])(Z) = (ad(X) ad(Y )((Z)− (ad(Y ) ad(X))(Z)

Using our expression for ad as a commutator, we find

[[X,Y ], Z] = [X, [Y,Z]]− [Y, [X,Z]]

This is called the Jacobi identity. It could have been more simply derived asan identity about matrix multiplication, but here we see that it is true for amore abstract reason, reflecting the existence of the adjoint representation. Itcan be written in other forms, rearranging terms using antisymmetry of thecommutator, with one example the sum of cyclic permutations

[[X,Y ], Z] + [[Z,X], Y ] + [[Y,Z], X] = 0

One can define Lie algebras much more abstractly as follows

54

Definition (Abstract Lie algebra). An abstract Lie algebra over a field k is avector space A over k, with a bilinear operation

[·, ·] : (X,Y ) ∈ A×A→ [X,Y ] ∈ A

satisfying

1. Antisymmetry:[X,Y ] = −[Y,X]

2. Jacobi identity:

[[X,Y ], Z] + [[Z,X], Y ] + [[Y, Z], X] = 0

Such Lie algebras do not need to be defined as matrices, and their Lie bracketoperation does not need to be defined in terms of a matrix commutator (al-though the same notation continues to be used). Later on in this course wewill encounter important examples of Lie algebras that are defined in this moreabstract way.

5.4 Complexification

The way we have defined a Lie algebra g, it is a real vector space, not a complexvector space. Even if G is a group of complex matrices, when it is not GL(n,C)itself but some subgroup, its tangent space at the identity will not necessarilybe a complex vector space. Consider for example the cases G = U(1) andG = SU(2), where u(1) = R and su(2) = R3. While the tangent space to thegroup of all invertible complex matrices is a complex vector space, imposingsome condition such as unitarity picks out a subspace which generally is just areal vector space, not a complex one. So the adjoint representation (Ad, g) is ingeneral not a complex representation, but a real representation, with

Ad(g) ∈ GL(g) = GL(dim g,R)

The derivative of this is the Lie algebra representation

ad : X ∈ g→ ad(X) ∈ gl(dim g,R)

and once we pick a basis of g, we can identify gl(dim g,R) = M(dim g,R). So,for each X ∈ g we get a real linear operator on a real vector space.

We would however often like to work with not real representations, butcomplex representations, since it is for these that Schur’s lemma applies, andrepresentation operators can be diagonalized. To get from a real Lie algebrarepresentation to a complex one, we can “complexify”, extending the action ofreal scalars to complex scalars. If we are working with real matrices, complex-ification is nothing but allowing complex entries and using the same rules formultiplying scalars as before.

More generally, for any real vector space we can define:

55

Definition. The complexification VC of a real vector space V is the space ofpairs (v1, v2) of elements of V with multiplication by a+ bi ∈ C given by

(a+ ib)(v1, v2) = (av1 − bv2, av2 + bv1)

One should think of the complexification of V as

VC = V + iV

with v1 in the first copy of V , v2 in the second copy. Then the rule for mul-tiplication by a complex number comes from the standard rules for complexmultiplication. In the cases we will be interested in this level of abstraction isnot really needed, since V will be given as a subspace of a complex space, and VCwill just be the larger subspace you get by taking complex linear combinationsof elements of V .

Given a real Lie algebra g, the complexification gC is pairs of elements(X1, X2) of g, with the above rule for multiplication by complex scalars. TheLie bracket on g extends to a Lie bracket on gC by the rule

[(X1, X2), (Y1, Y2)] = ([X1, X2]− [Y1, Y2], [X1, Y2] + [X2, Y1])

and gC is a Lie algebra over the complex numbers. In many cases this definitionis isomorphic to something just defined in terms of complex matrices, with thesimplest case

gl(n,R)C = gl(n,C)

Recalling our discussion of u(n), a real Lie algebra, with elements certain (skew-Hermitian) complex matrices, one can see that complexifying will just give allcomplex matrices so

u(n)C = gl(n,C)

This example shows that two different real Lie algebras may have the samecomplexification. For yet another example, since so(n) is the Lie algebra ofall real anti-symmetric matrices, so(n)C is the Lie algebra of all complex anti-symmetric matrices.

We can extend the operators ad(X) on g by complex linearity to turn adinto a complex representation of gC on the vector space gC itself

ad : Z ∈ gC → ad(Z)

Here Z is now a complex linear combination of elements of Xj ∈ g, and ad(Z)is the corresponding complex linear combination of the real matrices ad(Xj).


The material of this section is quite conventional mathematics, with many goodexpositions, although most aimed at a higher level than this course. An exampleat the level of this course is the book Naive Lie Theory [56]. It covers basics ofLie groups and Lie algebras, but without representations. The notes [27] andbook [26] of Brian Hall are a good source to study from. Some parts of theproofs given here are drawn from those notes.

56

Chapter 6

The Rotation and SpinGroups in 3 and 4Dimensions

Among the basic symmetry groups of the physical world is the orthogonal groupSO(3) of rotations about a point in three-dimensional space. The observablesone gets from this group are the components of angular momentum, and under-standing how the state space of a quantum system behaves as a representationof this group is a crucial part of the analysis of atomic physics examples andmany others. This is a topic one will find in some version or other in everyquantum mechanics textbook.

Remarkably, it turns out that the quantum systems in nature are oftenrepresentations not of SO(3), but of a larger group called Spin(3), one that hastwo elements corresponding to every element of SO(3). Such a group exists inany dimension n, always as a “doubled” version of the orthogonal group SO(n),one that is needed to understand some of the more subtle aspects of geometryin n dimensions. In the n = 3 case it turns out that Spin(3) ' SU(2) and wewill study in detail the relationship of SO(3) and SU(2). This appearance ofthe unitary group SU(2) is special to geometry in 3 and 4 dimensions, and wewill see that quaternions provide an explanation for this.

6.1 The rotation group in three dimensions

In R2 rotations about the origin are given by elements of SO(2), with a counter-clockwise rotation by an angle θ given by the matrix

R(θ) =

(cos θ − sin θsin θ cos θ

)57

This can be written as an exponential, R(θ) = eθL = cos θ1 + L sin θ for

L =

(0 −11 0

)Here SO(2) is a commutative Lie group with Lie algebra so(2) = R (it is one-dimensional, with trivial Lie bracket, all elements of the Lie algebra commute).Note that we have a representation on V = R2 here, but it is a real representa-tion, not one of the complex ones we have when we have a representation on aquantum mechanical state space.

In three dimensions the group SO(3) is 3-dimensional and non-commutative.Choosing a unit vector w and angle θ, one gets an element R(θ,w) of SO(3),rotation by θ about the w axis. Using standard basis vectors ej , rotations aboutthe coordinate axes are given by

R(θ, e1) =

1 0 00 cos θ − sin θ0 sin θ cos θ

, R(θ, e2) =

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

R(θ, e3) =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

A standard parametrization for elements of SO(3) is in terms of 3 “Euler angles”φ, θ, ψ with a general rotation given by

R(φ, θ, ψ) = R(ψ, e3)R(θ, e1)R(φ, e3)

i.e. first a rotation about the z-axis by an angle φ, then a rotation by an angleθ about the new x-axis, followed by a rotation by ψ about the new z-axis.Multiplying out the matrices gives a rather complicated expression for a rotationin terms of the three angles, and one needs to figure out what range to choosefor the angles to avoid multiple counting.

The infinitesimal picture near the identity of the group, given by the Liealgebra structure on so(3), is much easier to understand. Recall that for orthog-onal groups the Lie algebra can be identified with the space of anti-symmetricmatrices, so one in this case has a basis

l1 =

0 0 00 0 −10 1 0

l2 =

0 0 10 0 0−1 0 0

l3 =

0 −1 01 0 00 0 0

which satisfy the commutation relations

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

Note that these are exactly the same commutation relations satisfied bythe basis vectors X1, X2, X3 of the Lie algebra su(2), so so(3) and su(2) are

58

isomorphic Lie algebras. They both are the vector space R3 with the same Liebracket operation on pairs of vectors. This operation is familiar in yet anothercontext, that of the cross-product of standard basis vectors ej in R3:

e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2

We see that the Lie bracket operation

(X,Y ) ∈ R3 ×R3 → [X,Y ] ∈ R3

that makes R3 a Lie algebra so(3) is just the cross-product on vectors in R3.So far we have three different isomorphic ways of putting a Lie bracket on

R3, making it into a Lie algebra:

1. Identify R3 with anti-symmetric real 3 by 3 matrices and take the matrixcommutator as Lie bracket.

2. Identify R3 with skew-adjoint, traceless, complex 2 by 2 matrices and takethe matrix commutator as Lie bracket.

3. Use the vector cross-product on R3 to get a Lie bracket, i.e. define

[v,w] = v ×w

Something very special that happens for orthogonal groups only in three di-mensions is that the vector representation (the defining representation of SO(n)matrices on Rn) is isomorphic to the adjoint representation. Recall that anyLie group G has a representation (Ad, g) on its Lie algebra g. so(n) can beidentified with the anti-symmetric n by n matrices, so is of (real) dimensionn2−n

2 . Only for n = 3 is this equal to n, the dimension of the representationon vectors in Rn. This corresponds to the geometrical fact that only in 3 di-mensions is a plane (in all dimensions rotations are built out of rotations invarious planes) determined uniquely by a vector (the vector perpendicular tothe plane). Equivalently, only in 3 dimensions is there a cross-product v × wwhich takes two vectors determining a plane to a unique vector perpendicularto the plane.

The isomorphism between the vector representation (πvector,R3) on column

vectors and the adjoint representation (Ad, so(3)) on antisymmetric matrices isgiven by v1

v2

v3

↔ v1l1 + v2l2 + v3l3 =

0 −v3 v2

v3 0 −v1

−v2 v1 0

or in terms of bases by

ej ↔ lj

For the vector representation on column vectors, πvector(g) = g and π′vector(X) =X, where X is an antisymmetric 3 by 3 matrix, and g = eX is an orthogonal 3by 3 matrix. Both act on column vectors by the usual multiplication.

59

For the adjoint representation on antisymmetric matrices, one has

Ad(g)

0 −v3 v2

v3 0 −v1

−v2 v1 0

= g

0 −v3 v2

v3 0 −v1

−v2 v1 0

g−1

The corresponding Lie algebra representation is given by

ad(X)

0 −v3 v2

v3 0 −v1

−v2 v1 0

= [X,

0 −v3 v2

v3 0 −v1

−v2 v1 0

]

where X is a 3 by 3 antisymmetric matrix.One can explicitly check that these representations are isomorphic, for in-

stance by calculating how basis elements lj ∈ so(3) act. On vectors, these lj actby matrix multiplication, giving for instance, for j = 1

l1e1 = 0, l1e2 = e3, l1e3 = −e2

On antisymmetric matrices one has instead the isomorphic relations

(ad(l1))(l1) = 0, (ad(l1))(l2) = l3, (ad(l1))(l3) = −l2

6.2 Spin groups in three and four dimensions

A remarkable property of the orthogonal groups SO(n) is that they come with anassociated group, called Spin(n), with every element of SO(n) correspondingto two distinct elements of Spin(n). If you have seen some topology, whatis at work here is that (for n > 2) the fundamental group of SO(n) is non-trivial, with π1(SO(n)) = Z2 (this means there is a non-contractible loop inSO(n), contractible if you go around it twice). Spin(n) is topologically thesimply-connected double-cover of SO(n), and one can choose the covering mapΦ : Spin(n)→ SO(n) to be a group homomorphism. Spin(n) is a Lie group ofthe same dimension as SO(n), with an isomorphic tangent space at the identity,so the Lie algebras of the two groups are isomorphic: so(n) ' spin(n).

In chapter 25 we will explicitly construct the groups Spin(n) for any n buthere we will just do this for n = 3 and n = 4, using methods specific to these twocases. In the cases n = 5 (where Spin(5) = Sp(2), the 2 by 2 norm-preservingquaternionic matrices) and n = 6 (where Spin(6) = SU(4)) one can also usespecial methods to identify Spin(n) with other matrix groups. For n > 6 thegroup Spin(n) will be something truly distinct.

Given such a construction of Spin(n), we also need to explicitly constructthe homomorphism Φ, and show that its derivative Φ′ is an isomorphism ofLie algebras. We will see that the simplest construction of the spin groupshere uses the group Sp(1) of unit-length quaternions, with Spin(3) = Sp(1)and Spin(4) = Sp(1)× Sp(1). By identifying quaternions and pairs of complexnumbers, we can show that Sp(1) = SU(2) and thus work with these spin groupsas either 2 by 2 complex matrices (for Spin(3)), or pairs of such matrices (forSpin(4)).

60

6.2.1 Quaternions

The quaternions are a number system (denoted by H) generalizing the complexnumber system, with elements q ∈ H that can be written as

q = q0 + q1i + q2j + q3k, qi ∈ R

with i, j,k ∈ H satisfying

i2 = j2 = k2 = −1, ij = −ji = k,ki = −ik = j, jk = −kj = i

and a conjugation operation that takes

q → q = q0 − q1i− q2j− q3k

This operation satisfies (for u, v ∈ H)

uv = vu

As a vector space over R, H is isomorphic with R4. The length-squaredfunction on this R4 can be written in terms of quaternions as

|q|2 = qq = q20 + q2

1 + q22 + q2

3

and is multiplicative since

|uv|2 = uvuv = uvvu = |u|2|v|2

Usingqq

|q|2= 1

one has a formula for the inverse of a quaternion

q−1 =q

|q|2

The length one quaternions thus form a group under multiplication, calledSp(1). There are also Lie groups called Sp(n) for larger values of n, consistingof invertible matrices with quaternionic entries that act on quaternionic vectorspreserving the quaternionic length-squared, but these play no significant role inquantum mechanics so we won’t study them further. Sp(1) can be identifiedwith the three-dimensional sphere since the length one condition on q is

q20 + q2

1 + q22 + q2

3 = 1

the equation of the unit sphere S3 ⊂ R4.

61

6.2.2 Rotations and spin groups in four dimensions

Pairs (u, v) of unit quaternions give the product group Sp(1) × Sp(1). Anelement of this group acts on H = R4 by

q → uqv

This action preserves lengths of vectors and is linear in q, so it must correspondto an element of the group SO(4). One can easily see that pairs (u, v) and(−u,−v) give the same linear transformation of R4, so the same element ofSO(4). One can show that SO(4) is the group Sp(1) × Sp(1), with the twoelements (u, v) and (−u,−v) identified. The name Spin(4) is given to the Liegroup Sp(1)× Sp(1) that “double covers” SO(4) in this manner.

6.2.3 Rotations and spin groups in three dimensions

Later on in the course we’ll encounter Spin(4) and SO(4) again, but for nowwe’re interested in the subgroup Spin(3) that only acts non-trivially on 3 of thedimensions, and double-covers not SO(4) but SO(3). To find this, consider thesubgroup of Spin(4) consisting of pairs (u, v) of the form (u, u−1) (a subgroupisomorphic to Sp(1), since elements correspond to a single unit length quaternionu). This subgroup acts on quaternions by conjugation

q → uqu−1

an action which is trivial on the real quaternions, but nontrivial on the “pureimaginary” quaternions of the form

q = ~v = v1i + v2j + v3k

An element u ∈ Sp(1) acts on ~v ∈ R3 ⊂ H as

~v → u~vu−1

This is a linear action, preserving the length |~v|, so corresponds to an elementof SO(3). We thus have a map (which can easily be checked to be a homomor-phism)

Φ : u ∈ Sp(1)→ ~v → u~vu−1 ∈ SO(3)

62

Both u and −u act in the same way on ~v, so we have two elements inSp(1) corresponding to the same element in SO(3). One can show that Φ is asurjective map (one can get any element of SO(3) this way), so it is what is calleda “covering” map, specifically a two-fold cover. It makes Sp(1) a double-cover ofSO(3), and we give this the name “Spin(3)”. This also allows us to characterizemore simply SO(3) as a geometrical space. It is S3 = Sp(1) = Spin(3) withopposite points on the three-sphere identified. This space is known as RP(3),real projective 3-space, which can also be thought of as the space of lines throughthe origin in R4 (each such line intersects S3 in two opposite points).

63

For those who have seen some topology, note that the covering map Φ isan example of a topologically non-trivial cover. It is just not true that topo-logically S3 ' RP3 × (+1,−1). S3 is a connected space, not two disconnectedpieces. This topological non-triviality implies that globally there is no possiblehomomorphism going in the opposite direction from Φ (i.e. SO(3)→ Spin(3)).One can do this locally, picking a local patch in SO(3) and taking the inverseof Φ to a local patch in Spin(3), but this won’t work if we try and extend itglobally to all of SO(3).

The identification R2 = C allowed us to represent elements of the unit circlegroup U(1) as exponentials eiθ, where iθ was in the Lie algebra u(1) = R ofU(1). For Sp(1) one can do much the same thing, with the Lie algebra sp(1)now the space of all pure imaginary quaternions, which one can identify withR3 by

w =

w1

w2

w3

∈ R3 ↔ ~w = w1i + w2j + w3k ∈ H

Unlike the U(1) case, there’s a non-trivial Lie bracket, just the commutator ofquaternions.

Elements of the group Sp(1) are given by exponentiating such Lie algebraelements, which we will write in the form

u(θ,w) = eθ ~w = cos θ + ~w sin θ

64

where θ ∈ R and ~w is a purely imaginary quaternion of unit length. Taking θas a parameter, these give paths in Sp(1) going through the identity at θ = 0,with velocity vector ~w since

d

dθu(θ,w)|θ=0 = (− sin θ + ~w cos θ)|θ=0 = ~w

We can explicitly evaluate the homomorphism Φ on such elements u(θ,w) ∈Sp(1), with the result that Φ takes u(θ,w) to a rotation by an angle 2θ aroundthe axis w:

Theorem 6.1.

Φ(u(θ,w)) = R(2θ,w)

Proof. First consider the special case w = e3 of rotations about the 3-axis.

u(θ, e3) = eθk = cos θ + k sin θ

and

u(θ, e3)−1 = e−θk = cos θ − k sin θ

so Φ(u(θ, e3)) is the rotation that takes v (identified with the quaternion ~v =v1i + v2j + v3k) to

u(θ, e3)~vu(θ, e3)−1 =(cos θ + k sin θ)(v1i + v2j + v3k)(cos θ − k sin θ)

=(v1(cos2 θ − sin2 θ)− v2(2 sin θ cos θ))i

+ (2v1 sin θ cos θ + v2(cos2 θ − sin2 θ))j + v3k

=(v1 cos 2θ − v2 sin 2θ)i + (v1 sin 2θ + v2 cos 2θ)j + v3k

This is the orthogonal transformation of R3 given by

v =

v1

v2

v3

→cos 2θ − sin 2θ 0

sin 2θ cos 2θ 00 0 1

v1

v2

v3

(6.1)

One can readily do the same calculation for the cases of e1 and e2, then usethe fact that a general u(θ,w) can be written as a product of the cases alreadyworked out.

Notice that as θ goes from 0 to 2π, u(θ,w) traces out a circle in Sp(1). Thehomomorphism Φ takes this to a circle in SO(3), one that gets traced out twiceas θ goes from 0 to 2π, explicitly showing the nature of the double coveringabove that particular circle in SO(3).

The derivative of the map Φ will be a Lie algebra homomorphism, a linearmap

Φ′ : sp(1)→ so(3)

65

It takes the Lie algebra sp(1) of pure imaginary quaternions to the Lie algebraso(3) of 3 by 3 antisymmetric real matrices. One can compute it easily on basisvectors, using for instance equation 6.1 above to find for the case ~w = k

Φ′(k) =d

dθΦ(cos θ + k sin θ)|θ=0

=

−2 sin 2θ −2 cos 2θ 02 cos 2θ −2 sin 2θ 0

0 0 0

|θ=0

=

0 −2 02 0 00 0 0

= 2l3

Repeating this on other basis vectors one finds that

Φ′(i) = 2l1,Φ′(j) = 2l2,Φ

′(k) = 2l3

Thus Φ′ is an isomorphism of sp(1) and so(3) identifying the bases

i

2,

j

2,k

2and l1, l2, l3

Note that it is the i2 ,

j2 ,

k2 that satisfy simple commutation relations

[i

2,

j

2] =

k

2, [

j

2,k

2] =

i

2, [

k

2,

i

2] =

j

2

6.2.4 The spin group and SU(2)

Instead of doing calculations using quaternions with their non-commutativityand special multiplication laws, it is more conventional to choose an isomorphismbetween quaternions H and a space of 2 by 2 complex matrices, and work justwith matrix multiplication and complex numbers. The Pauli matrices can beused to gives such an isomorphism, taking

1→ 1 =

(1 00 1

), i→ −iσ1 =

(0 −i−i 0

), j→ −iσ2 =

(0 −11 0

)

k→ −iσ3 =

(−i 00 i

)The correspondence between H and 2 by 2 complex matrices is then given

by

q = q0 + q1i + q2j + q3k↔(q0 − iq3 −q2 − iq1

q2 − iq1 q0 + iq3

)Since

det

(q0 − iq3 −q2 − iq1

q2 − iq1 q0 + iq3

)= q2

0 + q21 + q2

2 + q23

66

we see that the length-squared function on quaternions corresponds to the de-terminant function on 2 by 2 complex matrices. Taking q ∈ Sp(1), so of lengthone, the corresponding complex matrix is in SU(2).

Under this identification of H with 2 by 2 complex matrices, we have anidentification of Lie algebras sp(1) = su(2) between pure imaginary quaternionsand skew-Hermitian trace-zero 2 by 2 complex matrices

~w = w1i + w2j + w3k↔(−iw3 −w2 − iw1

w2 − iw1 iw3

)= −iw · σ

The basis i2 ,

j2 ,

k2 gets identified with a basis for the Lie algebra su(2) which

written in terms of the Pauli matrices is

Xj = −iσj2

with the Xj satisfying the commutation relations

[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

which are precisely the same commutation relations as for so(3)

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

We now have no less than three isomorphic Lie algebras sp(1) = su(2) =so(3), with elements that get identified as follows

(w1

i2 + w2

j2 + w3

k2

)↔ − i

2

(w3 w1 − iw2

w1 + iw2 −w3

)↔

0 −w3 w2

w3 0 −w1

−w2 w1 0

This isomorphism identifies basis vectors by

i

2↔ −iσ1

2↔ l1

etc. The first of these identifications comes from the way we chose to identifyH with 2 by 2 complex matrices. The second identification is Φ′, the derivativeat the identity of the covering map Φ.

On each of these isomorphic Lie algebras we have adjoint Lie group (Ad)and Lie algebra (ad) representations. Ad si given by conjugation with the cor-responding group elements in Sp(1), SU(2) and SO(3). ad is given by takingcommutators in the respective Lie algebras of pure imaginary quaternions, skew-Hermitian trace-zero 2 by 2 complex matrices and 3 by 3 real antisymmetricmatrices.

Note that these three Lie algebras are all three-dimensional real vectorspaces, so these are real representations. If one wants a complex representa-tion, one can complexify and take complex linear combinations of elements.This is less confusing in the case of su(2) than for sp(1) since taking complex

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linear combinations of skew-Hermitian trace-zero 2 by 2 complex matrices justgives all trace-zero 2 by 2 matrices (the Lie algebra sl(2,C)).

In addition, recall that there is a fourth isomorphic version of this repre-sentation, the representation of SO(3) on column vectors. This is also a realrepresentation, but can straightforwardly be complexified. Since so(3) and su(2)are isomorphic Lie algebras, their complexifications so(3)C and sl(2,C) will alsobe isomorphic.

In terms of 2 by 2 complex matrices, one can exponentiate Lie algebra ele-ments to get group elements in SU(2) and define

Ω(θ,w) = eθ(w1X1+w2X2+w3X3) = e−iθ2w·σ (6.2)

= cos(θ

2)1− i(w · σ) sin(

θ

2) (6.3)

Transposing the argument of theorem 6.1 from H to complex matrices, one findsthat, identifying

v↔ v · σ =

(v3 v1 − iv2

v1 + iv2 −v3

)one has

Φ(Ω(θ,w)) = R(θ,w)

with Ω(θ,w) acting by conjugation, taking

v · σ → Ω(θ,w)(v · σ)Ω(θ,w)−1 = (R(θ,w)v) · σ (6.4)

Note that in changing from the quaternionic to complex case, we are treatingthe factor of 2 differently, since in the future we will want to use Ω(θ,w) toperform rotations by an angle θ. In terms of the identification SU(2) = Sp(1),we have Ω(θ,w) = u( θ2 ,w).

Recall that any SU(2) matrix can be written in the form(α β

−β α

)α = q0 − iq3, β = −q2 − iq1

with α, β ∈ C arbitrary complex numbers satisfying |α|2+|β|2 = 1. One can alsowrite down a somewhat unenlightening formula for the map Φ : SU(2)→ SO(3)in terms of such explicit SU(2) matrices, getting

Φ(

(α β

−β α

)) =

Im(β2 − α2) Re(α2 + β2) 2Im(αβ)Re(β2 − α2) Im(α2 + β2) 2Re(αβ)

2Re(αβ) −2Im(αβ) |α|2 − |β|2

See [54], page 123-4, for a derivation.

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6.3 A summary

To summarize, we have shown that for three dimensions we have two distinctLie groups:

• Spin(3), which geometrically is the space S3. Its Lie algebra is R3 withLie bracket the cross-product. We have seen two different explicit con-structions of Spin(3), in terms of unit quaternions (Sp(1)), and in termsof 2 by 2 unitary matrices of determinant 1 (SU(2)).

• SO(3), with the same Lie algebra R3 with the same Lie bracket.

There is a group homomorphism Φ that takes the first group to the second,which is a two-fold covering map. Its derivative Φ′ is an isomorphism of the Liealgebras of the two groups.

We can see from these constructions two interesting irreducible representa-tions of these groups:

• A representation on R3 which can be constructed in two different ways: asthe adjoint representation of either of the two groups, or as the definingrepresentation of SO(3). This is known to physicists as the “spin 1”representation.

• A representation of the first group on C2, which is most easily seen asthe defining representation of SU(2). It is not a representation of SO(3),since going once around a non-contractible loop starting at the identitytakes one to minus the identity, not back to the identity as required. Thisis called the “spin 1/2 or “spinor” representation and will be studied inmore detail in chapter 7.


For another discussion of the relationship of SO(3) and SU(2) as well as aconstruction of the map Φ, see [54], sections 4.2 and 4.3, as well as [3], chapter8, and [56] Chapters 2 and 4.

69

70

Chapter 7

Rotations and the Spin 12

Particle in a Magnetic Field

The existence of a non-trivial double-cover Spin(3) of the three-dimensionalrotation group may seem to be a somewhat obscure mathematical fact. Re-markably though, experiment shows that it is Spin(3) rather than SO(3) thatis the symmetry group corresponding to rotations of fundamental quantum sys-tems. Ignoring the degrees of freedom describing their motion in space, whichwe will examine in later chapters, states of elementary particles such as theelectron are described by a state space H = C2, with rotations acting on thisspace by the two-dimensional irreducible representation of SU(2) = Spin(3).

This is the same two-state system studied in chapter 3, with the SU(2)action found there now acquiring an interpretation as corresponding to thedouble-cover of rotations of physical space. In this chapter we will revisit thatexample, emphasizing the relation to rotations.

7.1 The spinor representation

In chapter 6 we examined in great detail various ways of looking at a particularthree-dimensional irreducible real representation of the groups SO(3), SU(2)and Sp(1). This was the adjoint representation for those three groups, andisomorphic to the vector representation for SO(3). In the SU(2) and Sp(1)cases, there is an even simpler non-trivial irreducible representation than theadjoint: the representation of 2 by 2 complex matrices in SU(2) on columnvectors C2 by matrix multiplication or the representation of unit quaternions inSp(1) on H by scalar multiplication. Choosing an identification C2 = H theseare isomorphic representations on C2 of isomorphic groups, and for calculationalconvenience we will use SU(2) and its complex matrices rather than dealing withquaternions. This irreducible representation is known as the “spinor” or “spin”representation of Spin(3) = SU(2). The homomorphism πspinor defining therepresentation is just the identity map from SU(2) to itself.

71

The spin representation of SU(2) is not a representation of SO(3). Thedouble cover map Φ : SU(2) → SO(3) is a homomorphism, so given a rep-resentation (π, V ) of SO(3) one gets a representation (π Φ, V ) of SU(2) bycomposition. One cannot go in the other direction: there is no homomorphismSO(3) → SU(2) that would allow us to make the standard representation ofSU(2) on C2 into an SO(3) representation.

One could try and define a representation of SO(3) by

π : g ∈ SO(3)→ π(g) = πspinor(g) ∈ SU(2)

where g is some choice of one of the elements g ∈ SU(2) satisfying Φ(g) = g.The problem with this is that we won’t quite get a homomorphism. Changingour choice of g will introduce a minus sign, so π will only be a homomorphismup to sign

π(g1)π(g2) = ±π(g1g2)

The nontrivial nature of the double-covering ensures that there is no way tocompletely eliminate all minus signs, no matter how we choose g. Exampleslike this, which satisfy the representation property only one up to a sign ambi-guity, are known as “projective representations”. So, the spinor representationof SU(2) = Spin(3) is only a projective representation of SO(3), not a truerepresentation of SO(3).

Quantum mechanics texts sometimes deal with this phenomenon by notingthat physically there is an ambiguity in how one specifies the space of states H,with multiplication by an overall scalar not changing the eigenvalues of operatorsor the relative probabilities of observing these eigenvalues. As a result, the signambiguity has no physical effect. It seems more straightforward though to nottry and work with projective representations, but just use the larger groupSpin(3), accepting that this is the correct symmetry group reflecting the actionof rotations on three-dimensional quantum systems.

Spinors are more fundamental objects than vectors, in the sense that spinorscannot be constructed out of vectors, but vectors can be described in terms ofspinors. We have seen this in the identification of R3 with 2 by 2 complexmatrices, a more precise statement is possible using the tensor product, tobe discussed in chapter 9. Note that taking spinors as fundamental entailsabandoning the descriptions of three-dimensional geometry purely in terms ofreal numbers. While the vector representation is a real representation of SO(3)or Spin(3), the spinor representation is a complex representation.

7.2 The spin 1/2 particle in a magnetic field

In chapter 3 we saw that a general quantum system with H = C2 could beunderstood in terms of the action of U(2) on C2. The self-adjoint observablescorrespond (up to a factor of i) to the corresponding Lie algebra representation.The U(1) ⊂ U(2) subgroup commutes with everything else and can be analyzedseparately, so we will just consider the SU(2) subgroup. For an arbitrary such

72

system, the group SU(2) has no particular geometric significance. When itoccurs in its role as double-cover of the rotational group, the quantum systemis said to carry “spin”, in particular “spin one-half” (in chapter 8 will discussstate spaces of higher spin values).

As before, we take as a standard basis for the Lie algebra su(2) the operatorsXj , j = 1, 2, 3, where

Xj = −iσj2


[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

To make contact with the physics formalism, we’ll define self-adjoint operators

Sj = iXj =σj2

We could have chosen the other sign, but this is the standard convention ofthe physics literature. In general, to a skew-adjoint operator (which is whatone gets from a unitary Lie algebra representation and what exponentiates tounitary operators) we will associate a self-adjoint operator by multiplying byi. These self-adjoint operators have real eigenvalues (in this case ± 1

2 ), so arefavored by physicists as observables since such eigenvalues will be related toexperimental results. In the other direction, given a physicist’s observable self-adjoint operator, we will multiply by −i to get a skew-adjoint operator that canbe exponentiated to get a unitary representation.

Note that the conventional definition of these operators in physics textsincludes a factor of ~:

Sphysj = i~Xj =~σj2

A compensating factor of 1/~ is then introduced when exponentiating to getgroup elements

Ω(θ,w) = e−iθ~w·Sphys ∈ SU(2)

which do not depend on ~. The reason for this convention has to do with theaction of rotations on functions on R3 (see chapter 17) and the appearance of~ in the definition of the momentum operator. Our definitions of Sj and ofrotations using (see equation 6.2)

Ω(θ,w) = e−iθw·S = eθw·X

will not include these factors of ~, but in any case they will be equivalent tothe physics text definitions when we make our standard choice of working withunits such that ~ = 1.

States in H = C2 that have a well-defined value of the observable Sj will bethe eigenvectors of Sj , with value for the observable the corresponding eigen-value, which will be ± 1

2 . Measurement theory postulates that if we perform themeasurement corresponding to Sj on an arbitrary state |ψ〉, then we will

73

• with probability c+ get a value of +12 and leave the state in an eigenvector

|j,+ 12 〉 of Sj with eigenvalue + 1

2

• with probability c− get a value of − 12 and leave the state in an eigenvector

|j,− 12 〉 of Sj with eigenvalue − 1

2

where if

|ψ〉 = α|j,+1

2〉+ β|j,−1

2〉

we have

c+ =|α|2

|α|2 + |β|2, c− =

|β|2

|α|2 + |β|2After such a measurement, any attempt to measure another Sk, k 6= j will give± 1

2 with equal probability and put the system in a corresponding eigenvectorof Sk.

If a quantum system is in an arbitrary state |ψ〉 it may not have a well-defined value for some observable A, but one can calculate the “expected value”of A. This is the sum over a basis of H consisting of eigenvectors (which willall be orthogonal) of the corresponding eigenvalues, weighted by the probabilityof their occurrence. The calculation of this sum in this case (A = Sj) usingexpansion in eigenvectors of Sj gives

〈ψ|A|ψ〉〈ψ|ψ〉

=(α〈j,+ 1

2 |+ β〈j,− 12 |)A(α|j,+ 1

2 〉+ β|j,− 12 〉)

(α〈j,+ 12 |+ β〈j,− 1

2 |)(α|j,+12 〉+ β|j,− 1

2 〉)

=|α|2(+ 1

2 ) + |β|2(− 12 )

|α|2 + |β|2

=c+(+1

2) + c−(−1

2)

One often chooses to simplify such calculations by normalizing states so thatthe denominator 〈ψ|ψ〉 is 1. Note that the same calculation works in generalfor the probability of measuring the various eigenvalues of an observable A, aslong as one has orthogonality and completeness of eigenvectors.

In the case of a spin one-half particle, the group Spin(3) = SU(2) acts onstates by the spinor representation with the element Ω(θ,w) ∈ SU(2) acting as

|ψ〉 → Ω(θ,w)|ψ〉

As we saw in chapter 6, the Ω(θ,w) also act on self-adjoint matrices by conju-gation, an action that corresponds to rotation of vectors when one makes theidentification

v↔ v · σ(see equation 6.4). Under this identification the Sj correspond (up to a factorof 2) to the basis vectors ej . Putting the Sj in a column vector one can writetheir transformation rule asS1

S2

S3

→ Ω(θ,w)

S1

S2

S3

Ω(θ,w)−1 = R(θ,w)T

S1

S2

S3

74

Note that, recalling the discussion in section 4.1, rotations on sets of basisvectors like this involve the transpose R(θ,w)T of the matrix R(θ,w) that actson coordinates.

In chapter 40 we will get to the physics of electromagnetic fields and howparticles interact with them in quantum mechanics, but for now all we need toknow is that for a spin one-half particle, the spin degree of freedom that we aredescribing by H = C2 has a dynamics described by the Hamiltonian

H = −µ ·B

Here B is the vector describing the magnetic field, and

µ = g−e

2mcS

is an operator called the magnetic moment operator. The constants that appearare: −e the electric charge, c the speed of light, m the mass of the particle, and g,a dimensionless number called the “gyromagnetic ratio”, which is approximately2 for an electron, about 5.6 for a proton.

The Schrodinger equation is

d

dt|ψ(t)〉 = −i(−µ ·B)|ψ(t)〉

with solution|ψ(t)〉 = U(t)|ψ(0)〉

where

U(t) = eitµ·B = eit−ge2mcS·B = et

ge2mcX·B = et

ge|B|2mc X· B

|B|

The time evolution of a state is thus given at time t by the same SU(2) elementthat, acting on vectors, gives a rotation about the axis w = B

|B| by an angle

ge|B|t2mc

so is a rotation about w taking place with angular velocity ge|B|2mc .

The amount of non-trivial physics that is described by this simple system isimpressive, including:

• The Zeeman effect: this is the splitting of atomic energy levels that occurswhen an atom is put in a constant magnetic field. With respect to theenergy levels for no magnetic field, where both states in H = C2 have thesame energy, the term in the Hamiltonian given above adds

±ge|B|4mc

to the two energy levels, giving a splitting between them proportional tothe size of the magnetic field.

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• The Stern-Gerlach experiment: here one passes a beam of spin one-halfquantum systems through an inhomogeneous magnetic field. We have notyet discussed particle motion, so more is involved here than the simpletwo-state system. However, it turns out that one can arrange this in sucha way as to pick out a specific direction w, and split the beam into twocomponents, of eigenvalue + 1

2 and − 12 for the operator w · S.

• Nuclear magnetic resonance spectroscopy: one can subject a spin one-halfsystem to a time-varying magnetic field B(t), and such a system will bedescribed by the same Schrodinger equation, although now the solutioncannot be found just by exponentiating a matrix. Nuclei of atoms providespin one-half systems that can be probed with time and space-varyingmagnetic fields, allowing imaging of the material that they make up.

• Quantum computing: attempts to build a quantum computer involve try-ing to put together multiple systems of this kind (qubits), keeping themisolated from perturbations by the environment, but still allowing inter-action with the system in a way that preserves its quantum behavior.The 2012 Physics Nobel prize was awarded for experimental work makingprogress in this direction.

7.3 The Heisenberg picture

So far in this course we’ve been describing what is known as the Schrodingerpicture of quantum mechanics. States in H are functions of time, obeyingthe Schrodinger equation determined by a Hamiltonian observable H, whileobservable self-adjoint operators A are time-independent. Time evolution isgiven by a unitary transformation

U(t) = e−itH , |ψ(t)〉 = U(t)|ψ(0)〉

One can instead use U(t) to make a unitary transformation that puts thetime-dependence in the observables, removing it from the states, as follows:

|ψ(t)〉 → |ψ(t)〉H = U−1(t)|ψ(t)〉 = |ψ(0)〉, A→ AH(t) = U−1(t)AU(t)

where the “H” subscripts for “Heisenberg” indicate that we are dealing with“Heisenberg picture” observables and states. One can easily see that the physi-cally observable quantities given by eigenvalues and expectations values remainthe same:

H〈ψ(t)|AH |ψ(t)〉H = 〈ψ(t)|U(t)(U−1(t)AU(t))U−1(t)|ψ(t)〉 = 〈ψ(t)|A|ψ(t)〉

In the Heisenberg picture the dynamics is given by a differential equationnot for the states but for the operators. Recall from our discussion of the adjointrepresentation (see equation 5.1) the formula

d

dt(etXY e−tX) = (

d

dt(etXY ))e−tX + etXY (

d

dte−tX)

= XetXY e−tX − etXY e−tXX

76

Using this withY = A, X = iH

we findd

dtAH(t) = [iH,AH(t)] = i[H,AH(t)]

and this equation determines the time evolution of the observables in the Heisen-berg picture.

Applying this to the case of the spin one-half system in a magnetic field, andtaking for our observable S (the Sj , taken together as a column vector) we find

d

dtSH(t) = i[H,SH(t)] = i

eg

2mc[SH(t) ·B,SH(t)]

We know from the discussion above that the solution will be

SH(t) = U(t)SH(0)U(t)−1

for

U(t) = e−itge|B|2mc S· B

|B|

and thus the spin vector observable evolves in the Heisenberg picture by rotating

about the magnetic field vector with angular velocity ge|B|2mc .

7.4 The Bloch sphere and complex projectivespace

There is a different approach one can take to characterizing states of a quantumsystem with H = C2. Multiplication of vectors in H by a non-zero complexnumber does not change eigenvectors, eigenvalues or expectation values, so ar-guably has no physical effect. Multiplication by a real scalar just correspondsto a change in normalization of the state, and we will often use this freedomto work with normalized states, those satisfying 〈ψ|ψ〉 = 1. With normalizedstates, one still has the freedom to multiply states by a phase eiθ without chang-ing eigenvectors, eigenvalues or expectation values. In terms of group theory,the overall U(1) in the unitary group U(2) acts on H by a representation ofU(1), which can be characterized by an integer, the corresponding “charge”,but this decouples from the rest of the observables and is not of much interest.One is mainly interested in the SU(2) part of the U(2), and the observablesthat correspond to its Lie algebra.

Working with normalized states in this case corresponds to working withunit-length vectors in C2, which are given by points on the unit sphere S3. Ifwe don’t care about the overall U(1) action, we can imagine identifying all statesthat are related by a phase transformation. Using this equivalence relation wecan define a new set, whose elements are the “cosets”, elements of S3 ⊂ C2,with elements that differ just by multiplication by eiθ identified. The set of theseelements forms a new geometrical space, called the “coset space”, often written

77

S3/U(1). This structure is called a “fibering” of S3 by circles, and is knownas the “Hopf fibration”. Try an internet search for various visualizations of thegeometrical structure involved, a surprising decomposition of three-dimensionalspace into non-intersecting curves.

The same space can be represented in a different way, as C2/C∗, by takingall elements of C2 and identifying those related by muliplication by a non-zerocomplex number. If we were just using real numbers, R2/R∗ can be thought ofas the space of all lines in the plane going through the origin.

One sees that each such line hits the unit circle in two opposite points, sothis set could be parametrized by a semi-circle, identifying the points at thetwo ends. This space is given the name RP 1, the “real projective line”, andthe analog space of lines through the origin in Rn is called RPn−1. What weare interested in is the complex analog CP 1, which is often called the “complexprojective line”.

To better understand CP 1, one would like to put coordinates on it. Astandard way to choose such a coordinate is to associate to the vector(

z1

z2

)∈ C2

the complex number z1/z2. Overall multiplication by a complex number willdrop out in this ratio, so one gets different values for the coordinate z1/z2 foreach different coset element, and it appears that elements of CP 1 correspond topoints on the complex plane. There is however one problem with this coordinate:

78

the coset of

(10

)

does not have a well-defined value: as one approaches this point one moves offto infinity in the complex plane. In some sense the space CP 1 is the complexplane, but with a “point at infinity” added.

It turns out that CP 1 is best thought of not as a plane together with apoint, but as a sphere, with the relation to the plane and the point at infinitygiven by stereographic projection. Here one creates a one-to-one mapping byconsidering the lines that go from a point on the sphere to the north pole ofthe sphere. Such lines will intersect the plane in a point, and give a one-to-onemapping between points on the plane and points on the sphere, except for thenorth pole. Now, one can identify the north pole with the “point at infinity”,and thus the space CP 1 can be identified with the space S2. The picture lookslike this

and the equations relating coordinates (X1, X2, X3) on the sphere and the com-plex coordinate z1/z2 = z = x+ iy on the plane are given by

x =X1

1−X3, y =

X2

1−X3

79

and

X1 =2x

x2 + y2 + 1, X2 =

2y

x2 + y2 + 1, X3 =

x2 + y2 − 1

x2 + y2 + 1

The action of SU(2) on H by(z1

z2

)→(α β

−β α

)(z1

z2

)takes

z =z1

z2→ αz + β

−βz + α

Such transformations of the complex plane are conformal (angle-preserving)transformations known as “Mobius transformations”. One can check that thecorresponding transformation on the sphere is the rotation of the sphere in R3

corresponding to this SU(2) = Spin(3) transformation.

To mathematicians, this sphere identified with CP 1 is known as the “Rie-mann sphere”, whereas physicists often instead use the terminology of “Blochsphere”. It provides a useful parametrization of the states of the qubit system,up to scalar multiplication, which is supposed to be physically irrelevant. TheNorth pole is the “spin-up” state, the South pole is the “spin-down” state, andalong the equator one finds the two states that have definite values for S1, aswell as the two that have definite values for S2.

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Notice that the inner product on vectors in H does not correspond at allto the inner product of unit vectors in R3. The North and South poles of theBloch sphere correspond to orthogonal vectors in H, but they are not at allorthogonal thinking of the corresponding points on the Bloch sphere as vectorsin R3. Similarly, eigenvectors for S1 and S2 are orthogonal on the Bloch sphere,but not at all orthogonal in H.

Many of the properties of the Bloch sphere parametrization of states inH arespecial to the fact that H = C2. In the next class we will study systems of spinn2 , where H = Cn. In these cases there is still a two-dimensional Bloch sphere,but only certain states in H are parametrized by it. We will see other examplesof systems with “coherent states” analogous to the states parametrized by theBloch sphere, but the case H has the special property that all states (up toscalar multiplication) are such “coherent states”.


Just about every quantum mechanics textbook works out this example of a spin1/2 particle in a magnetic field. For one example, see chapter 14 of [53]. Foran inspirational discussion of spin and quantum mechanics, together with more

81

about the Bloch sphere, see chapter 22 of [41].

82

Chapter 8

Representations of SU(2)and SO(3)

For the case of G = U(1), in chapter 2 we were able to classify all complexirreducible representations by an element of Z and explicitly construct eachirreducible representation. We would like to do the same thing here for repre-sentations of SU(2) and SO(3). The end result will be that irreducible repre-sentations of SU(2) are classified by a non-negative integer n = 0, 1, 2, 3, · · · ,and have dimension n+ 1, so we’ll (hoping for no confusion with the irreduciblerepresentations (πn,C) of U(1)) denote them (πn,C

n+1). For even n these willalso be irreducible representations of SO(3), but this will not be true for oddn. It is common in physics to label these representations by s = n

2 = 0, 12 , 1, · · ·

and call the representation labeled by s the “spin s representation”. We alreadyknow the first three examples:

• Spin 0: (π0,C) is the trivial representation on V, with

π0(g) = 1 ∀g ∈ SU(2)

This is also a representation of SO(3). In physics, this is sometimes calledthe “scalar representation”. Saying that something transforms under ro-tations as the “scalar representation” just means that it is invariant underrotations.

• Spin 12 : Taking

π1(g) = g ∈ SU(2) ⊂ U(2)

gives the defining representation on C2. This is the spinor representationdiscussed in chapter 7. It is not a representation of SO(3).

• Spin 1: Since SO(3) is a group of 3 by 3 matrices, it acts on vectors in R3.This is just the standard action on vectors by rotation. In other words,the representation is (ρ,R3), with ρ the identity homomorphism

g ∈ SO(3)→ ρ(g) = g ∈ SO(3)

83

One can complexify to get a representation on C3, which in this case justmeans acting with SO(3) matrices on column vectors, replacing the realcoordinates of vectors by complex coordinates. This is sometimes calledthe “vector representation”, and we saw in chapter 6 that it is isomorphicto the adjoint representation.

One gets a representation (π2,C3) of SU(2) by just composing the homo-

morphisms Φ and ρ:

π2 = ρ Φ : SU(2)→ SO(3)

This is the adjoint representation of SU(2).

8.1 Representations of SU(2): classification

8.1.1 Weight decomposition

If we make a choice of a U(1) ⊂ SU(2), then given any representation (π, V ) ofSU(2) of dimension m, we get a representation (π|U(1), V ) of U(1) by restrictionto the U(1) subgroup. Since we know the classification of irreducibles of U(1),we know that

(π|U(1), V ) = Cq1 ⊕Cq2 ⊕ · · · ⊕Cqm

for q1, q2, · · · , qm ∈ Z, where Cq denotes the one-dimensional representationof U(1) corresponding to the integer q. These are called the “weights” of therepresentation V . They are exactly the same thing we discussed earlier as“charges”, but here we’ll favor the mathematician’s terminology since the U(1)here occurs in a context far removed from that of electromagnetism and itselectric charges.

Since our standard choice of coordinates (the Pauli matrices) picks out thez-direction and diagonalizes the action of the U(1) subgroup corresponding torotation about this axis, this is the U(1) subgroup we will choose to define theweights of the SU(2) representation V . This is the subgroup of elements ofSU(2) of the form (

eiθ 00 e−iθ

)Our decomposition of an SU(2) representation (π, V ) into irreducible repre-

sentations of this U(1) subgroup equivalently means that we can choose a basisof V so that

π

(eiθ 00 e−iθ

)=

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

Instead of the set of the weights qj of the representation (π, V ) one could justas well use the set of eigenvalues eiqjθ of the matrices

π

(eiθ 00 e−iθ

)84

and these two sets are each independent of the choice of basis of V .An important property of the set of integers qj is the following:

Theorem. If q is in the set qj, so is −q.

Proof. Recall that if we diagonalize a unitary matrix, the diagonal entries arethe eigenvalues, but their order is undetermined: acting by permutations onthese eigenvalues we get different diagonalizations of the same matrix. In thecase of SU(2) the matrix

P =

(0 1−1 0

)has the property that conjugation by it permutes the diagonal elements, inparticular

P

(eiθ 00 e−iθ

)P−1 =

(e−iθ 0

0 eiθ

)So

π(P )π(

(eiθ 00 e−iθ

))π(P )−1 = π(

(e−iθ 0

0 eiθ

))

and we see that π(P ) gives a change of basis of V such that the representationmatrices on the U(1) subgroup are as before, with θ → −θ. Changing θ → −θin the representation matrices is equivalent to changing the sign of the weightsqj . The elements of the set qj are independent of the basis, so the additionalsymmetry under sign change implies that for each element in the set there isanother one with the opposite sign.

Looking at our three examples so far, we see that the scalar or spin 0 repre-sentation of course is one-dimensional of weight 0

(π0,C) = C0

and the spinor or spin 12 representation decomposes into U(1) irreducibles of

weights −1,+1:(π1,C

2) = C−1 ⊕C+1

For the spin 1 representation, recall that our double-cover homomorphismΦ takes (

eiθ 00 e−iθ

)∈ SU(2)→

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

∈ SO(3)

Acting with the SO(3) matrix on the right on C3 will give a unitary transforma-tion of C3, so in the group U(3). One can show that the upper left diagonal 2 by2 block acts on C2 with weights −2,+2, whereas the bottom right element actstrivially on the remaining part of C3, which is a one-dimensional representationof weight 0. So, the spin 1 representation decomposes as

(π2,C3) = C−2 ⊕C0 ⊕C+2

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Recall that the spin 1 representation of SU(2) is often called the “vector” rep-resentation, since it factors in this way through the representation of SO(3) byrotations on three-dimensional vectors.

8.1.2 Lie algebra representations: raising and lowering op-erators

To proceed further in characterizing a representation (π, V ) of SU(2) we needto use not just the action of the chosen U(1) subgroup, but the action ofgroup elements in the other two directions away from the identity. The non-commutativity of the group keeps us from simultaneously diagonalizing thoseactions and assigning weights to them. We can however work instead with thecorresponding Lie algebra representation (π′, V ) of su(2). As in the U(1) case,the group representation is determined by the Lie algebra representation. Wewill see that for the Lie algebra representation, we can exploit the complexifica-tion (recall section 5.4) sl(2,C) of su(2) to further analyze the possible patternsof weights.

Recall that the Lie algebra can be thought of as the tangent space R3 toSU(2) at the identity element, with a basis given by the three skew-adjoint 2by 2 matrices

Xj = −i12σj


[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

We will often use the self-adjoint versions Sj = iXj that satisfy

[S1, S2] = iS3, [S2, S3] = iS1, [S3, S1] = iS2

A unitary representation (π, V ) of SU(2) of dimension m is given by a homo-morphism

π : SU(2)→ U(m)

and we can take the derivative of this to get a map between the tangent spacesof SU(2) and of U(m), at the identity of both groups, and thus a Lie algebrarepresentation

π′ : su(2)→ u(m)

which takes skew-adjoint 2 by 2 matrices to skew-adjoint m by m matrices,preserving the commutation relations.

We have seen in section 8.1.1 that restricting the representation (π, V ) tothe diagonal U(1) subgroup of SU(2) and decomposing into irreducibles tells usthat we can choose a basis of V so that

(π, V ) = (πq1 ,C)⊕ (πq2 ,C)⊕ · · · ⊕ (πqm ,C)

86

For our choice of U(1) as all matrices of the form

ei2θS3 =

(eiθ 00 e−iθ

)with eiθ going around U(1) once as θ goes from 0 to 2π, this means we canchoose a basis of V so that

π(ei2θS3) =

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

Taking the derivative of this representation to get a Lie algebra representation,using

π′(X) =d

dθπ(eθX)|θ=0

we find for X = i2S3

π′(i2S3) =d

dθ

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

|θ=0

=

iq1 0 · · · 00 iq2 · · · 0· · · · · ·0 0 · · · iqm

Recall that π′ is a real-linear map from a real vector space (su(2) = R3) to

another real vector space (u(n), the skew-Hermitian m by m complex matrices).We can use complex linearity to extend any such map to a complex-linear mapfrom su(2)C (the complexification of su(2)) to u(m)C (the complexification ofu(m)). su(2)C is all complex linear combinations of the skew-adjoint, trace-free 2 by 2 matrices: the Lie algebra sl(2,C) of all complex, trace-free 2 by 2matrices. u(m)C is M(m,C) = gl(m,C), the Lie algebra of all complex m bym matrices.

As an example, mutiplying X = i2S3 ∈ su(2) by −i2 , we have S3 ∈ sl(2,C)

and the diagonal elements in the matrix π′(i2S3) get also multiplied by −i2 (sinceπ′ is a linear map), giving

π′(S3) =

q12 0 · · · 00 q2

2 · · · 0· · · · · ·0 0 · · · qm

2

We see that π′(S3) will have half-integral values, and make the following

definitions

Definition (Weights and Weight Spaces). If π′(S3) has an eigenvalue k2 , we

say that k is a weight of the representation (π, V ).The subspace Vk ⊂ V of the representation V satisfying

v ∈ Vk =⇒ π′(S3)v =k

2v

87

is called the k’th weight space of the representation. All vectors in it are eigen-vectors of π′(S3) with eigenvalue k

2 .The dimension dim Vk is called the multiplicity of the weight k in the rep-

resentation (π, V ).

S1 and S2 don’t commute with S3, so they may not preserve the subspacesVk and we can’t diagonalize them simultaneously with S3. We can howeverexploit the fact that we are in the complexification sl(2,C) to construct twocomplex linear combinations of S1 and S2 that do something interesting:

Definition (Raising and lowering operators). Let

S+ = S1 + iS2 =

(0 10 0

), S− = S1 − iS2 =

(0 01 0

)We have S+, S− ∈ sl(2,C). These are neither self-adjoint nor skew-adjoint, butsatisfy

(S±)† = S∓

and similarly we haveπ′(S±)† = π′(S∓)

We call π′(S+) a “raising operator” for the representation (π, V ), and π′(S−)a “lowering operator”.

The reason for this terminology is the following calculation:

[S3, S+] = [S3, S1 + iS2] = iS2 + i(−iS1) = S1 + iS2 = S+

which implies (since π′ is a Lie algebra homomorphism)

π′(S3)π′(S+)− π′(S+)π′(S3) = π′([S3, S+]) = π′(S+)

For any v ∈ Vk, we have

π′(S3)π′(S+)v = π′(S+)π′(S3)v + π′(S+)v = (k

2+ 1)π′(S+)v

sov ∈ Vk =⇒ π′(S+)v ∈ Vk+2

The linear operator π′(S+) takes vectors with a well-defined weight to vectorswith the same weight, plus 2 (thus the terminology “raising operator”). Asimilar calculation shows that π′(S−) takes Vk to Vk−2, lowering the weight by2.

We’re now ready to classify all finite dimensional irreducible unitary repre-sentations (π, V ) of SU(2). We define

Definition (Highest weights and highest weight vectors). A non-zero vectorv ∈ Vn ⊂ V such that

π′(S+)v = 0

is called a highest weight vector, with highest weight n.

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Irreducible representations will be characterized by a highest weight vector,as follows

Theorem (Highest weight theorem). Finite dimensional irreducible represen-tations of SU(2) have weights of the form

−n,−n+ 2, · · · , n− 2, n

for n a non-negative integer, each with multiplicity 1, with n a highest weight.

Proof. Finite dimensionality implies there is a highest weight n, and we canchoose any highest weight vector vn ∈ Vn. Repeatedly applying π′(S−) to vnwill give new vectors

vn−2j = π′(S−)jvn ∈ Vn−2j

with weights n− 2j.Consider the span of the vn−2j , j ≥ 0. To show that this is a representation

one needs to show that the π′(S3) and π′(S+) leave it invariant. For π′(S3) thisis obvious, for π′(S+) one can show that

π′(S+)vn−2j = j(n− j + 1)vn−2(j−1)

by an induction argument. For j = 0 this is just the highest weight conditionon vn. Assuming validity for j, one can check validity for j + 1 by

π′(S+)vn−2(j+1) =π′(S+)π′(S−)vn−2j

=(π′([S+, S−] + π′(S−)π′(S+))vn−2j

=(π′(2S3) + π′(S−)π′(S+))vn−2j

=((n− 2j)vn−2j + π′(S−)j(n− j + 1)vn−2(j−1)

=((n− 2j) + j(n− j + 1))vn−2j

=(j + 1)(n− (j + 1) + 1)vn−2((j+1)−1)

where we have used the commutation relation

[S+, S−] = 2S3

The span of the vn−2j is not just a representation, but an irreducible one,since all the non-zero vn−2j arise by repeated application of π′(S−) to vn. Inthe sequence of vn−2j for increasing j, finite-dimensionality of V n implies thatat some point one one must hit a “lowest weight vector”, one annihilated byπ′(S−). From that point on, the vn−2j for higher j will be zero. Taking intoaccount the fact that the pattern of weights is invariant under change of sign,one finds that the only possible pattern of weights is

−n,−n+ 2, · · · , n− 2, n

89

Since we saw in section 8.1.1 that representations can be studied by lookingat by the set of their weights under the action of our chosen U(1) ⊂ SU(2), wecan label irreducible representations of SU(2) by a non-negative integer n, thehighest weight. Such a representation will be of dimension n+ 1, with weights

−n,−n+ 2, · · · , n− 2, n

Each weight occurs with multiplicity one, and we have

(π, V ) = C−n ⊕C−n+2 ⊕ · · ·Cn−2 ⊕Cn

Starting with a highest-weight or lowest-weight vector, one can generate abasis for the representation by repeatedly applying raising or lowering operators.The picture to keep in mind is this

where all the vector spaces are copies of C, and all the maps are isomorphisms(multiplications by various numbers).

In summary, we see that all irreducible finite dimensional unitary SU(2)representations can be labeled by a non-negative integer, the highest weight n.These representations have dimension n+ 1 and we will denote them (πn, V

n =Cn+1). Note that Vn is the n’th weight space, V n is the representation withhighest weight n. The physicist’s terminology for this uses not n, but n

2 andcalls this number the “spin”of the representation. We have so far seen the lowestthree examples n = 0, 1, 2, or spin s = n

2 = 0, 12 , 1, but there is an infinite class

of larger irreducibles, with dim V = n+ 1 = 2s+ 1.

8.2 Representations of SU(2): construction

The argument of the previous section only tells us what properties possiblefinite dimensional irreducible representations of SU(2) must have. It showshow to construct such representations given a highest-weight vector, but doesnot provide any way to construct such highest weight vectors. We would like tofind some method to explicitly construct an irreducible (πn, V

n) for each highestweight n. There are several possible constructions, but perhaps the simplest oneis the following, which gives a representation of highest weight n by looking atpolynomials in two complex variables, homogeneous of degree n.

90

Recall from our early discussion of representations that if one has an actionof a group on a space M , one can get a representation on functions f on M bytaking

(π(g)f)(x) = f(g−1 · x)

For SU(2), we have an obvious action of the group on M = C2 (by matricesacting on column vectors), and we look at a specific class of functions on thisspace, the polynomials. We can break up the infinite-dimensional space ofpolynomials on C2 into finite-dimensional subspaces as follows:

Definition (Homogeneous polynomials). The complex vector space of homoge-neous polynomials of degree m in two complex variables z1, z2 is the space offunctions on C2 of the form

f(z1, z2) = a0zn1 + a1z

n−11 z2 + · · ·+ an−1z1z

n−12 + anz

n2

The space of such functions is a complex vector space of dimension n+ 1.

Using the action of SU(2) on C2, we will see that this space of functions isexactly the representation space V n that we need. More explicitly, for

g =

(α β

−β α

), g−1 =

(α −ββ α

)we can construct the representation as follows:

(πn(g)f)(z1, z2) =f(g−1

(z1

z2

))

=f(αz1 − βz2, βz1 + αz2)

=

n∑k=0

ak(αz1 − βz2)n−k(βz1 + αz2)k

Taking the derivative, the Lie algebra representation is given by

π′n(X)f =d

dtπn(etX)f|t=0 =

d

dtf(e−tX

(z1

z2

))|t=0

By the chain rule this is

π′n(X)f =(∂f

∂z1,∂f

∂z2)(d

dte−tX

(z1

z2

))|t=0

=− ∂f

∂z1(X11z1 +X12z2)− ∂f

∂z2(X21z1 +X22z2)

where the Xij are the components of the matrix X.Computing what happens for X = S3, S+, S−, we get

(π′n(S3)f)(z1, z2) =1

2(− ∂f∂z1

z1 +∂f

∂z2z2)

91

so

π′n(S3) =1

2(−z1

∂

∂z1+ z2

∂

∂z2)

and similarly

π′n(S+) = −z2∂

∂z1, π′n(S−) = −z1

∂

∂z2

The zk1zn−k2 are eigenvectors for S3 with eigenvalue 1

2 (n− 2k) since

π′n(S3)zk1zn−k2 =

1

2(−kzk1zn−k2 + (n− k)zk1z

n−k2 ) =

1

2(n− 2k)zk1z

n−k2

zn2 will be an explicit highest weight vector for the representation (πn, Vn).

An important thing to note here is that the formulas we have found for π′

are not in terms of matrices. Instead we have seen that when we construct ourrepresentations using functions on C2, for any X ∈ su(2) (or its complexificationsl(2,C)), π′n(X) is given by a differential operator. Note that these differentialoperators are independent of n: one gets the same operator π′(X) on all theV n. This is because the original definition of the representation

(π(g)f)(x) = f(g−1 · x)

is on the full infinite dimensional space of polynomials on C2. While this spaceis infinite-dimensional, issues of analysis don’t really come into play here, sincepolynomial functions are essentially an algebraic construction. Later on in thecourse we will need to work with function spaces that require much more seriousconsideration of issues in analysis.

Restricting the differential operators π′(X) to a finite dimensional irreduciblesubspace V n, the homogeneous polynomials of degree n, if one chooses a basisof V n, then the linear operator π′(X) will be given by a n+ 1 by n+ 1 matrix.Clearly though, the expression as a simple first-order differential operator ismuch easier to work with. In the examples we will be studying in much of therest of the course, the representations under consideration will also be on func-tion spaces, with Lie algebra representations appearing as differential operators.Instead of using linear algebra techniques to find eigenvalues and eigenvectors,the eigenvector equation will be a partial differential equation, wtih our focuson using Lie groups and their representation theory to solve such equations.

One issue we haven’t addressed yet is that of unitarity of the representation.We need Hermitian inner products on the spaces V n, inner products that willbe preserved by the action of SU(2) that we have defined on these spaces. Astandard way to define a Hermitian inner product on functions on a space Mis to define them using an integral: for f , g functions on M , take their innerproduct to be

〈f, g〉 =

∫M

fg

While for M = C2 this gives an SU(2) invariant inner product on functions, itis useless for f, g polynomial, since such integrals diverge. What one can do in

92

this case is define an inner product on polynomial functions on C2 by

〈f, g〉 =1

π2

∫C2

f(z1, z2)g(z1, z2)e−(|z1|2+|z2|2)dx1dy1dx2dy2

Here z1 = x1 + iy1, z2 = x2 + iy2. One can do integrals of this kind fairly easilysince they factorize into separate integrals over z1 and z2, each of which can betreated using polar coordinates and standard calculus methods. One can checkby explicit computation that the polynomials

zj1zk2√

j!k!

will be an orthornormal basis of the space of polynomial functions with re-spect to this inner product, and the operators π′(X), X ∈ su(2) will be skew-Hermitian.

Working out what happens for the first few examples of irreducible SU(2)representations, one finds orthonormal bases for the representation spaces V n

of homogeneous polynomials as follows

• For n = s = 01

• For n = 1, s = 12

z1, z2

• For n = 2, s = 11√2z2

1 , z1z2,1√2z2

2

• For n = 3, s = 32

1√6z3

1 ,1√2z2

1z2,1√2z1z

22 ,

1√6z3

2

8.3 Representations of SO(3) and spherical har-monics

We would like to now use the classification and construction of representations ofSU(2) to study the representations of the closely related group SO(3). For anyrepresentation (ρ, V ) of SO(3), we can use the double-covering homomorphismΦ : SU(2)→ SO(3) to get a representation

π = ρ Φ

of SU(2). It can be shown that if ρ is irreducible, π will be too, so we musthave π = ρ Φ = πn, one of the irreducible representations of SU(2) found inthe last section. Using the fact that Φ(−1) = 1, we see that

πn(−1) = ρ Φ(−1) = 1

93

From knowing that the weights of πn are −n,−n+2, · · · , n−2, n, we know that

πn(−1) = πn

(eiπ 00 e−iπ

)=

einπ 0 · · · 0

0 ei(n−2)π · · · 0· · · · · ·0 0 · · · e−inπ

= 1

which will only be true for n even, not for n odd. Since the Lie algebra of SO(3)is isomorphic to the Lie algebra of SU(2), the same Lie algebra argument usingraising and lowering operators as in the last section also applies. The irreduciblerepresentations of SO(3) will be (ρl, V = C2l+1) for l = 0, 1, 2, · · · , of dimension2l + 1 and satisfying

ρl Φ = π2l

Just like in the case of SU(2), we can explicitly construct these representa-tions using functions on a space with an SO(3) action. The obvious space tochoose is R3, with SO(3) matrices acting on x ∈ R3 as column vectors, by theformula we have repeatedly used

(ρ(g)f)(x) = f(g−1 · x) = f(g−1

x1

x2

x3

)

Taking the derivative, the Lie algebra representation is given by

ρ′(X)f =d

dtρ(etX)f|t=0 =

d

dtf(e−tX

x1

x2

x3

)|t=0

where X ∈ so(3). Recall that a basis for so(3) is given by

l1 =

0 0 00 0 −10 1 0

l2 =

0 0 10 0 0−1 0 0

l3 =

0 −1 01 0 00 0 0


[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

Digression. A Note on ConventionsWe’re using the notation lj for the real basis of the Lie algebra so(3) =

su(2). For a unitary representation ρ, the ρ′(lj) will be skew-Hermitian linearoperators. For consistency with the physics literature, we’ll use the notationLj = iρ′(lj) for the self-adjoint version of the linear operator corresponding tolj in this representation on functions. The Lj satisfy the commutation relations

[L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2

We’ll also use elements l± = l1 ± il2 of the complexified Lie algebra to createraising and lowering operators L± = iρ′(l±).

94

As with the SU(2) case, we won’t include a factor of ~ as is usual in physics(e.g. the usual convention is Lj = i~ρ′(lj)), since for considerations of the actionof the rotation group it would just cancel out (physicists define rotations using

ei~ θLj ). The factor of ~ is only of significance when Lj is expressed in terms of

the momentum operator, a topic discussed in chapter 17.

In the SU(2) case, the π′(Sj) had half-integral eigenvalues, with the eigen-values of π′(2S3) the integral weights of the representation. Here the Lj willhave integer eigenvalues, the weights will be the eigenvalues of 2L3, which willbe even integers.

Computing ρ′(l1) we find

ρ′(l1)f =d

dtf(e

−t

0 0 00 0 −10 1 0

x1

x2

x3

)|t=0 (8.1)

=d

dtf(

0 0 00 cos t sin t0 − sin t cos t

x1

x2

x3

)|t=0 (8.2)

=d

dtf(

0x2 cos t+ x3 sin t−x2 sin t+ x3 cos t

)|t=0 (8.3)

=(∂f

∂x1,∂f

∂x2,∂f

∂x3) ·

0x3

−x2

(8.4)

=x3∂f

∂x2− x2

∂f

∂x3(8.5)

so

ρ′(l1) = x3∂

∂x2− x2

∂

∂x3

and similar calculations give

ρ′(l2) = x1∂

∂x3− x3

∂

∂x1, ρ′(l3) = x2

∂

∂x1− x1

∂

∂x2

The space of all functions on R3 is much too big: it will give us an infinity ofcopies of each finite dimensional representation that we want. Notice that whenSO(3) acts on R3, it leaves the distance to the origin invariant. If we work inspherical coordinates (r, θ, φ) (see picture)

95

we will have

x1 =r sin θ cosφ

x2 =r sin θ sinφ

x3 =r cos θ

Acting on f(r, φ, θ), SO(3) will leave r invariant, only acting non-trivially onθ, φ. It turns out that we can cut down the space of functions to somethingthat will only contain one copy of the representation we want in various ways.One way to do this is to restrict our functions to the unit sphere, i.e. just lookat functions f(θ, φ). We will see that the representations we are looking for canbe found in simple trigonometric functions of these two angular variables.

We can construct our irreducible representations ρl by explicitly constructinga function we will call Y ll (θ, φ) that will be a highest weight vector of weightl. The weight l condition and the highest weight condition give two differentialequations for Y ll (θ, φ):

L3Yll = lY ll , L+Y

ll = 0

These will turn out to have a unique solution (up to scalars).

We first need to change coordinates from rectangular to spherical in ourexpressions for L3, L±. Using the chain rule to compute expressions like

∂

∂rf(x1(r, θ, φ), x2(r, θ, φ), x3(r, θ, φ))

we find ∂∂r∂∂θ∂∂φ

=

sin θ cosφ sin θ sinφ cos θr cos θ cosφ r cos θ sinφ − sin θ−r sin θ sinφ r sin θ cosφ 0

∂∂x1∂∂x2∂∂x3

96

so ∂∂r

1r∂∂θ

1r sin θ

∂∂φ

=

sin θ cosφ sin θ sinφ cos θcos θ cosφ cos θ sinφ − sin θ− sinφ cosφ 0

∂∂x1∂∂x2∂∂x3

This is an orthogonal matrix, so one can invert it by taking its transpose, to get ∂

∂x1∂∂x2∂∂x3

=

sin θ cosφ cos θ cosφ − sinφsin θ sinφ cos θ sinφ cosφ

cos θ − sin θ 0

∂∂r

1r∂∂θ

1r sin θ

∂∂φ

So we finally have

L1 = iρ′(l1) = i(x3∂

∂x2− x2

∂

∂x3) = i(sinφ

∂

∂θ+ cot θ cosφ

∂

∂φ)

L2 = iρ′(l2) = i(x1∂

∂x3− x3

∂

∂x1) = i(− cosφ

∂

∂θ+ cot θ sinφ

∂

∂φ)

L3 = iρ′(l3) = i(x1∂

∂x3− x3

∂

∂x1) = −i ∂

∂φ

and

L+ = iρ′(l+) = eiφ(∂

∂θ+ i cot θ

∂

∂φ), L− = iρ′(l−) = e−iφ(− ∂

∂θ+ i cot θ

∂

∂φ)

Now that we have expressions for the action of the Lie algebra on functions inspherical coordinates, our two differential equations saying our function Y ll (θ, φ)is of weight l and in the highest-weight space are

L3Yll (θ, φ) = −i ∂

∂φY ll (θ, φ) = lY ll (θ, φ)

and

L+Yll (θ, φ) = eiφ(

∂

∂θ+ i cot θ

∂

∂φ)Y ll (θ, φ) = 0

The first of these tells us that

Y ll (θ, φ) = eilφFl(θ)

for some function Fl(θ), and using the second we get

(∂

∂θ− l cot θ)Fl(θ)

with solutionFl(θ) = Cll sin

l θ

for an arbitrary constant Cll. Finally

Y ll (θ, φ) = Clleilφ sinl θ

97

This is a function on the sphere, which is also a highest weight vector in a2l + 1 dimensional irreducible representation of SO(3). To get functions whichgive vectors spanning the rest of the weight spaces, one just repeatedly appliesthe lowering operator L−, getting functions

Y ml (θ, φ) =Clm(L−)l−mY ll (θ, φ)

=Clm(e−iφ(− ∂

∂θ+ i cot θ

∂

∂φ))l−meilφ sinl θ

for m = l, l − 1, l − 2 · · · ,−l + 1,−lThe functions Y ml (θ, φ) are called “spherical harmonics”, and they span the

space of complex functions on the sphere in much the same way that the einθ

span the space of complex valued functions on the circle. Unlike the case ofpolynomials on C2, for functions on the sphere, one gets finite numbers byintegrating such functions over the sphere. So one can define an inner producton these representations for which they are unitary by simply setting

〈f, g〉 =

∫S2

fg sin θdθdφ =

∫ 2π

φ=0

∫ π

θ=0

f(θ, φ)g(θ, φ) sin θdθdφ

We will not try and show this here, but for the allowable values of l,m theY ml (θ, φ) are mutually orthogonal with respect to this inner product.

One can derive various general formulas for the Y ml (θ, φ) in terms of Leg-endre polynomials, but here we’ll just compute the first few examples, withthe proper constants that give them norm 1 with respect to the chosen innerproduct.

• For the l = 0 representation

Y 00 (θ, φ) =

√1

4π

• For the l = 1 representation

Y 11 = −

√3

8πsin θeiφ, Y 0

1 =

√3

4πcos θ, Y −1

1 =

√3

8πsin θe−iφ

(one can easily see that these have the correct eigenvalues for ρ′(L3) =−i ∂∂φ ).

• For the l = 2 representation one has

Y 22 =

√15

32πsin2 θei2φ, Y 1

2 = −√

15

8πsin θ cos θeiφ

Y 02 =

√5

16π(3 cos2 θ − 1)

Y −12 =

√15

8πsin θ cos θe−iφ, Y −2

2 =

√15

32πsin2 θe−i2φ

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We will see later that these functions of the angular variables in sphericalcoordinates are exactly the functions that give the angular dependence of wavefunctions for the physical system of a particle in a spherically symmetric poten-tial. In such a case the SO(3) symmetry of the system implies that the statespace (the wave functions) will provide a unitary representation π of SO(3), andthe action of the Hamiltonian operator H will commute with the action of theoperators L3, L±. As a result all of the states in an irreducible representationcomponent of π will have the same energy. States are thus organized into “or-bitals”, with singlet states called “s” orbitals (l = 0), triplet states called “p”orbitals (l = 1), multiplicity 5 states called “d” orbitals (l = 2), etc.

8.4 The Casimir operator

For both SU(2) and SO(3), we have found that all representations can beconstructed out of function spaces, with the Lie algebra acting as first-orderdifferential operators. It turns out that there is also a very interesting second-order differential operator that comes from these Lie algebra representations,known as the Casimir operator. For the case of SO(3)

Definition (Casimir operator for SO(3)). The Casimir operator for the repre-sentation of SO(3) on functions on S2 is the second-order differential operator

L2 = L21 + L2

2 + L23

A straightforward calculation using the commutation relations satisfied bythe Lj shows that

[L2, ρ′(X)] = 0

for anyX ∈ so(3). Knowing this, a version of Schur’s lemma says that L2 will acton an irreducible representation as a scalar (i.e. all vectors in the representationare eigenvectors of L2, with the same eigenvalue). This eigenvalue can be usedto characterize the irreducible representation.

The easiest way to compute this eigenvalue turns out to be to act with L2 ona highest weight vector. First one rewrites L2 in terms of raising and loweringoperators.

L−L+ =(L1 − iL2)(L1 + iL2)

=L21 + L2

2 + i[L1, L2]

=L21 + L2

2 − L3

so

L2 = L21 + L2

2 + L23 = L−L+ + L3 + L2

3

For the representation ρ of SO(3) on functions on S2 constructed above,we know that on a highest weight vector of the irreducible representation ρl

99

(restriction of ρ to the 2l+ 1 dimensional irreducible subspace of functions thatare linear combinations of the Y ml (θ, φ)), we have the two eigenvalue equations

L+f = 0, L3f = lf

with solution the functions proportional to Y ll (θ, φ). Just from these conditionsand our expression for L2 we can immediately find the scalar eigenvalue of L2

sinceL2f = L−L+f + (L3 + L2

3)f = 0 + l + l2 = l(l + 1)

We have thus shown that our irreducible representation ρl can be characterizedas the representation on which L2 acts by the scalar l(l + 1).

In summary, we have two different sets of partial differential equations whosesolutions provide a highest weight vector for and thus determine the irreduciblerepresentation ρl:

•L+f = 0, L3f = lf

which are first order equations, with the first using complexification andsomething like a Cauchy-Riemann equation, and

•L2f = l(l + 1)f, L3f = lf

where the first equation is a second order equation, something like aLaplace equation.

That a solution of the first set of equations gives a solution of the second setis obvious. Much harder to show is that a solution of the second set gives asolution of the first set. Note that just looking at the solution space of

L2f = l(l + 1)f

for l a non-negative integer gives explicitly a 2l + 1-dimensional vector space,with an action of SO(3), providing the spin l representation.

One can compute the explicit second order differential operator L2 in the ρrepresentation on functions, it is

L2 =L21 + L2

2 + L23

=(i(sinφ∂

∂θ+ cot θ cosφ

∂

∂φ))2 + (i(− cosφ

∂

∂θ+ cot θ sinφ

∂

∂φ))2 + (−i ∂

∂φ)2

=− (1

sin θ

∂

∂θ(sin θ

∂

∂θ) +

1

sin2 θ

∂2

∂φ2)

We will re-encounter this operator later on in the course as the angular part ofthe Laplace operator on R3.

For the group SU(2) we can also find irreducible representations as solutionspaces of differential equations on functions on C2. In that case, the differentialequation point of view is much less useful, since the solutions we are looking forare just the homogeneous polynomials, which are more easily studied by purelyalgebraic methods.

100


The classification of SU(2) representations is a standard topic in all textbooksthat deal with Lie group representations. A good example is [27], which coversthis material well, and from which the discussion here of the construction ofrepresentations as homogeneous polynomials is drawn (see pages 77-79). Thecalculation of the Lj and the derivation of expressions for spherical harmonicsas Lie algebra representations of so(3) appears in most quantum mechanicstextbooks in one form or another (for example, see Chapter 12 of [53]). Anothersource used here for the explicit constructions of representations is [13], Chapters27-30.

101

102

Chapter 9

Tensor Products,Entanglement, andAddition of Spin

If one has two independent quantum systems, with state spaces H1 and H2,the combined quantum system has a description that exploits the mathematicalnotion of a “tensor product”, with the combined state space the tensor productH1 ⊗ H2. Because of the ability to take linear combinations of states, thiscombined state space will contain much more than just products of independentstates, including states that are described as “entangled”, and responsible forsome of the most counter-intuitive behavior of quantum physical systems.

This same tensor product construction is a basic one in representation the-ory, allowing one to construct a new representation (φ1 ⊗ φ2,W1 ⊗W2) out ofrepresentations (φ1,W1) and (φ2,W2). When we take the tensor product ofstates corresponding to two irreducible representations of SU(2) of spins s1, s2,we will get a new representation (π2s1 ⊗π2s2 , V

2s1 ⊗V 2s2). It will be reducible,a direct sum of representations of various spins, a situation we will analyze indetail.

Starting with a quantum system with state space H that describes a singleparticle, one can describe a system of N particles by taking an N -fold tensorproduct H⊗N = H ⊗ H ⊗ · · · ⊗ H. A deep fact about the physical worldis that for identical particles, we don’t get the full tensor product space, butonly the subspaces either symmetric or antisymmetric under the action of thepermutation group by permutations of the factors, depending on whether ourparticles are “bosons” or “fermions”. An even deeper fact is that elementaryparticles of half-integral spin s must behave as fermions, those of integral spin,bosons.

Digression. When physicists refer to “tensors”, they generally mean the “ten-sor fields” used in general relativity or other geometry-based parts of physics,

103

not tensor products of state spaces. A tensor field is a function on a manifold,taking values in some tensor product of copies of the tangent space and its dualspace. The simplest tensor fields are just vector fields, functions taking valuesin the tangent space. A more non-trivial example is the metric tensor, whichtakes values in the dual of the tensor product of two copies of the tangent space.

9.1 Tensor products

Given two vector spaces V and W (over C, but one could instead take R or anyother field), one can easily construct the direct sum vector space V ⊕W , just bytaking pairs of elements (v, w) for v ∈ V,w ∈W , and giving them a vector spacestructure by the obvious addition and multiplication by scalars. This space willhave dimension

dim(V ⊕W ) = dimV + dimW

If e1, e2, . . . , edimV are a basis of V , and f1, f2, . . . , fdimW a basis of W , the

(e1, 0), (e2, 0), . . . , (edimV , 0), (0, f1), (0, f2), . . . , (0, fdimW )

will be a basis of V ⊕W .A less trivial construction is the tensor product of the vector spaces V and

W . This will be a new vector space called V ⊗W , of dimension

dim(V ⊗W ) = (dimV )(dimW )

One way to motivate the tensor product is to think of vector spaces as vectorspaces of functions. Elements

v = v1e1 + v2e2 + · · ·+ vdimV edimV ∈ V

can be thought of as functions on the dimV points ei, taking values vi at ei. Ifone takes functions on the union of the sets ei and fj one gets elements ofV ⊕W . The tensor product V ⊗W will be what one gets by taking all functionson not the union, but the product of the sets ei and fj. This will be theset with (dimV )(dimW ) elements, which we will write ei ⊗ fj , and elementsof V ⊗W will be functions on this set, or equivalently, linear combinations ofthese basis vectors.

This sort of definition is less than satisfactory, since it is tied to an explicitchoice of bases for V and W . We won’t however pursue more details of thisquestion or a better definition here. For this, one can consult pretty much anyadvanced undergraduate text in abstract algebra, but here we will take as giventhe following properties of the tensor product that we will need:

• Given vectors v ∈ V,w ∈W we get an element v⊗w ∈ V ⊗W , satisfyingbilinearity conditions (c1, c2 ∈ C)

v ⊗ (c1w1 + c2w2) = c1(v ⊗ w1) + c2(v ⊗ w2)

(c1v1 + c2v2)⊗ w = c1(v1 ⊗ w) + c2(v2 ⊗ w)

104

• There are natural isomorphisms

C⊗ V ' V, V ⊗W 'W ⊗ V

andU ⊗ (V ⊗W ) ' (U ⊗ V )⊗W

for vector spaces U, V,W

• Given a linear operator A on V and another linear operator B on W , wecan define a linear operator A⊗B on V ⊗W by

(A⊗B)(v ⊗ w) = Av ⊗Bw

for v ∈ V,w ∈W .

With respect to the bases ei, fj of V and W , A will be a (dimV ) by(dimV ) matrix, B will be a (dimW ) by (dimW ) matrix and A⊗ B willbe a (dimV )(dimW ) by (dimV )(dimW ) matrix (which one can think ofas a (dimV ) by (dimV ) matrix of blocks of size (dimW )).

• One often wants to consider tensor products of vector spaces and dual vec-tor spaces. An important fact is that there is an isomorphism between thetensor product V ∗⊗W and linear maps from V to W given by identifyingl ⊗ w (l ∈ V ∗) with the linear map

v ∈ V → l(v)w ∈W

9.2 Composite quantum systems and tensor prod-ucts

Consider two quantum systems, one defined by a state space H1 and a set ofoperators O1 on it, the second given by a state spaceH2 and set of operators O2.One can describe the composite quantum system corresponding to consideringthe two quantum systems as a single one, with no interaction between them, byjust taking as a new state space

HT = H1 ⊗H2

with operators of the form

A⊗ Id + Id⊗B

with A ∈ O1, B ∈ O2. To describe an interacting quantum system, one can usethe state space HT , but with a more general class of operators.

If H is the state space of a quantum system, one can think of this as de-scribing a single particle, and then to describe a system of N such particles, oneuses the multiple tensor product

H⊗N = H⊗H⊗ · · · ⊗ H ⊗H︸︷︷︸N times

105

The symmetric group SN acts on this state space, and one has a repre-sentation (π,H⊗N ) of SN as follows. For σ ∈ SN a permutation of the set1, 2, . . . , N of N elements, on a tensor product of vectors one has

π(σ)(v1 ⊗ v2 ⊗ · · · ⊗ vN ) = vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(N)

The representation of SN that this gives is in general reducible, containingvarious components with different irreducible representations of the group SN .

A fundamental axiom of quantum mechanics is that ifH⊗N describesN iden-tical particles, then all physical states occur as one-dimensional representationsof SN , which are either symmetric (“bosons”) or anti-symmetric (“fermions”)where

Definition. A state v ∈ H⊗N is called

• symmetric, or bosonic if ∀σ ∈ SNπ(σ)v = v

The space of such states is denoted SN (H).

• anti-symmetric, or fermionic if ∀σ ∈ SNπ(σ)v = (−1)|σ|v

The space of such states is denoted ΛN (H). Here |σ| is the minimal num-ber of transpositions that by composition give σ.

Note that in the fermionic case, for σ a transposition interchanging twoparticles, the antisymmetric representation π acts on the factor H ⊗ H by in-terchanging vectors, taking

w ⊗ w ∈ H ⊗H

to itself. Antisymmetry requires that this state go to its negative, so the statecannot be non-zero. So one cannot have non-zero states in H⊗N describing twoidentical particles in the same state w ∈ H, a fact that is known as the “PauliPrinciple”.

While the symmetry or anti-symmetry of states of multiple identical particlesis a separate axiom when such particles are described in this way as tensorproducts, we will see later on (chapter 32) that this phenomenon instead findsa natural explanation when particles are described in terms of quantum fields.

9.3 Indecomposable vectors and entanglement

If one is given a function f on a space X and a function g on a space Y , onecan form a product function fg on the product space X × Y by taking (forx ∈ X, y ∈ Y )

(fg)(x, y) = f(x)g(y)

However, most functions on X × Y are not decomposable in this manner. Sim-ilarly, for a tensor product of vector spaces, one has:

106

Definition (Decomposable and indecomposable vectors). A vector in V ⊗Wis called decomposable if it is of the form v ⊗ w for some v ∈ V,w ∈ W . If itcannot be put in this form it is called indecomposable.

Note that our basis vectors of V ⊗W are all decomposable since they areproducts of basis vectors of V and W . Linear combinations of these basis vectorshowever are in general indecomposable. If we think of an element of V ⊗Was a dimV by dimW matrix, with entries the coordinates with respect to ourbasis vectors for V ⊗W , then for decomposable vectors we get a special classof matrices, those of rank one.

In the physics context, the language used is:

Definition (Entanglement). An indecomposable state in the tensor productstate space HT = H1 ⊗H2 is called an entangled state.

The phenomenon of entanglement is responsible for some of the most surpris-ing and subtle aspects of quantum mechanical system. The Einstein-Podolsky-Rosen paradox concerns the behavior of an entangled state of two quantumsystems, when one moves them far apart. Then performing a measurement onone system can give one information about what will happen if one performsa measurement on the far removed system, introducing a sort of unexpectednon-locality.

Measurement theory itself involves crucially an entanglement between thestate of a system being measured, thought of as in a state space Hsystem, andthe state of the measurement apparatus, thought of as lying in a state spaceHapparatus. The laws of quantum mechanics presumably apply to the totalsystem Hsystem⊗Happaratus, with the counter-intuitive nature of measurementsappearing due to this decomposition of the world into two entangled parts: theone under study, and a much larger for which only an approximate descriptionin classical terms is possible. For much more about this, a recommended readingis Chapter 2 of [48].

9.4 Tensor products of representations

Given two representations of a group, one can define a new representation, thetensor product representation, by

Definition (Tensor product representation of a group). For (πV , V ) and (πW ,W )representations of a group G, one has a tensor product representation (πV⊗W , V⊗W ) defined by

(πV⊗W (g))(v ⊗ w) = πV (g)v ⊗ πW (g)w

One can easily check that πV⊗W is a homomorphism.

To see what happens for the corresponding Lie algebra representation, one

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computes (for X in the Lie algebra)

π′V⊗W (X)(v ⊗ w) =d

dtπV⊗W (etX)(v ⊗ w)t=0

=d

dt(πV (etX)v ⊗ πW (etX)w)t=0

=((d

dtπV (etX)v)⊗ πW (etX)w)t=0 + (πV (etX)v ⊗ (

d

dtπW (etX)w))t=0

=(π′V (X)v)⊗ w + v ⊗ (π′W (X)w)

which could also be written

π′V⊗W (X) = (π′V (X)⊗ 1W ) + (1V ⊗ π′W (X))

9.4.1 Tensor products of SU(2) representations

Given two representations (πV , V ) and (πW ,W ) of a group G, we can decom-pose each into irreducibles. To do the same for the tensor product of the tworepresentations, we need to know how to decompose the tensor product of twoirreducibles. This is a fundamental non-trivial problem for a group G, with theanswer for G = SU(2) as follows:

Theorem 9.1 (Clebsch-Gordan decomposition). The tensor product (πn1⊗

πn2, V n1 ⊗ V n2) decomposes into irreducibles as

(πn1+n2, V n1+n2)⊕ (πn1+n2−2, V

n1+n2−2)⊕ · · · ⊕ (π|n1−n2|, V|n1−n2|)

Proof. One way to prove this result is to use highest-weight theory, raising andlowering operators, and the formula for the Casimir operator. We will not tryand show the details of how this works out, but in the next section give asimpler argument using characters. However, in outline (for more details, seefor instance section 5.2 of [46]), here’s how one could proceed:

One starts by noting that if vn1∈ Vn1

, vn2∈ Vn2

are highest weight vectorsfor the two representations, vn1

⊗vn2will be a highest weight vector in the tensor

product representation (i.e. annihilated by π′n1+n2(S+)), of weight n1 + n2.

So (πn1+n2 , Vn1+n2) will occur in the decomposition. Applying π′n1+n2

(S−) tovn1⊗vn2

one gets a basis of the rest of the vectors in (πn1+n2, V n1+n2). However,

at weight n1 +n2−2 one can find another kind of vector, a highest-weight vectororthogonal to the vectors in (πn1+n2

, V n1+n2). Applying the lowering operatorto this gives (πn1+n2−2, V

n1+n2−2). As before, at weight n1 + n2 − 4 one findsanother, orthogonal highest weight vector, and gets another representation, withthis process only terminating at weight |n1 − n2|.

9.4.2 Characters of representations

A standard tool for dealing with representations that we have ignored so far isthat of associating to a representation an invariant called its character. This

108

will be a conjugation-invariant function on the group that only depends on theequivalence class of the representation. Given two representations constructedin very different ways, one can often check whether they are isomorphic just byseeing if their character functions match. The problem of identifying the possibleirreducible representations of a group can be attacked by analyzing the possiblecharacter functions of irreducible representations. We will not try and enterinto the general theory of characters here, but will just see what the charactersof irreducible representations are for the case of G = SU(2). These can be usedto give a simple argument for the Clebsch-Gordan decomposition of the tensorproduct of SU(2) representations. For this we don’t need general theoremsabout the relations of characters and representations, but can directly checkthat the irreducible representations of SU(2) correspond to distinct characterfunctions which are easily evaluated.

Definition (Character). The character of a representation (π, V ) of a group Gis the function on G given by

χV (g) = Tr(π(g))

Since the trace of a matrix is invariant under conjugation, χV in general willbe a complex valued, conjugation-invariant function on G. One can easily checkthat it will satisfy the relations

χV⊕W = χV + χW , χV⊗W = χV χW

For the case of G = SU(2), any element can be conjugated to be in thesubgroup U(1) of diagonal matrices. Knowing the weights of the irreduciblerepresentations (πn, V

n) of SU(2), we know the characters to be the functions

χV n(

(eiθ 00 e−iθ

)) = einθ + ei(n−2)θ + · · ·+ e−i(n−2)θ + e−inθ (9.1)

As n gets large, this becomes an unwieldy expression, but one has

Theorem (Weyl character formula).

χV n(

(eiθ 00 e−iθ

)) =

ei(n+1)θ − e−i(n+1)θ

eiθ − e−iθ=

sin((n+ 1)θ)

sin(θ)

Proof. One just needs to use the identity

(einθ + ei(n−2)θ + · · ·+ e−i(n−2)θ + e−inθ)(eiθ − e−iθ) = ei(n+1)θ − e−i(n+1)θ

and equation 9.1 for the character.

To get a proof of 9.1, one can compute the character of the tensor producton the diagonal matrices using the Weyl character formula for the second factor

109

(ordering things so that n2 > n1)

χV n1⊗V n2 =χV n1χV n2

=(ein1θ + ei(n1−2)θ + · · ·+ e−i(n1−2)θ + e−in1θ)ei(n2+1)θ − e−i(n2+1)θ

eiθ − e−iθ

=(ei(n1+n2+1)θ − e−i(n1+n2+1)θ) + · · ·+ (ei(n2−n1+1)θ − e−i(n2−n1+1)θ)

eiθ − e−iθ=χV n1+n2 + χV n1+n2−2 + · · ·+ χV n2−n1

So, when we decompose the tensor product of irreducibles into a direct sum ofirreducibles, the ones that must occur are exactly those of theorem 9.1.

9.4.3 Some examples

Some simple examples of how this works are:

• Tensor product of two spinors:

V 1 ⊗ V 1 = V 2 ⊕ V 0

This says that the four complex dimensional tensor product of two spinorrepresentations (which are each two complex dimensional) decomposesinto irreducibles as the sum of a three dimensional vector representationand a one dimensional trivial (scalar) representation.

Using the basis

(10

),

(01

)for V 1, the tensor product V 1⊗V 1 has a basis(

10

)⊗(

10

),

(10

)⊗(

01

),

(01

)⊗(

10

),

(01

)⊗(

01

)The vector

1√2

(

(10

)⊗(

01

)−(

01

)⊗(

10

)) ∈ V 1 ⊗ V 1

is clearly antisymmetric under permutation of the two factors of V 1⊗V 1.One can show that this vector is invariant under SU(2), by computingeither the action of SU(2) or of its Lie algebra su(2). So, this vectoris a basis for the component V 0 in the decomposition of V 1 ⊗ V 1 intoirreducibles.

The other component, V 2, is three dimensional, and has a basis(10

)⊗(

10

),

1√2

(

(10

)⊗(

01

)+

(01

)⊗(

10

)),

(01

)⊗(

01

)These three vectors span one-dimensional complex subspaces of weightsq = 2, 0,−2 under the U(1) ⊂ SU(2) subgroup(

eiθ 00 e−iθ

)110

They are symmetric under permutation of the two factors of V 1 ⊗ V 1.

We see that if we take two identical quantum systems with H = V 1 = C2

and make a composite system out of them, if they were bosons we wouldget a three dimensional state space V 2 = S2(V 1), transforming as a vector(spin one) under SU(2). If they were fermions, we would get a one-dimensional state space V 0 = Λ2(V 1) of spin zero (invariant under SU(2)).Note that in this second case we automatically get an entangled state, onethat cannot be written as a decomposable product.

• Tensor product of three or more spinors:

V 1⊗V 1⊗V 1 = (V 2⊕V 0)⊗V 1 = (V 2⊗V 1)⊕ (V 0⊗V 1) = V 3⊕V 1⊕V 1

This says that the tensor product of three spinor representations decom-poses as a four dimensional (“spin 3/2”) representation plus two copies ofthe spinor representation.

One can clearly generalize this and considerN -fold tensor products (V 1)⊗N

of the spinor representation. Taking N high enough one can get any ir-reducible representation of SU(2) that one wants this way, giving an al-ternative to our construction using homogeneous polynomials. Doing thishowever gives the irreducible as just one component of something larger,and one needs a method to project out the component one wants. Onecan do this using the action of the symmetric group SN on (V 1)⊗N andan understanding of the irreducible representations of SN . This relation-ship between irreducible representations of SU(2) and those of SN comingfrom looking at how both groups act on (V 1)⊗N is known as “Schur-Weylduality”, and generalizes to the case of SU(n), where one looks at N -foldtensor products of the defining representation of SU(n) matrices on Cn.For SU(n) this provides perhaps the most straight-forward constructionof all irreducible representations of the group.

9.5 Bilinear forms and tensor products

A different sort of application of tensor products that will turn out to be im-portant is to the description of bilinear forms, which generalize the dual spaceV ∗ of linear forms on V . We have

Definition (Bilinear forms). A bilinear form B on a vector space V over a fieldk (for us, k = R or C) is a map

B : (u, u′) ∈ V × V → B(u, u′) ∈ k

that is bilinear in both entries, i.e.

B(u+ u′, u′′) = B(u, u′′) +B(u′, u′′), B(cu, u′) = cB(u, u′)

B(u, u′ + u′′) = B(u, u′) +B(u, u′′), B(u, cu′) = cB(u, u′)

111

where c ∈ k.If B(u′, u) = B(u, u′) the bilinear form is called symmetric, if B(u′, u) =

−B(u, u′) it is anti-symmetric.

The relation to tensor products is

Theorem 9.2. The space of bilinear forms on V is isomorphic to V ∗ ⊗ V ∗.

Proof. Given two linear forms α ∈ V ∗, β ∈ V ∗, one has a map

α⊗ β ∈ V ∗ ⊗ V ∗ → B : B(u, u′) = α(u)β(u′)

Choosing a basis ej of V , the coordinate functions vj provide a basis of V ∗, sothe vj⊗vk will be a basis of V ∗⊗V ∗. The map above takes linear combinationsof these to bilinear forms, and is easily seen to be one-to-one and surjective forsuch linear combinations.

Given a basis ej of V and dual basis vj of V ∗ (the coordinates), one canwrite the element of V ∗ ⊗ V ∗ corresponding to B as the sum∑

j,k

Bjkvj ⊗ vk

This expresses the bilinear form B in terms of a matrix B with entries Bjk,which can be computed as

Bjk = B(ej , ek)

In terms of the matrix B, the bilinear form is computed as

B(u, u′) =(u1 . . . ud

)B11 . . . B1d

......

...Bd1 . . . Bdd

u′1...u′d

= u ·Bu′

The symmetric bilinear forms lie in S2(V ∗) ⊂ V ∗ ⊗ V ∗ and correspond tosymmetric matrices. Elements of V ∗ give linear functions on V , and one canget quadratic functions on V from elements B ∈ S2(V ∗) by taking

u ∈ V → B(u, u) = u ·Bu

That one gets quadratic functions by multiplying two linear functions corre-sponds in terms of tensor products to

(α, β) ∈ V ∗ × V ∗ → 1

2(α⊗ β + β ⊗ α) ∈ S2(V ∗)

We will not give the details here, but one can generalize the above frombilinear forms (isomorphic to V ∗⊗V ∗) to multi-linear forms with N arguments(isomorphic to (V ∗)⊗N ). Evaluating such a multi-linear form with all argu-ments set to u ∈ V gives a homogeneous polynomial of degree N , and one

112

has an isomorphism between symmetric multi-linear forms in SN (V ∗) and suchpolynomials.

Anti-symmetric bilinear forms lie in Λ2(V ∗) ⊂ V ∗ ⊗ V ∗ and correspond toanti-symmetric matrices. One can define a multiplication (called the “wedgeproduct”) on V ∗ that takes values in Λ2(V ∗) by

(α, β) ∈ V ∗ × V ∗ → α ∧ β =1

2(α⊗ β − β ⊗ α) ∈ Λ2(V ∗)

One can use this to get a product on the space of anti-symmetric multilinearforms of different degrees, giving something in many ways analogous to thealgebra of polynomials. This plays a role in the description of fermions and willbe considered in more detail in chapter 26.


For more about the tensor product and tensor product of representations, seesection 6 of [61], or appendix B of [55]. Almost every quantum mechanics text-book will contain an extensive discussion of the Clebsch-Gordan decompositionfor the tensor product of two irreducible SU(2) representations.

113

114

Chapter 10

Energy, Momentum andTranslation Groups

We’ll now turn to the problem that conventional quantum mechanics coursesgenerally begin with: that of the quantum system describing a free particlemoving in physical space R3. This is something quite different than the classicalmechanical decription of a free particle, which will be reviewed in chapter 12.A common way of motivating this is to begin with the 1924 suggestion by deBroglie that, just as photons may behave like particles or waves, the same shouldbe true for matter particles. Photons carry an energy given by E = ~ω, whereω is the angular frequency, and de Broglie’s proposal was that matter particleswould behave like a wave with spatial dependence

eik·x

where x is the spatial position, and the momentum of the particle is p = ~k.This proposal was realized in Schrodinger’s early 1926 discovery of a version

of quantum mechanics, in which the state space H is a space of complex-valuedfunctions on R3, called “wave-functions”. The operator

P = −i~∇

will have eigenvalues ~k, the de Broglie momentum, so it can be identified asthe momentum operator.

In this chapter our discussion will emphasize the central role of the momen-tum operator. This operator will have the same relationship to spatial trans-lations as the Hamiltonian operator does to time translations. In both cases,the operators are given by the Lie algebra representation corresponding to aunitary representation on the quantum state space H of groups of translations(translation in the three space and one time directions respectively).

One way to motivate the quantum theory of a free particle is that, whateverit is, it should have the same sort of behavior as in the classical case under thetranslational and rotational symmetries of space-time. In chapter 12 we will

115

see that in the Hamiltonian form of classical mechanics, the momentum vectorgives a basis of the Lie algebra of the spatial translation group R3, the energy abasis of the Lie algebra of the time translation group R. Invoking the classicalrelationship between energy and momentum

E =|p2|2m

used in non-relativistic mechanics relates the Hamiltonian and momentum op-erators, giving the conventional Schrodinger differential equation for the wave-function of a free particle. We will examine the solutions to this equation, begin-ning with the case of periodic boundary conditions, where spatial translationsin each direction are given by the compact group U(1) (whose representationswe have studied in detail).

10.1 Energy, momentum and space-time trans-lations

We have seen that it is a basic axiom of quantum mechanics that the observ-able operator responsible for infinitesimal time translations is the Hamiltonianoperator H, a fact that is expressed as the Schrodinger equation

i~d

dt|ψ〉 = H|ψ〉

When H is time-independent, one can understand this equation as reflecting theexistence of a unitary representation (U(t),H) of the group R of time transla-tions on the state space H. For the case of H infinite-dimensional, this is knownas Stone’s theorem for one-parameter unitary groups, see for instance chapter10.2 of [28] for details.

When H is finite-dimensional, the fact that a differentiable unitary repre-sentation U(t) of R on H is of the form

U(t) = e−i~ tH

for H a self-adjoint matrix follows from the same sort of argument as in theorem2.1. Such a U(t) provides solutions of the Schrodinger equation by

|ψ(t)〉 = U(t)|ψ(0)〉

The Lie algebra of R is also R and we get a Lie algebra representation of Rby taking the time derivative of U(t), which gives us

~d

dtU(t)|t=0 = −iH

Since this Lie algebra representation comes from taking the derivative of a uni-tary representation, −iH will be skew-adjoint, so H will be self-adjoint. The

116

minus sign is a convention, for reasons that will be explained in the discussionof momentum to come later.

Note that if one wants to treat the additive group R as a matrix group,related to its Lie algebra R by exponentiation of matrices, one can describe thegroup as the group of matrices of the form(

1 a0 1

)since (

1 a0 1

)(1 b0 1

)=

(1 a+ b0 1

)Since

e

0 a0 0

=

(1 a0 1

)the Lie algebra is just matrices of the form(

0 a0 0

)We will mostly though write the group law in additive form. We are inter-

ested in the group R as a group of translations acting on a linear space, and thecorresponding infinite dimensional representation induced on functions on thespace. The simplest case is when R acts on itself by translation. Here a ∈ Racts on q ∈ R (where q is a coordinate on R) by

q → a · q = q + a

and the induced representation π on functions uses

π(g)f(q) = f(g−1 · q)

to getπ(a)f(q) = f(q − a)

In the Lie algebra version of this representation, we will have

π′(a) = −a ddq

since

π(a)f = eπ′(a)f = e−a

ddq f(q) = f(q)− adf

dq+a2

2!

d2f

dq2+ · · · = f(q − a)

which for functions with appropriate properties is just Taylor’s formula. Notethat here the same a labels points of the Lie algebra and of the group. We arenot treating the group R as a matrix group, since we want an additive group

117

law. So Lie algebra elements are not defined as for matrix groups (things oneexponentiates to get group elements). Instead, we think of the Lie algebra asthe tangent space to the group at the identity, and then simply identify R as thetangent space at 0 (the Lie algebra) and R as the additive group. Note howeverthat the representation obeys a multiplicative law, with the homomorphismproperty

π(a+ b) = π(a)π(b)

so there is an exponential in the relation between π and π′.Since we now want to describe quantum systems that depend not just on

time, but on space variables q = (q1, q2, q3), we will have an action by unitarytransformations of not just the group R of time translations, but also the groupR3 of spatial translations. We will define the corresponding Lie algebra rep-resentations using self-adjoint operators P1, P2, P3 that play the same role forspatial translations that the Hamiltonian plays for time translations:

Definition (Momentum operators). For a quantum system with state spaceH given by complex valued functions of position variables q1, q2, q3, momentumoperators P1, P2, P3 are defined by

P1 = −i~ ∂

∂q1, P2 = −i~ ∂

∂q2, P3 = −i~ ∂

∂q3

These are given the name “momentum operators” since we will see that theireigenvalues have an intepretation as the components of the momentum vectorfor the system, just as the eigenvalues of the Hamiltonian have an interpretationas the energy. Note that while in the case of the Hamiltonian the factor of ~ kepttrack of the relative normalization of energy and time units, here it plays thesame role for momentum and length units. It can be set to one if appropriatechoices of units of momentum and length are made.

The differentiation operator is skew-adjoint since, using integration by partsone has for ψ ∈ H∫ +∞

−∞ψ(

d

dqψ)dq =

∫ +∞

−∞(d

dq(ψψ)− (

d

dqψ)ψ)dq = −

∫ +∞

−∞(d

dqψ)ψdq

The Pj are thus self-adjoint operators, with real eigenvalues as expected for anobservable operator. Multiplying by −i to get the corresponding skew-adjointoperator of a unitary Lie algebra representation we find

−iPj = −~ ∂

∂qj

Up to the ~ factor that depends on units, these are exactly the Lie algebrarepresentation operators on basis elements for the action of R3 on functions onR3 induced from translation:

π(a1, a2, a3)f(q1, q2, q3) = f(q1 − a1, q2 − a2, q3 − a3)

118

π′(a1, a2, a3) = a1(−iP1) +a2(−iP2) +a3(−iP3) = −~(a1∂

∂q1+a2

∂

∂q2+a3

∂

∂q3)

Note that the convention for the sign choice here is the opposite from thecase of the Hamiltonian (−iP = −~ d

dq vs. −iH = ~ ddt ). This means that the

conventional sign choice we have been using for the Hamiltonian makes it minusthe generator of translations in the time direction. The reason for this comesfrom considerations of special relativity, where the inner product on space-timehas opposite signs for the space and time dimensions . We will review thissubject in chapter 35 but for now we just need the relationship special relativitygives between energy and momentum. Space and time are put together in“Minkowski space”, which is R4 with indefinite inner product

< (u0, u1, u2, u3), (v0, v1, v2, v3) >= −u0v0 + u1v1 + u2v2 + u3v3

Energy and momentum are the components of a Minkowski space vector (p0 =E, p1, p2, p3) with norm-squared given by minus the mass-squared:

< (E, p1, p2, p3), (E, p1, p2, p3) >= −E2 + |p|2 = −m2

This is the formula for a choice of space and time units such that the speed oflight is 1. Putting in factors of the speed of light c to get the units right onehas

E2 − |p|2c2 = m2c4

Two special cases of this are:

• For photons, m = 0, and one has the energy momentum relation E = |p|c

• For velocities v small compared to c (and thus momenta |p| small com-pared to mc), one has

E =√|p|2c2 +m2c4 = c

√|p|2 +m2c2 ≈ c|p|2

2mc+mc2 =

|p|2

2m+mc2

In the non-relativistic limit, we use this energy-momentum relation todescribe particles with velocities small compared to c, typically droppingthe momentum-independent constant term mc2.

In later chapters we will discuss quantum systems that describe photons,as well as other possible ways of constructing quantum systems for relativisticparticles. For now though, we will stick to the non-relativistic case. To describea quantum non-relativistic particle, we write the Hamiltonian operator in termsof the components of the momentum operator, in such a way that the eigenvaluesof the operators (the energy and momentum) will satisfy the classical equation

E = |p|22m :

H =1

2m(P 2

1 + P 22 + P 2

3 ) =1

2m|P|2 =

−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)

119

The Schrodinger equation then becomes:

i~∂

∂tψ(q, t) =

−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)ψ(q, t) =−~2

2m∇2ψ(q, t)

This is an easily solved simple constant coefficient second-order partial differ-ential equation. One method of solution is to separate out the time-dependence,by first finding solutions ψE to the time-independent equation

HψE(q) =−~2

2m∇2ψE(q) = EψE(q)

with eigenvalue E for the Hamiltonian operator and then use the fact that

ψ(q, t) = ψE(q)e−i~ tE

will give solutions to the full-time dependent equation

i~∂

∂tψ(q, t) = Hψ(q, t)

The solutions ψE(q) to the time-independent equation are just complex ex-ponentials proportional to

ei(k1q1+k2q2+k3q3) = eik·q

satisfying−~2

2m(−i)2|k|2 =

~2|k|2

2m= E

We have found solutions to the Schrodinger equation given by linear combina-tions of states |k〉 labeled by a vector k which are eigenstates of the momentumand Hamiltonian operators with

Pj |k〉 = ~kj |k〉, H|k〉 =~2

2m|k|2|k〉

These are states with well-defined momentum and energy

pj = ~kj , E =|p|2

2m

so they satisfy exactly the same energy-momentum relations as those for a clas-sical non-relativistic particle.

While the quantum mechanical state space H contains states with the clas-sical energy-momentum relation, it also contains much, much more since itincludes linear combinations of such states. At t = 0 one has

|ψ〉 =∑k

ckeik·q

120

where ck are complex numbers, and the general time-dependent state will be

|ψ(t)〉 =∑k

ckeik·qe−it~

|k|22m

or, equivalently in terms of momenta p = ~k

|ψ(t)〉 =∑p

cpei~p·qe−

i~|p|22m t

10.2 Periodic boundary conditions and the groupU(1)

We have not yet discussed the inner product on our space of states when theyare given as wave-functions on R3, and there is a significant problem with doingthis. To get unitary representations of translations, we need to use a translationinvariant, Hermitian inner product on wave-functions, and this will have to beof the form

< ψ1, ψ2 >= C

∫R3

ψ1(q)ψ2(q)d3q

for some constant C. But if we try and compute the norm-squared of one ofour basis states |k〉 we find

〈k|k〉 = C

∫R3

(e−ik·q)(eik·q)d3q = C

∫R3

1 d3q =∞

As a result there is no value of C which will give these states a unit norm.In the finite dimensional case, a linear algebra theorem assure us that given a

self-adjoint operator, we can find an orthonormal basis of its eigenvectors. In thisinfinite dimensional case this is no longer true, and a much more sophisticatedformalism (the “spectral theorem for self-adjoint operators”) is needed to replacethe linear algebra theorem. This is a standard topic in treatments of quantummechanics aimed at mathematicians emphasizing analysis, but we will not tryand enter into this here. One place to find such a discussion is section 2.1 of[59].

One way to deal with the normalization problem is to replace the non-compact space by one of finite volume. We’ll consider first the simplified case ofa single spatial dimension, since once one sees how this works for one dimension,treating the others the same way is straight-forward. In this one dimensionalcase, one replaces R by the circle S1. This is equivalent to the physicist’s methodof imposing “periodic boundary conditions”, meaning to define the theory onan interval, and then identify the ends of the interval. One can then think ofthe position variable q as an angle φ and define the inner product as

< ψ1, ψ2 >=1

2π

∫ 2π

0

ψ1(φ)ψ2(φ)dφ

121

The state space is thenH = L2(S1)

the space of complex-valued square-integrable functions on the circle.Instead of the translation group R, we have the standard action of the

group SO(2) on the circle. Elements g(θ) of the group are rotations of the circlecounterclockwise by an angle θ, or if we parametrize the circle by an angle φ,just shifts

φ→ φ+ θ

Recall that in general we can construct a representation on functions from agroup action on a space by

π(g)f(x) = f(g−1 · x)

so we see that this rotation action on the circle gives a representation on H

π(g(θ))ψ(φ) = ψ(φ− θ)

If X is a basis of the Lie algebra so(2) (for instance taking the circle as the unit

circle in R2, rotations 2 by 2 matrices, X =

(0 −11 0

), g(θ) = eθX) then the

Lie algebra representation is given by taking the derivative

π′(X)f(φ) =d

dθf(φ− θ)|θ=0 = −f ′(φ)

so we have

π′(X) = − d

dφ

This operator is defined on a dense subspace of H = L2(S1) and is skew-adjoint,since (using integration by parts)

< ψ1, ψ′2 >=

1

2π

∫ 2π

0

ψ1d

dφψ2dφ

=1

2π

∫ 2π

0

(d

dφ(ψ1ψ2)− (

d

dφψ1)ψ2)dφ

=− < ψ′1, ψ2 >

The eigenfunctions of π′(X) are just the einφ, for n ∈ Z, which we will alsowrite as state vectors |n〉. These are also a basis for the space L2(S1), a basisthat corresponds to the decomposition into irreducibles of

L2(S1)

as a representation of SO(2) described above. One has

(π, L2(S1)) = ⊕n∈Z(πn,C)

122

where πn are the irreducible one-dimensional representations given by

πn(g(θ)) = einθ

The theory of Fourier series for functions on S1 says that one can expand anyfunction ψ ∈ L2(S1) in terms of this basis, i.e.

|ψ〉 = ψ(φ) =

+∞∑n=−∞

cneinφ =

+∞∑n=−∞

cn|n〉

where cn ∈ C. The condition that ψ ∈ L2(S1) corresponds to the condition

+∞∑n=−∞

|cn|2 <∞

on the coefficients cn. Using orthonormality of the |n〉 we find

cn = 〈n|ψ〉 =1

2π

∫ 2π

0

e−inφψ(φ)dφ

The Lie algebra of the group S1 is the same as that of the group (R,+),and the π′(X) we have found for the S1 action on functions is related to themomentum operator in the same way as in the R case. So, we can use the samemomentum operator

P = −i~ d

dφ

which satisfiesP |n〉 = ~n|n〉

By changing space to the compact S1 we now have momenta that instead oftaking on any real value, can only be integral numbers times ~. Solving theSchrodinger equation

i~∂

∂tψ(φ, t) =

P 2

2mψ(φ, t) =

−~2

2m

∂2

∂φ2ψ(φ, t)

as before, we find

EψE(φ) =−~2

2m

d2

dφ2ψE(φ)

an eigenvector equation, which has solutions |n〉, with

E =~2n2

2m

Writing a solution to the Schrodinger equation as

ψ(φ, t) =

+∞∑n=−∞

cneinφe−i

~n2

2m t

123

the cn will be determined from the initial condition of knowing the wave-functionat time t = 0, according to the Fourier coefficient formula

cn =1

2π

∫ 2π

0

e−inφψ(φ, 0)dφ

To get something more realistic, we need to take our circle to have an ar-bitrary circumference L, and we can study our original problem by consideringthe limit L → ∞. To do this, we just need to change variables from φ to φL,where

φL =L

2πφ

The momentum operator will now be

P = −i~ d

dφL

and its eigenvalues will be quantized in units of 2π~L . The energy eigenvalues

will be

E =2π2~2n2

mL2

10.3 The group R and the Fourier transform

In the previous section, we imposed periodic boundary conditions, replacing thegroup R of translations by a compact group S1, and then used the fact thatunitary representations of this group are labeled by integers. This made theanalysis rather easy, with H = L2(S1) and the self-adjoint operator P = −i~ ∂

∂φbehaving much the same as in the finite-dimensional case: the eigenvectors ofP give a countable orthonormal basis of H. If one wants to, one can think of Pas an infinite-dimensional matrix.

Unfortunately, in order to understand many aspects of quantum mechanics,we can’t get away with this trick, but need to work with R itself. One reasonfor this is that the unitary representations of R are labeled by the same group,R, and we will find it very important to exploit this and treat positions andmomenta on the same footing (see the discussion of the Heisenberg group inchapter 11). What plays the role here of |n〉 = einφ, n ∈ Z will be the |k〉 = eikq,k ∈ R. These are functions on R that are irreducible representations under thetranslation action (π(a) acts on functions of q by taking q → q − a)

π(a)eikq = eik(q−a) = e−ikaeikq

We can try and mimic the Fourier series decomposition, with the coefficientscn that depend on the labels of the irreducibles replaced by a function f(k)depending on the label k of the irreducible representation of R.

Definition (Fourier transform). The Fourier transform of a function ψ is givenby

Fψ = ψ(k) ≡ 1√2π

∫ ∞−∞

e−ikqψ(q)dq

124

The definition makes sense for ψ ∈ L1(R), Lebesgue integrable functions onR. For the following, it is convenient to instead restrict to the Schwartz spaceS(R) of functions ψ such that the function and its derivatives fall off faster thanany power at infinity (which is a dense subspace of L2(R)). For more detailsabout the analysis and proofs of the theorems quoted here, one can refer to astandard textbook such as [58].

Given the Fourier transform of ψ, one can recover ψ itself:

Theorem (Fourier Inversion). http://www.math.columbia.edu/

For ψ ∈ S(R) the Fourier transform of a function ψ ∈ S(R), one has

ψ(q) = F ψ =1√2π

∫ +∞

−∞eikqψ(k)dk

Note that F is the same linear operator as F , with a change in sign of theargument of the function it is applied to. Note also that we are choosing oneof various popular ways of normalizing the definition of the Fourier transform.In others, the factor of 2π may appear instead in the exponent of the complexexponential, or just in one of F or F and not the other.

The operators F and F are thus inverses of each other on S(R). One has

Theorem (Plancherel). F and F extend to unitary isomorphisms of L2(R)with itself. In other words∫ ∞

−∞|ψ(q)|2dq =

∫ ∞−∞|ψ(k)|2dk

Note that we will be using the same inner product on functions on R

< ψ1, ψ2 >=

∫ ∞−∞

ψ1(q)ψ2(q)dq

both for functions of q and their Fourier transforms, functions of k.An important example is the case of Gaussian functions where

Fe−αq2

2 =1√2π

∫ +∞

−∞e−ikqe−α

q2

2 dq

=1√2π

∫ +∞

−∞e−

α2 ((q+i 2k

α )2−( ikα )2)dq

=1√2πe−

k2

2α

∫ +∞

−∞e−

α2 q′2dq′

=1√αe−

k2

2α

A crucial property of the unitary operator F on H is that it diagonalizesthe differentiation operator and thus the momentum operator P . Under Fouriertransform, differential operators become just multiplication by a polynomial,

125

giving a powerful technique for solving differential equations. Computing theFourier transform of the differentiation operator using integration by parts, wefind

dψ

dq=

1√2π

∫ +∞

−∞e−ikq

dψ

dqdq

=1√2π

∫ +∞

−∞(d

dq(e−ikqψ)− (

d

dqe−ikq)ψ)dq

=ik1√2π

∫ +∞

−∞e−ikqψdq

=ikψ(k)

So under Fourier transform, differentiation by q becomes multiplication by ik.This is the infinitesimal version of the fact that translation becomes multiplica-tion by a phase under Fourier transform. If ψa(q) = ψ(q + a), one has

ψa(k) =1√2π

∫ +∞

−∞e−ikqψ(q + a)dq

=1√2π

∫ +∞

−∞e−ik(q′−a)ψ(q′)dq′

=eikaψ(k)

Since p = ~k, we can easily change variables and work with p instead of k,and often will do this from now on. As with the factors of 2π, there’s a choiceof where to put the factors of ~ in the normalization of the Fourier transform.We’ll make the following choices, to preserve symmetry between the formulasfor Fourier transform and inverse Fourier transform:

ψ(p) =1√2π~

∫ +∞

−∞e−i

pq~ dq

ψ(q) =1√2π~

∫ +∞

−∞eipq~ dp

Note that in this case we have lost an important property that we had forfinite dimensional H and had managed to preserve by using S1 rather thanR as our space. If we take H = L2(R), the eigenvectors for the operator P(the functions eikq) are not square-integrable, so not in H. The operator Pis an unbounded operator and we no longer have a theorem saying that itseigenvectors give an orthornormal basis of H. As mentioned earlier, one wayto deal with this uses a general spectral theorem for self-adjoint operators on aHilbert space, for more details see Chapter 2 of [59].

10.3.1 Delta functions

One would like to think of the eigenvectors of the operator P as in some sensecontinuing to provide an orthonormal basis for H. One problem is that these

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eigenvectors are not square-integrable, so one needs to expand one’s notion ofstate space H beyond a space like L2(R). Another problem is that Fouriertransforms of such eigenvectors (which will be eigenvectors of the position op-erator) gives something that is not a function but a distribution. The propergeneral formalism for handling state spacesH which include eigenvectors of bothposition and momentum operators seems to be that of “rigged Hilbert spaces”which this author confesses to never have mastered (the standard reference is[21]). As a result we won’t here give a rigorous discussion, but will use non-normalizable functions and distributions in the non-rigorous form in which theyare used in physics. The physics formalism is set up to work as if H was finitedimensional and allows easy manipulations which don’t obviously make sense.Our interest though is not in the general theory, but in very specific quantumsystems, where everything is determined by their properties as unitary grouprepresentations. For such systems, the general theory of rigged Hilbert spacesis not needed, since for the statements we are interested in various ways canbe found to make them precise (although we will generally not enter into thecomplexities needed to do so).

Given any function g(q) on R, one can try and define an element of the dualspace of the space of functions on R by integration, i.e by the linear operator

f →∫ +∞

−∞g(q)f(q)dq

(we won’t try and specify which condition on functions f or g is chosen to makesense of this). There are however some other very obvious linear functionals onsuch a function space, for instance the one given by evaluating the function atq = c:

f → f(c)

Such linear functionals correspond to generalized functions, objects which whenfed into the formula for integration over R give the desired linear functional.The most well-known of these is the one that gives this evaluation at q = c, itis known as the “delta function” and written as δ(q− c). It is the object which,if it were a function, would satisfy∫ +∞

−∞δ(q − c)f(q)dq = f(c)

To make sense of such an object, one can take it to be a limit of actual functions.For the δ-function, consider the limit as ε→ 0 of

gε =1√2πε

e−(q−c)2

2ε

which satisfy ∫ +∞

−∞gε(q)dq = 1

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for all ε > 0 (one way to see this is to use the formula given earlier for theFourier transform of a Gaussian).

Heuristically (ignoring obvious problems of interchange of integrals thatdon’t make sense), one can write the Fourier inversion formula as follows

ψ(x) =1√2π

∫ +∞

−∞eikqψ(k)dk

=1√2π

∫ +∞

−∞eikq(

1√2π

∫ +∞

−∞e−ikq

′ψ(q′)dq′)dk

=1

2π

∫ +∞

−∞(

∫ +∞

−∞eik(q−q′)ψ(q′)dk)dq′

=

∫ +∞

−∞δ(q′ − q)ψ(q′)dq′

Taking the delta function to be an even function (so δ(x′ − x) = δ(x− x′)),one can interpret the above calculation as justifying the formula

δ(q − q′) =1

2π

∫ +∞

−∞eik(q−q′)dk

One then goes on to consider the eigenvectors

|k〉 =1√2πeikq

of the momentum operator as satisfying a replacement for the finite-dimensionalorthonormality relation, with the δ-function replacing the δij :

〈k′|k〉 =

∫ +∞

−∞(

1√2πeik′q)(

1√2πeikq)dq =

1

2π

∫ +∞

−∞ei(k−k

′)qdq = δ(k − k′)

As mentioned before, we will usually work with the variable p = ~k, in whichcase we have

|p〉 =1√2π~

eipq~

and

δ(p− p′) =1

2π~

∫ +∞

−∞ei

(p−p′)q~ dq

For a more mathematically legitimate version of this calculation, one placeto look is Lecture 6 in the notes on physics by Dolgachev [12].


Every book about quantum mechanics covers this example of the free quantumparticle somewhere very early on, in detail. Our discussion here is unusual

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just in emphasizing the role of the spatial translation groups and its unitaryrepresentations. Discussions of quantum mechanics for mathematicians (suchas [59]) typically emphasize the development of the functional analysis neededfor a proper description of the Hilbert space H and of the properties of generalself-adjoint operators on this state space. In this class we’re restricting attentionto a quite limited set of such operators coming from Lie algebra representations,so will avoid the general theory.

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130

Chapter 11

The Heisenberg group andthe SchrodingerRepresentation

In our discussion of the free particle, we used just the actions of the groups R3 ofspatial translations and the group R of time translations, finding correspondingobservables, the self-adjoint momentum (P ) and Hamiltonian (H) operators.We’ve seen though that the Fourier transform involves a perfectly symmetricaltreatment of position and momentum variables. This allows us to introduce aposition operator Q acting on our state space H. We will analyze in detail inthis chapter the implications of extending the algebra of observable operators inthis way, most of the time restricting to the case of a single spatial dimension,since the physical case of three dimensions is an easy generalization.

The P and Q operators generate an algebra named the Heisenberg algebra,since Werner Heisenberg and collaborators used it in the earliest work on a fullquantum-mechanical formalism in 1925. It was quickly recognized by HermannWeyl that this algebra comes from a Lie algebra representation, with a cor-responding group (called the Heisenberg group by mathematicians, the Weylgroup by physicists). The state space of a quantum particle, either free or mov-ing in a potential, will be a unitary representation of this group, with the groupof spatial translations a subgroup. Note that this particular use of a groupand its representation theory in quantum mechanics is both at the core of thestandard axioms and much more general than the usual characterization of thesignificance of groups as “symmetry groups”. The Heisenberg group does not inany sense correspond to a group of invariances of the physical situation (thereare no states invariant under the group), and its action does not commute withany non-zero Hamiltonian operator. Instead it plays a much deeper role, withits unique unitary representation determining much of the structure of quantummechanics.

Note: beginning with this chapter, we will always assume units for position

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and momentum chosen so that ~ = 1 and no longer keep track of how thisdimensional constant appears in equations.

11.1 The position operator and the HeisenbergLie algebra

In the description of the state space H as functions of a position variable q, themomentum operator is

P = −i ddq

The Fourier transform F provides a unitary transformation to a description ofH as functions of a momentum variable p in which the momentum operator Pis just multiplication by p. Exchanging the role of p and q, one gets a positionoperator Q that acts as

Q = id

dp

when states are functions of p (the sign difference comes from the sign change

in F vs. F), or as multiplication by q when states are functions of q.

11.1.1 Position space representation

In the position space representation, taking as position variable q′, one hasnormalized eigenfunctions describing a free particle of momentum p

|p〉 =1√2πeipq

′

which satisfy

P |p〉 = −i ddq′

(1√2πeipq

′) = p(

1√2πeipq

′) = p|p〉

The operator Q in this representation is just the multiplication operator

Qψ(q′) = q′ψ(q′)

that multiplies a function of the position variable q′ by q′. The eigenvectors |q〉of this operator will be the δ-functions δ(q′ − q) since

Q|q〉 = q′δ(q′ − q) = qδ(q′ − q)

A standard convention in physics is to think of a state written in the notation|ψ〉 as being representation independent. The wave-function in the positionspace representation can then be found by taking the coefficient of |ψ〉 in theexpansion of a state in Q eigenfunctions |q〉, so

〈q|ψ〉 =

∫ +∞

−∞δ(q − q′)ψ(q′)dq′ = ψ(q)

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and in particular

〈q|p〉 =1√2πeipq

11.1.2 Momentum space representation

In the momentum space description of H as functions of p′, the state is theFourier transform of the state in the position space representation, so the state|p〉 will be the function on momentum space

F(1√2πeipq

′) =

1

2π

∫ +∞

−∞e−ip

′q′eipq′dq′ =

1

2π

∫ +∞

−∞ei(p−p

′)q′dq′ = δ(p− p′)

These are eigenfunctions of the operator P , which is a multiplication operatorin this representation

P |p〉 = p′δ(p′ − p) = pδ(p′ − p)

The position eigenfunctions are also given by Fourier transform

|q〉 = F(δ(q − q′)) =1√2π

∫ +∞

−∞e−ip

′q′δ(q − q′)dq′ =1√2πe−ip

′q

The position operator is

Q = id

dp

and |q〉 is an eigenvector with eigenvalue q

Q|q〉 = id

dp′(

1√2π~

e−ip′q) = q(

1√2πe−ip

′q) = q|q〉

Another way to see that this is the correct operator is to use the unitary trans-formation F and its inverse F that relate the position and momentum spacerepresentations. Going from position space to momentum space one has

Q→ FQF

and one can check that this transformed Q operator will act as i ddp′ on functions

of p′.

One can express momentum space wavefunctions as coefficients of the ex-pansion of a state ψ in terms of momentum eigenvectors

〈p|ψ〉 =

∫ +∞

−∞(

1√2πe−ipq

′)ψ(q′)dq′ = F(ψ(q)) = ψ(p)

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11.1.3 Physical interpretation

With now both momentum and position operators on H, we have the standardset-up for describing a non-relativistic quantum particle that is discussed exten-sively early on in any quantum mechanics textbook, and one of these should beconsulted for more details and for explanations of the physical interpretation ofthis quantum system. The classically observable quantity corresponding to theoperator P is the momentum, and eigenvectors of P are the states that have well-defined values for this, values that will have the correct non-relativistic energymomentum relationship. Note that for the free particle P commutes with the

Hamiltonian H = P 2

2m so there is a conservation law: states with a well-definedmomentum at one time always have the same momentum. This corresponds toan obvious physical symmetry, the symmetry under spatial translations.

The operator Q on the other hand does not correspond to a physical sym-metry, since it does not commute with the Hamiltonian. We will see that itdoes generate a group action, and from the momentum space picture we cansee that this is a shift in the momentum, but such shifts are not symmetries ofthe physics and there is no conservation law for Q. The states in which Q has awell-defined numerical value are the ones such that the position wave-functionis a delta-function. If one prepares such a state at a given time, it will notremain a delta-function, but quickly evolve into a wave-function that spreadsout in space.

Since the eigenfunctions of P and Q are non-normalizable, one needs aslightly different formulation of the measurement theory principle used for finitedimensional H. In this case, the probability of observing a position of a particlewith wave function ψ(q) in the interval [q1, q2] will be∫ q2

q1ψ(q)ψ(q)dq∫ +∞

−∞ ψ(q)ψ(q)dq

This will make sense for states |ψ〉 ∈ L2(R), which we will normalize to havenorm-squared one when discussing their physical interpretation. Then the sta-tistical expectation value for the measured position variable will be

〈ψ|Q|ψ〉

which can be computed with the same result in either the position or momentumspace representation.

Similarly, the probability of observing a momentum of a particle with momentum-space wave function ψ(q) in the interval [p1, p2] will be∫ p2

p1ψ(p)ψ(p)dp∫ +∞

−∞ ψ(p)ψ(p)dp

and for normalized states the statistical expectation value of the measured mo-mentum is

〈ψ|P |ψ〉

134

Note that states with a well-defined position (the delta-function states inthe position-space representation) are equally likely to have any momentumwhatsoever. Physically this is why such states quickly spread out. States witha well-defined momentum are equally likely to have any possible position. Theproperties of the Fourier transform imply the so-called “Heisenberg uncertaintyprinciple” that gives a lower bound on the product of a measure of uncertaintyin position times the same measure of uncertainty in momentum. Examplesof this that take on the lower bound are the Gaussian shaped functions whoseFourier transforms were computed earlier.

For much more about these questions, again most quantum mechanics text-books will contain an extensive discussion.

11.2 The Heisenberg Lie algebra

In either the position or momentum space representation the operators P andQ satisfy the relation

[Q,P ] = i1

Soon after this commutation relation appeared in early work on quantum me-chanics, Weyl realized that it can be interpreted as the relation between oper-ators one would get from a representation of a three-dimensional Lie algebra,now called the Heisenberg Lie algebra.

Definition (Heisenberg Lie algebra, d = 1). The Heisenberg Lie algebra h3 isthe vector space R3 with the Lie bracket defined by its values on a basis (X,Y, Z)by

[X,Y ] = Z, [X,Z] = [Y,Z] = 0

Writing a general element of h3 in terms of this basis as xX + yY + zZ, theLie bracket is given by

[xX + yY + zZ, x′X + y′Y + z′Z] = (xy′ − yx′)Z

Note that this is a non-trivial Lie algebra, but only minimally so. All Liebrackets of Z are zero. All Lie brackets of Lie brackets are also zero (as a result,this is an example of what is known as a “nilpotent” Lie algebra).

The Heisenberg Lie algebra is isomorphic to the Lie algebra of 3 by 3 strictlyupper triangular real matrices, with Lie bracket the matrix commutator, by thefollowing isomorphism:

X ↔

0 1 00 0 00 0 0

, Y ↔

0 0 00 0 10 0 0

, Z ↔

0 0 10 0 00 0 0

xX + yY + zZ ↔

0 x z0 0 y0 0 0

135

since one has

[

0 x z0 0 y0 0 0

,

0 x′ z′

0 0 y′

0 0 0

] =

0 0 xy′ − x′y0 0 00 0 0

The generalization of this to higher dimensions is

Definition (Heisenberg Lie algebra). The Heisenberg Lie algebra h2d+1 is thevector space R2d+1 with the Lie bracket defined by its values on a basis Xj , Yj , Z, (j =1, . . . d) by

[Xj , Yk] = δjkZ, [Xj , Z] = [Yj , Z] = 0

In the physical case is d = 3, elements of the Heisenberg Lie algebra can bewritten

0 x1 x2 x3 z0 0 0 0 y3

0 0 0 0 y2

0 0 0 0 y1

0 0 0 0 0

11.3 The Heisenberg group

One can easily see that exponentiating matrices in h3 gives

exp

0 x z0 0 y0 0 0

=

1 x z + 12xy

0 1 y0 0 1

so the group with Lie algebra h3 will be the group of upper triangular 3 by 3 realmatrices with ones on the diagonal, and this group will be the Heisenberg groupH3. For our purposes though, it is better to work in exponential coordinates(i.e. labeling a group element with the Lie algebra element that exponentiatesto it).

Matrix exponentials in general satisfy the Baker-Campbell-Hausdorff for-mula, which says

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

12 [A,[A,B]]− 112 [B,[A,B]]+···

where the higher terms can all be expressed as repeated commutators. Thisprovides one way of showing that the Lie group structure is determined (forgroup elements expressible as exponentials) by knowing the Lie bracket. Forthe full formula and a detailed proof, see chapter 3 of [26]. One can easily checkthe first few terms in this formula by expanding the exponentials, but it is notat all obvious why all the terms can be organized in terms of commutators.

For the case of the Heisenberg Lie algebra, since all multiple commutatorsvanish, the Baker-Campbell-Hausdorff formula implies for exponentials of ele-ments of h3

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

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(a proof of this special case of Baker-Campbell-Hausdorff is in section 3.1 of [26]).We can use this to explicitly write the group law in exponential coordinates:

Definition (Heisenberg group, d = 1). The Heisenberg group H3 is the spaceR3 with the group law

(x, y, z) · (x′, y′, z′) = (x+ x′, y + y′, z + z′ +1

2(xy′ − yx′)) (11.1)

Note that the Lie algebra basis elements X,Y, Z each generate subgroupsof H3 isomorphic to R. Elements of the first two of these subgroups generatethe full group, and elements of the third subgroup are “central”, meaning theycommute with all group elements. Also notice that the non-commutative natureof the Lie algebra or group depends purely on the factor xy′ − yx′.

The generalization of this to higher dimensions is:

Definition (Heisenberg group). The Heisenberg group H2d+1 is the space R2d+1

with the group law

(x,y, z) · (x′,y′, z′) = (x + x′,y + y′, z + z′ +1

2(x · y′ − y · x′))

where the vectors here all live in Rd.

Note that in these exponential coordinates the exponential map relating theHeisenberg Lie algebra h2d+1 and the Heisenberg Lie group H2d+1 is just theidentity map.

11.4 The Schrodinger representation

Since it can be defined in terms of 3 by 3 matrices, the Heisenberg group H3

has an obvious representation on C3, but this representation is not unitary andnot of physical interest. What is of great interest is the infinite dimensionalrepresentation on functions of q for which the Lie algebra version is given bythe Q, P and unit operators:

Definition (Schrodinger representation, Lie algebra version). The Schrodingerrepresentation of the Heisenberg Lie algebra h3 is the representation (Γ′S , L

2(R))satisfying

Γ′S(X)ψ(q) = −iQψ(q) = iqψ(q), Γ′S(Y )ψ(q) = −iPψ(q) = − d

dqψ(q)

Γ′S(Z)ψ(q) = −iψ(q)

Factors of i have been chosen to make these operators skew-Hermitian. Theycan be exponentiated, and using the fact that the group H3 is generated byelements of the form exX , eyY , ezZ one has

137

Definition (Schrodinger representation, Lie group version). The Schrodingerrepresentation of the Heisenberg Lie group H3 is the representation (ΓS , L

2(R))satisfying

ΓS(exX)ψ(q) = e−xiQψ(q) = e−ixqψ(q)

ΓS(eyY )ψ(q) = e−yiPψ(q) = e−yddqψ(q) = ψ(q − y)

ΓS(ezZ)ψ(q) = e−izψ(q)

The group analog of the Heisenberg commutation relations (often called the“Weyl form” of the commutation relations) is the relation

ΓS(exX)ΓS(eyY ) = e−ixyΓS(eyY )ΓS(exX)

or, more explicitlye−ixQe−iyP = e−ixye−iyP e−ixQ

One can derive this by calculating (using the Baker-Campbell-Hausdorff for-mula)

ΓS(exX)ΓS(eyY ) = e−ixQe−iyP = e−i(xQ+yP )+ 12 [−ixQ,−iyP ]) = e−i

xy2 e−i(xQ+yP )

as well as the same product in the opposite order, and then comparing theresults.

We have seen that the Fourier transform F takes the Schrodinger represen-tation to a unitarily equivalent representation of H3, in terms of functions of p(the momentum space representation). The operators change as

ΓS(g)→ F ΓS(g)F

when one makes the unitary transformation to the momentum space represen-tation.

In typical physics quantum mechanics textbooks, one often sees calculationsmade just using the Heisenberg commutation relations, without picking a spe-cific representation of the operators that satisfy these relations. This turns outto be justified by the remarkable fact that, for the Heisenberg group, once onepicks the constant with which Z acts, all irreducible representations are uni-tarily equivalent. By unitarity this constant is −ic, c ∈ R. We have chosenc = 1, but other values of c would correspond to different choices of units. In asense, the representation theory of the Heisenberg group is very simple: there’sjust one irreducible representation. This is very different than the theory foreven the simplest compact Lie groups (U(1) and SU(2)) which have an infinityof inequivalent irreducibles labeled by weight or by spin. Representations of aHeisenberg group will appear in different guises (we’ve seen two, will see an-other in the discussion of the harmonic oscillator, and there are yet others thatappear in the theory of theta-functions), but they are all unitarily equivalent.This statement is known as the Stone-von Neumann theorem.

So far we’ve been modestly cavalier about the rigorous analysis needed tomake precise statements about the Schrodinger representation. In order to prove

138

a theorem like the Stone-von Neumann theorem, which tries to say somethingabout all possible representations of a group, one needs to invoke a great dealof analysis. Much of this part of analysis was developed precisely to be able todeal with general quantum mechanical systems and prove theorems about them.The Heisenberg group, Lie algebra and its representations are treated in detailin many expositions of quantum mechanics for mathematicians. Some goodreferences for this material are [59], and [28]. In depth discussions devoted tothe mathematics of the Heisenberg group and its representations can be foundin [32] and [19].

In these references can be found a proof of the (not difficult)

Theorem. The Schrodinger representation ΓS described above is irreducible.

and the much more difficult

Theorem (Stone-von Neumann). Any irreducible representation π of the groupH3 on a Hilbert space, satisfying

π′(Z) = −i1

is unitarily equivalent to the Schrodinger representation (ΓS , L2(R))

Note that all of this can easily be generalized to the case of d spatial di-mensions, for d finite, with the Heisenberg group now H2d+1 and the Stone-vonNeumann theorem still true. In the case of an infinite number of degrees offreedom, which is the case of interest in quantum field theory, the Stone-vonNeumann theorem no longer holds and one has an infinity of inequivalent irre-ducible representations, leading to quite different phenomena. For more on thistopic see chapter 31.

It is also important to note that the Stone-von Neumann theorem is for-mulated for Heisenberg group representations, not for Heisenberg Lie algebrarepresentations. For infinite-dimensional representations in cases like this, thereare representations of the Lie algebra that are “non-integrable”: they aren’t thederivatives of Lie group representations. For representations of the Heisen-berg Lie algebra, i.e. the Heisenberg commutator relations, there are counter-examples to the Stone von-Neumann theorem. It is only for integrable rep-resentations that the theorem holds and one has a unique sort of irreduciblerepresentation.


For a lot more detail about the mathematics of the Heisenberg group, its Liealgebra and the Schrodinger representation, see [8], [32], [19] and [60]. An ex-cellent historical overview of the Stone-von Neumann theorem [47] by JonathanRosenberg is well worth reading.

139

140

Chapter 12

The Poisson Bracket andSymplectic Geometry

We have seen that the quantum theory of a free particle corresponds to the con-struction of a representation of the Heisenberg Lie algebra in terms of operatorsQ and P . One would like to use this to produce quantum systems with a similarrelation to more non-trivial classical mechanical systems than the free particle.During the earliest days of quantum mechanics it was recognized by Dirac thatthe commutation relations of the Q and P operators somehow corresponded tothe Poisson bracket relations between the position and momentum variables inclassical mechanics in the Hamiltonian formalism. In this chapter we’ll beginby outlining the topic of Hamiltonian mechanics and the Poisson bracket, andgo on to study the Lie algebra structure determined by the Poisson bracket.

The Heisenberg Lie algebra h2d+1 is usually thought of as quintessentiallyquantum in nature, but it is already present in classical mechanics, as the Liealgebra of degree zero and one polynomials on phase space, with Lie bracketthe Poisson bracket. The full Lie algebra of all functions on phase space (withLie bracket the Poisson bracket) is infinite dimensional, so not the sort of finitedimensional Lie algebra given by matrices that we have studied so far (although,historically, it is this kind of infinite dimensional Lie algebra that motivated thediscovery of the theory of Lie groups and Lie algebras by Sophus Lie during the1870s).

12.1 Classical mechanics and the Poisson bracket

In classical mechanics in the Hamiltonian formalism, the space M = R2d

that one gets by putting together positions and the corresponding momentais known as “phase space”. Points in phase space can be thought of as uniquelyparametrizing possible initial conditions for classical trajectories, so another in-terpretation of phase space is that it is the space that parametrizes solutionsof the equations of motion of a given classical mechanical system. Functions

141

on phase space carry the following structure (for simplicity we’ll start by justwriting things for d = 1):

Definition (Poisson bracket). There is a bilinear operation on functions on thephase space M = R2 (with coordinates (q, p)) called the Poisson bracket, givenby

(f1, f2)→ f1, f2 =∂f1

∂q

∂f2

∂p− ∂f1

∂p

∂f2

∂q(12.1)

In Hamiltonian mechanics, the state of the system at a fixed time is givenby a point of phase space. Observables are functions f on phase space, andtheir time dependence is determined by a distinguished one, the Hamiltonianfunction h, according to Hamilton’s equations

q =∂h

∂p

p = −∂h∂q

One hasdf

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂h

∂p− ∂f

∂p

∂h

∂q

so an observable f varies along the classical trajectories determined by a Hamil-tonian h according to

df

dt= f, h

This relation is equivalent to Hamilton’s equations since it implies them bytaking f = q and f = p

q = q, h =∂h

∂p

p = p, h = −∂h∂q

For a non-relativistic free particle, h = p2

2m and these equations become

q =p

m, p = 0

which just says that the momentum is the mass times the velocity, and is con-served. For a particle subject to a potential V (q) one has

h =p2

2m+ V (q)

and the trajectories are the solutions to

q =p

m, p = −∂V

∂q

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which adds Newton’s second law

F = −∂V∂q

= ma = mq

to the definition of momentum in terms of velocity.One can easily check that the Poisson bracket has the properties

• Anti-symmetry

f1, f2 = −f2, f1

• Jacobi identity

f1, f2, f3+ f3, f1, f2+ f2, f3, f1 = 0

These two properties, together with the bilinearity, show that the Poissonbracket fits the definition of a Lie bracket, making the space of functions onphase space into an infinite dimensional Lie algebra. This Lie algebra is respon-sible for much of the structure of the subject of Hamiltonian mechanics, and itwas historically the first sort of Lie algebra to be studied.

The conservation laws of classical mechanics are best understood using thisLie algebra. From the fundamental dynamical equation

df

dt= f, h

we see that

f, h = 0 =⇒ df

dt= 0

and in this case the function f is called a “conserved quantity”, since it doesnot change under time evolution. Note that if we have two functions f1 and f2

on phase space such that

f1, h = 0, f2, h = 0

then using the Jacobi identity we have

f1, f2, h = −h, f1, f2 − f2, h, f1 = 0

This shows that if f1 and f2 are conserved quantities, so is f1, f2.

12.2 Poisson brackets and vector fields

We have seen that a choice of Hamiltonian function h on phase space provides asystem of first-order differential equations for the classical trajectories, Hamil-ton’s equations. One can formulate this kind of system of equation in terms ofvector fields:

143

Definition (Vector Fields on R2). A vector field X on R2 is given by twofunctions Fq, Fp on R2 and can be written

X = Fqeq + Fpep

or

X = Fq∂

∂q+ Fp

∂

∂p

where (q, p) are coordinates on R2.

The two ways of writing X correspond to two equivalent ways of thinkingof the vector field X. One can think of X as a map assigning to each point inR2 a vector F varying over R2, with components (Fq, Fp). Or one can think ofit as a first-order differentiation operator: the dot product

F · ∇

of the vector at each point with the gradient operator.One can also characterize a vector field X by its integral curves. These are

the trajectories (q(t), p(t)) in R2 with velocity vector X, i.e. satisfying

dq

dt= Fq,

dp

dt= Fp

Given a function f on the phase space M = R2, consider the vector field

Xf =∂f

∂p

∂

∂q− ∂f

∂q

∂

∂p

The integral curves for Xf satisfy

dq

dt=∂f

∂p,dp

dt= −∂f

∂q

which are exactly Hamilton’s equations for f the Hamiltonian function. Not allvector fields are of the form Xf for some function f . A vector field of this formis referred to as a “Hamiltonian vector field”.

For each function f we get a unique vector field Xf , but f+C will correspondto the same vector field as f , for any constant C (f is the integral of Xf so the“+ C” ambiguity of first-year calculus integrals appears).

The simplest non-zero Hamiltonian vector fields are those for f a linearfunction. For cq, cp constants, if

f = cqq + cpp

then

Xf = cp∂

∂q− cq

∂

∂p

Note that this mapf → Xf

144

provides an isomorphism of the dual space M∗ with M itself by

q ↔ Xq = − ∂

∂p, p↔ Xp =

∂

∂q(12.2)

Here we are thinking of q and p as coordinate functions on M . The q andp are linear functions on M that provide a basis of M∗. The constant vectorfields ∂

∂q and ∂∂p provide a basis for M , when one identifies the vector space M

with the constant vector fields on M , with p and q then the dual basis. Onecan write

eq =∂

∂q, ep =

∂

∂p, e∗q = q, e∗p = p

We have seen that an inner product on M provides an isomorphism of Mand M∗, but the isomorphism 12.2 is something different, with its origin not aninner product, but the Poisson bracket and Hamilton’s equations instead.

For other examples, one can look at quadratic functions, with homogeneousquadratic functions giving vector fields with coefficients linear functions of q andp:

q2 → Xq2 = −2q∂

∂p

p2 → Xp2 = 2p∂

∂q

qp→ Xqp = q∂

∂q− p ∂

∂p

The relation of vector fields to the Poisson bracket is given by

f1, f2 = Xf2(f1) = −Xf1(f2)

so one hasf, · = −Xf (·)

and in particular

q, f =∂f

∂p, p, f = −∂f

∂q

This relation of the Poisson bracket to vector fields explains another of itsfundamental properties, the

• Leibniz rule

f1f2, f = f1, ff2 + f1f2, f, f, f1f2 = f, f1f2 + f1f, f2

This property says that taking Poisson bracket with a function f acts on aproduct of functions in a way that satisfies the Leibniz rule for what happenswhen you take the derivative of a product. This is exactly what one expects,since taking Poisson bracket with f corresponds to differentiation by Xf (up toa minus sign). Note that, unlike anti-symmetry and the Jacobi identity, which

145

reflect the Lie algebra structure on functions, the Leibniz property describes therelation of the Lie algebra structure to multiplication of functions.

The Leibniz rule also implies that one can calculate the value of the Poissonbracket on any polynomial just from knowing what its values are on linearfunctions on phase space. So

Theorem 12.1. The Poisson bracket on polynomial functions on M is deter-mined by the Leibniz rule and the anti-symmetric bilinear form Ω on M∗, where

Ω(u, u′) = u, u′

for u, u′ ∈M∗

Here u and u′ are two different linear functions on M , so can be written interms of coordinate functions as

u = cqq + cpp, u′ = c′qq + c′pp

for some constants cq, cp, c′q, c′p. One has

Ω(u, u′) = cqq + cpp, c′qq + c′pp = cqc

′p − cpc′q (12.3)

On basis elements of M∗ one has

Ω(q, q) = Ω(p, p) = 0, Ω(q, p) = −Ω(p, q) = 1

Recalling the discussion of bilinear forms from section 9.5, a bilinear formon a vector space V can be identified with an element of V ∗ ⊗ V ∗. TakingV = M∗, the bilinear form Ω on M∗ defined above is antisymmetric and thus inΛ2((M∗)∗) = Λ2(M). Using our identification of M with constant vector fields,it can be written

Ω =∂

∂q⊗ ∂

∂p− ∂

∂p⊗ ∂

∂q

We can use the isomorphism 12.2 between M and M∗ to get an isomorphismbetween bilinear forms on M∗ and bilinear forms on M . This takes Ω ∈ Λ2(M)to ω ∈ Λ2(M∗), where

ω = q ⊗ p− p⊗ q

which satisfiesω(Xu, Xu′) = Ω(u, u′) = u, u′

for u, u′ ∈M∗. On basis elements of M one has

ω(∂

∂q,∂

∂q) = ω(

∂

∂p,∂

∂p) = 0, ω(

∂

∂q,∂

∂p) = −ω(

∂

∂p,∂

∂q) = 1

Recall from section 4.4 that a symmetric non-degenerate bilinear form (aninner product), provides an isomorphism between a vector space V and its dualV ∗ by taking

v ∈ V → 〈v, ·〉 ∈ V ∗

146

Here we are using an anti-symmetric non-degenerate bilinear form ω in the sameway, with the isomorphism 12.2 given by

∂

∂q∈M → ω(

∂

∂q, ·) = p ∈M∗, ∂

∂p∈M → ω(

∂

∂p, ·) = −q ∈M∗

For arbitrary d, we have 2d-dimensional vector spaces M and M∗. The qj , pjcoordinates give a basis of M∗, the ∂

∂qj, ∂∂pj

a basis of M . The Poisson bracket

is

f1, f2 =

d∑j=1

(∂f1

∂qj

∂f2

∂pj− ∂f1

∂pj

∂f2

∂qj) (12.4)

and one has antisymmetric bilinear forms Ω on M∗ and ω on M given by

ω =

d∑j=1

(qj ⊗ pj − pj ⊗ qj), Ω =

d∑j=1

(∂

∂qj⊗ ∂

∂pj− ∂

∂pj⊗ ∂

∂qj)

Functions on M determine vector fields on M by

f → Xf =

d∑j=1

(∂f

∂pj

∂

∂qj− ∂f

∂qj

∂

∂pj) (12.5)

For linear functions f this gives an isomorphism M∗ = M , which on basiselements is

qj ↔ −∂

∂pj, pj ↔

∂

∂qj

The definition we have given here of Xf

Xf (g) = g, f

carries with it a choice of how to deal with a confusing sign issue. One canshow that vector fields on M form a Lie algebra, when one takes as Lie bracketthe commutator (thinking of the vector field as a differential operator). Thecommutator of two first-order differential operators is a third one since higherorder derivatives will cancel, using equality of mixed partial derivatives. Anatural question is that of how this Lie algebra is related to the Lie algebra offunctions on M (with Lie bracket the Poisson bracket).

The Jacobi identity implies

f, f1, f2 = f, f1, f2+ f2, f, f1 = f, f1, f2 − f, f2, f1

so

Xf1,f2 = Xf2Xf1−Xf1

Xf2= −[Xf1

, Xf2] (12.6)

This shows that the map

f → Xf

147

that we defined between these Lie algebras is not quite a Lie algebra homo-morphism because of the - sign in equation 12.6 (it is called a Lie algebra“anti-homomorphism”). The map that is a Lie algebra homomorphism is

f → −Xf

Recall that when we have a Lie group G acting on a space, the way we get anaction of g ∈ G on functions φ on the space is by

π(g)φ(x) = φ(g−1 · x)

and this inverse is the unavoidable source of the confusing − sign. One needsto keep straight the difference between

1. The functions on phase space M are a Lie algebra, with adjoint action

ad(f)(·) = f, ·

and

2. The functions f provide vector fields Xf acting on functions on M , where

Xf (·) = ·, f = −f, ·

The first of these is most relevant when we quantize functions on M to getoperators, preserving the Lie algebra structure. The second is what one natu-rally gets from the geometrical action of a Lie group G on the phase space M(taking the derivative, the infinitesimal version of this action is a map from theLie algebra of G to vector fields on M). As a simple example, the function psatisfies

p, F (q) = −∂F∂q

so

p, · = −∂(·)∂q

is the infinitesimal action of translation in q on functions, whereas ∂∂q is the

vector field on M corresponding to infinitesimal translation in the position co-ordinate.

12.3 Symplectic geometry

We have seen that the Poisson bracket provides an isomorphism of M∗ and M ,together with a specific antisymmetric bilinear form ω on M . One can definemore generally the class of bilinear forms which can play the same role as ω:

Definition (Symplectic form). A symplectic form ω on a vector space V is abilinear map

ω : V × V → R

such that

148

• ω is antisymmetric, i.e. ω(u, u′) = −ω(u, u′)

• ω is nondegenerate, i.e. if u 6= 0, then ω(u, ·) ∈ V ∗ is non-zero.

A vector space V with a symplectic form ω is called a symplectic vectorspace. The study of vector spaces with such a non-degenerate antisymmetricbilinear form is in many ways analogous to the usual geometry of vector spaceswith an inner product (which is a non-degenerate symmetric bilinear form).In the symplectic case, one can show that the vector space must always beeven dimensional. The analog of Gram-Schmidt orthogonalization is that in thesymplectic case one can always find a basis of V with coordinates qj , pj suchthat ω is the anti-symmetric form determined by the standard Poisson bracket:

ω =d∑j=1

(qj ⊗ pj − pj ⊗ qj)

Digression. For those familiar with differential manifolds, vector fields anddifferential forms, the notion of a symplectic vector space can be extended to:

Definition (Symplectic manifold). A symplectic manifold M is a manifold witha differential two-form ω(·, ·) (called a symplectic two-form) satisfying the con-ditions

• ω is non-degenerate, i.e. for a nowhere zero vector field X, ω(X, ·) is anowhere zero one-form.

• dω = 0, in which case ω is said to be closed.

The cotangent bundle T ∗N of a manifold N (i.e. the space of pairs of apoint on N together with a linear function on the tangent space at that point)provides one class of symplectic manifolds, generalizing the linear case N = Rd,and corresponding physically to a particle moving on N . A simple example thatis neither linear nor a cotangent bundle is the sphere M = S2, with ω the areatwo-form.

Note that there is no assumption here that M has a metric (i.e. it maynot be a Riemannian manifold). A symplectic two-form ω is a structure on amanifold analogous to a metric but with opposite symmetry properties. Whereasa metric is a symmetric non-degenerate bilinear form on the tangent space ateach point, a symplectic form is an anti-symmetric non-degenerate bilinear formon the tangent space.

Using the symplectic form ω, for any symplectic manifold M , one has ageneralization of Hamilton’s equations. This is the following equality of one-forms, giving the relation between a Hamiltonian function h and the vector fieldXh that determines time evolution on M

iXhω = ω(Xh, ·) = dh

The Poisson bracket in this context can be defined as

f1, f2 = ω(Xf1, Xf2

)

149

The condition dω = 0 is then a condition on a three-form dω equivalentto the condition that the Poisson bracket satisfies the Jacobi identity (standarddifferential form computations allow one to express dω(Xf1

, Xf2, Xf3

) = 0 interms of the Poisson bracket and functions f1, f2, f3). The infinite dimensionalLie algebra of functions on M with Lie bracket the Poisson bracket is the Liealgebra for an infinite dimensional group, the group of all diffeomorphisms ofM that preserve ω (sometimes called the group of “symplectomorphisms”). Formuch more about symplectic geometry and Hamiltonian mechanics, see [2] and[8].

From the point of view of symplectic geometry, at each point the fundamentalstructure is the anti-symmetric bilinear form on tangent vectors given by ω.In discussions of quantization however, what is most important is the Poissonbracket which, in the linear approximation near each point, is an anti-symmetricbilinear form on linear combinations of the local coordinate functions (qi, pi).These lie not in the tangent space but in its dual space, and it is important tokeep that in mind as we go on to exploit symmetries based on the Poisson bracket,rather than the closely related symplectic form. Our discussion of quantizationwill rely crucially on having a linear structure on phase space, so will not applyto general symplectic manifolds.

A generalization of the notion of a symplectic manifold is that of a “Poissonmanifold”. Here one keeps the Poisson bracket Lie algebra structure on func-tions, but drops the non-degeneracy condition on ω. Then to the differerentialdf of a function f , there may be many vector fields Xf satisfying

ω(Xf , ·) = df

If one restricts to a submanifold on which ω is non-degenerate, this submanifoldwill be a symplectic manifold, and one can think of Poisson manifolds as beingdecomposable into families of “leaves” of symplectic manifolds.

12.4 Phase space and the Heisenberg Lie alge-bra

The Poisson brackets of linear functions on the phase space M = R2d aredetermined by the relations

q, p = −p, q = 1, q, q = p, p = 0

and it is these simple relations (together with the Leibniz rule) which determinethe entire infinite dimensional Lie algebra structure on polynomials on M . Thevector space of linear polynomials onM is three dimensional, with basis elementsq, p, 1 and this space is closed under Poisson bracket, so gives a three dimensionalLie algebra. This Lie algebra is isomorphic to the Heisenberg Lie algebra h3 withthe isomorphism given by (see section 11.2)

X ↔ q, Y ↔ p, Z ↔ 1

150

This isomorphism preserves the Lie bracket relations since

[X,Y ] = Z ↔ q, p = 1

In higher dimensions, choosing a basis for the phase space M = R2d gives abasis q1, · · · , qd, p1, · · · , pd for the dual space M∗ consisting of the linear coeffi-cient functions of vectors in M . It is convenient to choose its own notation forthe dual phase space, so we will often write M∗ =M. Taking as an additionalbasis element the constant function 1, we have a 2d+ 1 dimensional space

M⊕R

with basis q1, · · · , qd, p1, · · · , pd, 1. The Poisson bracket relations

qj , qk = pj , pk = 0, qj , pk = δjk

turn this space into a Lie algebra, the Heisenberg Lie algebra h2d+1.This Lie bracket on h2d+1 is explicitly given on linear combinations of the

basis elements by an antisymmetric, non-degenerate bilinear form Ω as follows.If (u, c) and (u′, c′) are elements of h2d+1 =M⊕R, with

u = cq1q1 + · · ·+ cqdqd + cp1p1 + · · ·+ cpdpd ∈M, c ∈ R

u′ = c′q1q1 + · · ·+ c′qdqd + c′p1p1 + · · ·+ c′pdpd ∈M, c′ ∈ R

then[(u, c), (u′, c′)] = (0,Ω(u, u′))

where

Ω(u, u′) =cq1q1 + cp1p1 + · · ·+ cqdqd + cpdpd, c

′q1q1 + c′p1

p1 + · · ·+ c′qdqd + c′pdpd=cq1c

′p1− cp1

c′q1 + · · · cqdc′pd − cpdc′qd

12.5 The Poisson bracket on the dual of a Liealgebra

We have been careful here to keep track of the difference between phase spaceM = R2d and its dual M = M∗, since it is M⊕R that is given the structureof a Lie algebra (in this case h2d+1) by the Poisson bracket, and it is this Liealgebra we want to use in chapter 14 when we define a quantization of theclassical system. From a purely classical point of view, one can generalize thenotion of phase space and Poisson bracket, defining them purely in terms of Liealgebras, by starting not with h2d+1, but with a general Lie algebra g. Since thedual of a dual is the original vector space, one can think of this g as the dual ofg∗, i.e.

g = (g∗)∗

and then the Lie bracket on g has an interpretation as a Poisson bracket onlinear functions on g∗. So, for any Lie algebra g, one automatically has a

151

Poisson bracket on linear functions on g∗, which by the Leibniz property canbe extended to all functions on g∗. This allows one to treat g∗ as a sort ofgeneralized classical phase space.

This Poisson bracket on linear functions on g∗ is an anti-symmetric bilinearform, but typically is not non-degenerate, so it will not provide the isomorphismbetween phase space and its dual that characterizes symplectic geometry andthe Hamiltonian formalism of mechanics. This bilinear form will become non-degenerate when restricted to certain subspaces, and g∗ can often be thought ofas a family of conventional phase spaces (sub-manifolds on which the bilinearform is non-degenerate on tangent spaces). For g = h2d+1 one has

h∗2d+1 = M ⊕R

so g∗ is a one-parameter family of the usual sort of phase space.

Digression. In the context of the earlier discussion of generalizing phase spacesto arbitrary symplectic manifolds, g∗ is an example of a Poisson manifold, whichcan be decomposed into families of symplectic manifolds. For a simple example,consider g = su(2), where

su(2)∗ = R3

can be thought of as the space of spheres S2, parametrized by their radius. Eachsphere is an example of a symplectic manifold, with symplectic form the areatwo-form.


Two good sources for discusssions of symplectic geometry and the geometricalformulation of Hamiltonian mechanics are Arnold’s Mathematical Methods ofClassical Mechanics [2] and Berndt’s An Introduction to Symplectic Geometry[8].

152

Chapter 13

The Symplectic Group andthe Moment Map

The Poisson bracket on functions on phase space M = R2d is determined by anantisymmetric bilinear form Ω on the dual phase spaceM = M∗. Just as thereis a group of linear transformations (the orthogonal group) leaving invariant aninner product, which is a symmetric bilinear form, here there is a group leavinginvariant Ω, the symplectic group Sp(2d,R). The Lie algebra sp(2d,R) of thisgroup can be identified with the Lie algebra of order-two polynomials onM , withLie bracket the Poisson bracket. Elements of Sp(2d,R) act on M, preservingΩ, and so provide a map of the Heisenberg Lie algebra h2d+1 =M⊕R to itself,preserving the Lie bracket (which is defined using Ω). The symplectic groupthus acts by automorphisms on h2d+1. This action has an infinitesimal version,reflected in the non-trivial Poisson brackets between order two and order onepolynomials on M .

For subgroups G of Sp(2d,R) or H2d+1, given in terms of their action onM , elements of the Lie algebra of G can be identified with quadratic polynomi-als f such that the corresponding vector field Xf is the infinitesimal action onM . This map from the Lie algebra of G to functions on M is called the “mo-ment map”, with special cases giving the momentum and angular momentumfunctions. It is these functions which provide distinguished classical observableswhen there is a group action, and after quantization we will find correspondinglinear operators that provide distinguished quantum observables.

13.1 The symplectic group for d = 1

In chapter 12 we saw that the Lie algebra of polynomials on phase space of degreezero and one was isomorphic to the Heisenberg Lie algebra h2d+1. Turning topolynomials of degree two (and for now considering d = 1), the homogeneousdegree two polynomials in p and q form a three-dimensional sub-Lie algebra ofthe Lie algebra of functions on phase space, since the non-zero Poisson bracket

153

relations between them on a basis q2

2 ,p2

2 , qp are

q2

2,p2

2 = qp qp, p2 = 2p2 qp, q2 = −2q2

This Lie algebra is isomorphic to the Lie algebra of trace-less 2 by 2 realmatrices of the form

L =

(a bc −a

)which is sl(2,R), the Lie algebra of the group SL(2,R). An explicit isomorphismis given by identifying basis elements as follows:

q2

2↔ E =

(0 10 0

)− p2

2↔ F =

(0 01 0

)− qp↔ G =

(1 00 −1

)(13.1)

The commutation relations amongst these matrices are

[E,F ] = G [G,E] = 2E [G,F ] = −2F

which are the same as the Poisson bracket relations between the correspondingquadratic polynomials.

We thus see that we have an isomorphism between the Lie algebra of degreetwo homogeneous polynomials with the Poisson bracket and the Lie algebraof 2 by 2 trace-zero real matrices with the commutator as Lie bracket. Theisomorphism on general elements of these Lie algebras is

µL = −aqp+bq2

2− cp2

2=

1

2

(q p

)( b −a−a −c

)(qp

)↔(a bc −a

)(13.2)

and this is a Lie algebra isomorphism since one can check that

µL, µL′ = µ[L,L′]

(there will be a proof of this for general d in section 13.3).Turning to the group SL(2,R), two important subgroups are

• The subgroup of elements one gets by exponentiating G, which is isomor-phic to the multiplicative group of positive real numbers

etG =

(et 00 e−t

)Here one can explicitly see that this group has elements going off to infinity.

• Exponentiating the Lie algebra element E−F gives rotations of the plane

eθ(E−F ) =

(cos θ sin θ− sin θ cos θ

)154

Note that the Lie algebra element being exponentiated here is

E − F ↔ 1

2(p2 + q2)

which we will later re-encounter as the Hamiltonian function for the har-monic oscillator.

The group SL(2,R) is non-compact and thus its representation theory isquite unlike the case of SU(2). In particular, all of its non-trivial irreducibleunitary representations are infinite-dimensional, forming an important topic inmathematics, but one that is beyond our scope. We will be studying just onesuch irreducible representation, and it is a representation only of a double-coverof SL(2,R), not SL(2,R) itself.

One way to understand the significance here of the group SL(2,R) is torecall (theorem 12.1) that our definition of the Poisson bracket depends just ona choice of an antisymmetric bilinear form Ω on R2 = M. This satisfies (seeequation 12.3)

Ω(cqq + cpp, c′qq + c′pp) = cqc

′p − cpc′q

which one can write as

Ω(cqq + cpp, c′qq + c′pp) =

(cq cp

)( 0 1−1 0

)(c′qc′p

)A change in basis of M will correspond to a linear map on the coefficients(

cqcp

)→(α βγ δ

)(cqcp

)The condition for Ω to be invariant under such a transformation is(

α βγ δ

)T (0 1−1 0

)(α βγ δ

)=

(0 1−1 0

)(13.3)

or (0 αδ − βγ

−αδ + βγ 0

)=

(0 1−1 0

)so

det

(α βγ δ

)= αδ − βγ = 1

which says that the change of basis matrix must be an element of SL(2,R).This shows that elements g of the group SL(2,R) act on M preserving Ω.

Denoting elements cqq + cpp + c of the Lie algebra h3 = M⊕ R (see section12.4) by

(

(cqcp

), c)

the Lie bracket is

[(

(cqcp

), c), (

(c′qc′p

), c′)] = (

(00

), cqc

′p − cpc′q) = (

(00

)Ω(

(cqcp

),

(c′qc′p

)))

155

This Lie bracket just depends on Ω, so acting on M by g ∈ SL(2,R) will givea map of h3 to itself preserving the Lie bracket. More explicitly, an SL(2,R)group element acts on the Lie algebra by

X = (

(cqcp

), c) ∈ h3 → g ·X = (

(α βγ δ

)(cqcp

), c)

This is an example of the general phenomenon of

Definition (Lie algebra automorphisms). If an action of a group G on a Liealgebra h

X ∈ h→ g ·X ∈ h

satisfies

[g ·X, g · Y ] = g · [X,Y ]

for all g ∈ G and X,Y ∈ h, the group is said to act on h by automorphisms.The action of an element g on h is an automorphism of h.

Two examples are

• In the case discussed above, SL(2,R) acts on h3 by automorphisms.

• If h is the Lie algebra of a Lie group H, the adjoint representation (Ad, h)gives an action

X ∈ h→ h ·X = Ad(h)(X) = hXh−1

of H on h by automorphisms.

For group elements g ∈ SL(2,R) near the identity, one can write g in theform g = etL where L is in the the Lie algebra sl(2,R). The condition that gacts on M preserving Ω implies that (using 13.3)

d

dt((etL)T

(0 1−1 0

)etL) = (etL)T (LT

(0 1−1 0

)+

(0 1−1 0

)L)etL = 0

so

LT(

0 1−1 0

)+

(0 1−1 0

)L = 0 (13.4)

This requires that L must be of the form

L =

(a bc −a

)which is what one expects: the Lie algebra sl(2,R) is the Lie algebra of 2 by 2real matrices with zero trace.

The SL(2,R) action on h3 by Lie algebra automorphisms has an infinitesimalversion (i.e. for group elements infinitesimally close to the identity), an action of

156

the Lie algebra of SL(2,R) on h3. This is defined for L ∈ sl(2,R) and X ∈ h3

by

L ·X =d

dt(etL ·X)|t=0 (13.5)

Computing this, one finds

L · ((cqcp

), c) = (L

(cqcp

), 0)

so L acts on h3 =M⊕R just by matrix multiplication on vectors in M.More generally, one has

Definition (Lie algebra derivations). If an action of a Lie algebra g on a Liealgebra h

X ∈ h→ Z ·X ∈ h

satisfies[Z ·X,Y ] + [X,Z · Y ] = Z · [X,Y ]

for all Z ∈ g and X,Y ∈ h, the Lie algebra g is said to act on h by derivations.The action of an element Z on h is a derivation of h.

Given an action of a Lie group G on a Lie algebra h by automorphisms,taking the derivative as in 13.5 gives an action of g on h by derivations since

Z · [X,Y ] =d

dt(etL · [X,Y ])|t=0 =

d

dt([etL ·X, etL ·Y ])|t=0 = [Z ·X,Y ]+[X,Z ·Y ]

We will often refer to this action of g on h as the infinitesimal version of theaction of G on h.Two examples are

• The case above, where sl(2,R) acts on h3 by derivations.

• The adjoint representation of a Lie algebra h on itself gives an action of hon itself by derivations, with

X ∈ h→ Z ·X = ad(Z)(X) = [Z,X]

13.2 The Lie algebra of quadratic polynomialsfor d = 1

We have found two three-dimensional Lie algebras (h3 and sl(2,R)) as subalge-bras of the infinite dimensional Lie algebra of functions on phase space:

• h3, the Lie algebra of linear polynomials on M , with basis 1, q, p.

• sl(2,R), the Lie algebra of order two homogeneous polynomials on M ,with basis q2, p2, qp.

157

Taking all quadratic polynomials, we get a six-dimensional Lie algebra withbasis elements 1, q, p, qp, q2, p2. This is not the product of h3 and sl(2,R) sincethere are nonzero Poisson brackets

qp, q = −q, qp, p = p

p2

2, q = −p, q

2

2, p = q

In chapter 15 we will see that this is an example of a general construction whichtakes two distinct Lie algebras, with one acting on the other by derivations, andgives a new Lie algebra called a semi-direct product. The notation for this casewill be

h3 o sl(2,R)

For a general L ∈ sl(2,R) and cqq + cpp+ c ∈ h3 we have

µL, cqq + cpp+ c = c′qq + c′pp,

(c′qc′p

)=

(acq + bcpccq − acp

)= L

(cqcp

)(13.6)

(here µL is given by 13.2). We see that this is just the action of sl(2,R) by deriva-tions on h3 of section 13.1, the infinitesimal version of the action of SL(2,R) onh3 by automorphisms. Note that in this larger Lie algebra, the action of sl(2,R)on h3 by derivations is part of the adjoint action of the Lie algebra on itself.

Recall from section 12.2 that we have a map from functions on M to vectorfields on M that takes

f → Xf

and is an anti-homomorphism of Lie algebras. For linear functions this givesconstant vector fields, corresponding to the infinitesimal translation action onM , taking

cq + cp + c→= −cq∂

∂p+ cp

∂

∂q

For degree two polynomials one has

µL → XµL = (−aq − cp) ∂∂q− (bq − ap) ∂

∂p

which are vector fields with coefficients linear in the coordinates, correspondingto the infinitesimal action of SL(2,R) on M .

13.3 The symplectic group for arbitary d

The group of linear transformations of M that leave Ω invariant (i.e. lineartransformations g such that

Ω(gu, gu′) = Ω(u, u′)

for all u ∈M) is called the symplectic group. Generalizing the explicit definitionin terms of matrices of the last section to arbitrary d we have

158

Definition (Symplectic group Sp(2d,R)). The group Sp(2d,R) is the group ofreal 2d by 2d matrices g that satisfy

gT(

0 1−1 0

)g =

(0 1−1 0

)where 0 is the d by d zero matrix, 1 the d by d unit matrix.

For d = 1 we have Sp(2,R) = SL(2,R), but for higher d the symplecticgroups are a new and distinct class of groups.

By a similar argument to the d = 1 case where the Lie algebra sp(2,R)was determined by the condition 13.4, sp(2d,R) is the Lie algebra of 2d by 2dmatrices L satisfying

LT(

0 1−1 0

)+

(0 1−1 0

)L = 0

Such matrices will be those with the block-diagonal form

L =

(A BC −AT

)(13.7)

where A,B,C are d by d real matrices, with B and C symmetric, i.e.

B = BT , C = CT

One can generalize the d = 1 relations 13.2 and 13.6 between quadraticpolynomials, the symplectic and Heisenberg Lie algebras, and the action of thefirst of these on the second by derivations as follows:

Theorem 13.1. The Lie algebra sp(2d,R) is isomorphic to the Lie algebra oforder two homogeneous polynomials on M = R2d by the isomorphism (using avector notation for the coefficient functions q1, · · · , qd, p1, · · · , pd)

L↔ µL

where

µL =1

2

(q p

)( B −A−AT −C

)(qp

)=

1

2(q ·Bq− 2q ·Ap− p · Cp) (13.8)

The sp(2d,R) action on h2d+1 =M⊕R by derivations is

L · (cq · q + cp · p + c) = µL, cq · q + cp · p + c = c′q · q + c′p · p (13.9)

where (c′qc′p

)= L

(cqcp

)or, equivalently (see section 4.1), on coordinate function basis vectors of M onehas

µL,(

qp

) = LT

(qp

)

159

Proof. We will prove the second part of this first, then use the Jacobi identityto see that the first part follows.

One can first prove 13.9 for the cases when only one of A,B,C is non-zero,then the general case follows by linearity. For instance, taking the special case

L =

(0 B0 0

), µL =

1

2q ·Bq

one can show that the action on coordinate functions (the basis vectors of M)is

1

2q ·Bq,

(qp

) = LT

(qp

)=

(0Bq

)by computing

1

2

∑j,k

qjBjkqk, pl =1

2

∑j,k

(qjBjkqk, pl+ qjBjk, plqk)

=1

2(∑j

qjBjl +∑k

Blkqk)

=∑j

Bljqj (since B = BT )

Repeating for A and C one finds that for general L one has

µL,(

qp

) = LT

(qp

)Since an element in M can be written as(

cq cp)LT(

qp

)= (L

(cqcp

))T(

qp

)we have (

c′qc′p

)= L

(cqcp

)Now turning to the first part of the theorem, the map

L→ µL

is clearly a vector space isomorphism of a space of matrices and one of quadraticpolynomials. To show that it is a Lie algebra isomorphism, one can use theJacobi identity for the Poisson bracket to show

µL, µL′ , cq ·q+cp ·p−µL′ , µL, cq ·q+cp ·p = µL, µL′, cq ·q+cp ·p

The left-hand side of this equation is c′′qq + c′′pp, where(c′′qc′′p

)= (LL′ − L′L)

(cqcp

)160

As a result, the right-hand side is the linear map given by

µL, µL′ = µ[L,L′]

Just as in the d = 1 case, here the Poisson brackets between order two andorder one polynomials give the infinitesimal version of the action of the Liegroup Sp(2d,R) on M, preserving Ω, and thus acting by automorphisms onh2d+1.

13.4 The moment map

We have seen that the Lie algebras h2d+1 and sp(2d,R) combine to form a largerLie algebra, which we’ll denote

h2d+1 o sp(2d,R)

and identify with the Lie algebra of all quadratic polynomials on phase spaceM = R2d. For each such polynomial f , we have a vector field Xf on M . Wewill be interested in various cases of Lie groups G acting on M where the Liealgebra g is a sub-Lie algebra

g ⊂ h2d+1 o sp(2d,R)

What we are generally given though is not this inclusion map, but the actionof G on M , from which we can construct an anti-homomorphism

L ∈ g→ XL

from g to vector fields on M by taking XL to be the vector field which acts onfunctions on M by

XLf(m) =d

dtf(etL ·m)|t=0

We would like to find the inclusion map by identifying for each L ∈ g thecorresponding polynomial function

µL ∈ h2d+1 o sp(2d,R)

and will do this by looking for a vector field XµL such that XµL = XL. Wedefine

Definition (Moment map). Given an action of a Lie group G on M , a map

L→ µL

from g to functions on M is said to be a co-moment map if

XL = XµL

161

Equivalently, for functions f on M , µL satisfies

XµLf = −µL, f = XLf

One can repackage this information as a map

µ : m ∈M → µ(m) ∈ g∗

called the moment map, where

(µ(m))(L) = µL(m)

We will use the terminology “moment map” to refer to either the co-moment ormoment map.

Note that µL is only defined up to a constant since µL and µL + C givethe same vector field XL. In most cases of interest to us, this constant can bechosen to make the moment map a Lie algebra homomorphism. When this fails,the G-symmetry of the classical phase space is said to have an “anomaly”.

For G = Sp(2d,R) the moment map is just the identification of its Liealgebra with order two homogeneous polynomials. For G = H2d+1 the momentmap takes

L ∈ h2d+1 → µL ∈M∗

This is a Lie algebra homomorphism, but note that it is not an isomorphismsince the constant functions in h2d+1 correspond to the zero vector field on M .

The Lie algebra sp(2d,R) has a subalgebra gl(d,R) consisting of matricesof the form

L =

(A 00 −AT

)or, in terms of polynomials, polynomials

−q ·Ap = −(ATp) · q

Here A is any real d by d matrix. This shows that one way to get symplectictransformations is to take any linear transformation of the position coordinates,together with the dual linear transformation on momentum coordinates. In thisway, any linear group of symmetries of the position space becomes a group ofsymmetries of phase-space. An example of this is the group SO(d) of spatialrotations, with Lie algebra so(d) ⊂ gl(d,R), the antisymmetric d by d matrices.In the case d = 3, µL gives the standard expression for the angular momentumas a function of the qj , pj coordinates on phase space.

This relationship to angular momentum is one justification for the terminol-ogy “moment map”. Another is that if, for d = 3 one takes the subgroup of H7

corresponding to translations of the spatial coordinates q1, q2, q3, this will havea Lie algebra R3 and a moment map which identifies infinitesimal translationin the j-direction with the momentum coordinate pj .

162

Digression. For the case of M a general symplectic manifold, one can still de-fine the moment map, whenever one has a Lie group G acting on M , preservingthe symplectic form ω. The infinitesimal condition for such a G action is that

LXω = 0

where LX is the Lie derivative along the vector field X. Using the formula

LX = (d+ iX)2 = diX + iXd

for the Lie derivative acting on differential forms (iX is contraction with thevector field X), one has

(diX + iXd)ω = 0

and since dω = 0 we havediXω = 0

When M is simply-connected, one-forms iXω whose differential is 0 (called“closed”) will be the differentials of some function (and called “exact”). Sothere will be a function µ such that

iXω(·, ·) = ω(X, ·) = dµ(·)

although such a µ is only unique up to a constant.Given an element L ∈ g, the G action on M gives a vector field XL. When

we can choose the constants appropriately and find functions µL satisfying

iXLω(·, ·) = dµL(·)

such that the mapL→ µL

taking Lie algebra elements to functions on M (with Lie bracket the Poissonbracket) is a Lie algebra homomorphism, then this is called the moment map.This same map can be repackaged as a map

µ : M → g∗

by defining(µ(m))(L) = µL(m)

An important class of symplectic manifolds M with an action of a Lie groupG, preserving the symplectic form, are the co-adjoint orbits Ml. These are thespaces one gets by acting on a chosen l ∈ g∗ by the co-adjoint action Ad∗,meaning the action of g on g∗ satisfying

(Ad∗(g) · l)(X) = l(Ad(g)X)

where X ∈ g, and Ad(g) is the usual adjoint action on g. For these cases, themoment map µ is just the inclusion map. A simple example is that of g = so(3),where the non-zero co-adjoint orbits are spheres, with radius the length of l.

163


See also [25], especially chapter 14, which discusses the isomorphism betweensp(2d,R) and homogeneous degree two polynomials.

164

Chapter 14

Quantization

Given any Hamiltonian classical mechanical system with phase space R2d, phys-ics textbooks have a standard recipe for producing a quantum system, by amethod known as “canonical quantization”. We will see that for linear functionson phase space, this is just the construction we have already seen of a unitaryrepresentation Γ′S of the Heisenberg Lie algebra, the Schrodinger representation,and the Stone-von Neumann theorem assures us that this is the unique suchconstruction, up to unitary equivalence. We will also see that this recipe canonly ever be partially successful, with the Schrodinger representation extendingto give us a representation of a sub-algebra of the algebra of all functions onphase space (the polynomials of degree two and below), and a no-go theoremshowing that this cannot be extended to a representation of the full infinitedimensional Lie algebra. Recipes for quantizing higher-order polynomials willalways suffer from a lack of uniqueness, a phenomenon known to physicists asthe existence of “operator ordering ambiguities”.

In later chapters we will see that this quantization prescription does giveunique quantum systems corresponding to some Hamiltonian systems (in par-ticular the harmonic oscillator and the hydrogen atom), and does so in a mannerthat allows a description of the quantum system purely in terms of representa-tion theory.

14.1 Canonical quantization

Very early on in the history of quantum mechanics, when Dirac first saw theHeisenberg commutation relations, he noticed an analogy with the Poissonbracket. One has

q, p = 1 and − i

~[Q,P ] = 1

as well asdf

dt= f, h and

d

dtO(t) = − i

~[O, H]

165

where the last of these equations is the equation for the time dependence of aHeisenberg picture observable O(t) in quantum mechanics. Dirac’s suggestionwas that given any classical Hamiltonian system, one could “quantize” it byfinding a rule that associates to a function f on phase space a self-adjointoperator Of (in particular Oh = H) acting on a state space H such that

Of,g = − i~

[Of , Og]

This is completely equivalent to asking for a unitary representation (π′,H)of the infinite dimensional Lie algebra of functions on phase space (with thePoisson bracket as Lie bracket). To see this, note that one can choose unitsfor momentum p and position q such that ~ = 1. Then, as usual getting askew-adjoint Lie algebra representation operator by multiplying a self-adjointoperator by −i, setting

π′(f) = −iOfthe Lie algebra homomorphism property

π′(f, g) = [π′(f), π′(g)]

corresponds to−iOf,g = [−iOf ,−iOg] = −[Of , Og]

so one has Dirac’s suggested relation.Recall that the Heisenberg Lie algebra is isomorphic to the three-dimensional

sub-algebra of functions on phase space given by linear combinations of the con-stant function, the function q and the function p. The Schrodinger representa-tion ΓS provides a unitary representation not of the Lie algebra of all functionson phase space, but of these polynomials of degree at most one, as follows

O1 = 1, Oq = Q, Op = P

so

Γ′S(1) = −i1, Γ′S(q) = −iQ = −iq, Γ′S(p) = −iP = − d

dq

Moving on to quadratic polynomials, these can also be quantized, as follows

O p2

2

=P 2

2, O q2

2

=Q2

2

For the function pq one can no longer just replace p by P and q by Q since theoperators P and Q don’t commute, and PQ or QP is not self-adjoint. Whatdoes work, satisfying all the conditions to give a Lie algebra homomorphism is

Opq =1

2(PQ+QP )

This shows that the Schrodinger representation Γ′S that was defined as arepresentation of the Heisenberg Lie algebra h3 extends to a unitary Lie algebra

166

representation of a larger Lie algebra, that of all quadratic polynomials on phasespace, a representation that we will continue to denote by Γ′S and refer to as theSchrodinger representation. On a basis of homogeneous order two polynomialswe have

Γ′S(p2

2) = −iP

2

2=i

2

d2

dq2

Γ′S(q2

2) = −iQ

2

2= − i

2q2

Γ′S(pq) =−i2

(PQ+QP )

Restricting Γ′S to just linear combinations of these homogeneous order two poly-nomials (which give the Lie algebra sl(2,R), recall equation 13.1) we get a Liealgebra representation of sl(2,R) called the metaplectic representation.

Restricted to the Heisenberg Lie algebra, the Schrodinger representation Γ′Sexponentiates to give a representation ΓS of the corresponding Heisenberg Liegroup (see 11.4). As an sl(2,R) representation however, one can show that Γ′Shas the same sort of problem as the spinor representation of su(2) = so(3), whichwas not a representation of SO(3), but only of its double cover SU(2) = Spin(3).To get a group representation, one must go to a double cover of the groupSL(2,R), which will be called the metaplectic group and denoted Mp(2,R).

The source of the problem is the subgroup of SL(2,R) generated by expo-nentiating the Lie algebra element

1

2(p2 + q2)↔ E − F =

(0 1−1 0

)When we study the Schrodinger representation using its action on the quantumharmonic oscillator state space H in chapter 19 we will see that the Hamiltonianis the operator

1

2(P 2 +Q2)

and this has half-integer eigenvalues. As a result, trying to exponentiate Γ′Sgives a representation of SL(2,R) only up to a sign, and one needs to go to thedouble cover Mp(2,R) to get a true representation.

One should keep in mind though that, since SL(2,R) acts non-trivially byautomorphisms on H3, elements of these two groups do not commute. TheSchrodinger representation is a representation not of the product group, but ofsomething called a “semi-direct product” which will be discussed in more detailin chapter 15.

14.2 The Groenewold-van Hove no-go theorem

If one wants to quantize polynomial functions on phase space of degree greaterthan two, it quickly becomes clear that the problem of “operator ordering am-biguities” is a significant one. Different prescriptions involving different ways

167

of ordering the P and Q operators lead to different Of for the same functionf , with physically different observables (although the differences involve thecommutator of P and Q, so higher-order terms in ~).

When physicists first tried to find a consistent prescription for producing anoperator Of corresponding to a polynomial function on phase space of degreegreater than two, they found that there was no possible way to do this consistentwith the relation

Of,g = − i~

[Of , Og]

for polynomials of degree greater than two. Whatever method one devises forquantizing higher degree polynomials, it can only satisfy that relation to lowestorder in ~, and there will be higher order corrections, which depend upon one’schoice of quantization scheme. Equivalently, it is only for the six-dimensional Liealgebra of polynomials of degree up to two that the Schrodinger representationgives one a Lie algebra representation, and this cannot be consistently extendedto a representation of a larger subalgebra of the functions on phase space. Thisproblem is made precise by the following no-go theorem

Theorem (Groenewold-van Hove). There is no map f → Of from polynomialson R2 to self-adjoint operators on L2(R) satisfying

Of,g = − i~

[Of , Og]

andOp = P, Oq = Q

for any Lie subalgebra of the functions on R2 larger than the subalgebra ofpolynomials of degree less than or equal to two.

Proof. For a detailed proof, see section 5.4 of [8], section 4.4 of [19], or chapter16 of [25]. In outline, the proof begins by showing that taking Poisson brack-ets of polynomials of degree three leads to higher order polynomials, and thatfurthermore for degree three and above there will be no finite-dimensional sub-algebras of polynomials of bounded degree. The assumptions of the theoremforce certain specific operator ordering choices in degree three. These are thenused to get a contradiction in degree four, using the fact that the same degreefour polynomial has two different expressions as a Poisson bracket:

q2p2 =1

3q2p, p2q =

1

9q3, p3

14.3 Canonical quantization in d dimensions

One can easily generalize the above to the case of d dimensions, with theSchrodinger representation ΓS now giving a unitary representation of the Heisen-berg group H2d+1, with the corresponding Lie algebra representation given by

Γ′S(qj) = −iQj , Γ′S(pj) = −iPj

168

which satisfy the Heisenberg relations

[Qj , Pk] = iδjk

Generalizing to quadratic polynomials in the phase space coordinate func-tions, we have

Γ′S(qjqk) = −iQjQk, Γ′S(pjpk) = −iPjPk, Γ′S(qjpk) = − i2

(QjPk + PkQj)

One can exponentiate these operators to get a representation of metaplecticMp(2d,R), a double cover of the symplectic group Sp(2d,R). Since this con-struction is based on starting with ΓS , a representation of the Heisenberg group,the Stone-von Neumann theorem assures us that this will give a unique irre-ducible representation of this group (up to a change of basis by unitary trans-formations). The Groenewold-van Hove theorem implies that we cannot find aunitary representation of a larger group of canonical transformations extendingthis one on the Heisenberg and metaplectic groups.


Just about all quantum mechanics textbooks contain some version of the discus-sion here of canonical quantization starting with classical mechanical systemsin Hamiltonian form. For discussions of quantization from the point of view ofrepresentation theory, see [8] and chapters 14-16 of [25]. For a detailed discus-sion of the Heisenberg group and Lie algebra, together with their representationtheory, see chapter 2 of [32].

169

170

Chapter 15

Semi-direct Products andtheir Representations

The theory of a free particle depends crucially on the group of symmetriesof three-dimensional space, a group which includes a subgroup R3 of spatialtranslations, and a subgroup SO(3) of rotations. The second subgroup acts non-trivially on the first, since the direction of a translation is rotated by an elementof SO(3). In later chapters dealing with special relativity, these symmetrygroups get enlarged to include a fourth dimension, time, and the theory of afree particle will again be determined by these symmetry groups. In chapters 12and 13 we saw that there are two groups acting on phase space: the Heisenberggroup H2d+1 and the symplectic group Sp(2d,R). Again the second one actsnon-trivially on the first by automorphisms (since H2d+1 just depends on thePoisson bracket, and Sp(2d,R) leaves the Poisson bracket invariant).

This situation of two groups, with one acting on the other, allows one toconstruct a kind of group as a sort of product of the two groups, called thesemi-direct product. and this will be the topic for this chapter. We’ll also beginthe study of representations of such groups, with chapter 16 describing how theSchrodinger representation becomes a representation of the semi-direct productof H2d+1 and Sp(2d,R). In chapter 17 we’ll consider the cases of the semi-directproduct of translations and rotations in two and three dimensions, and there seehow the irreducible representations are provided by the quantum state space ofa free particle.

15.1 An example: the Euclidean group

Given two groups G′ and G′′, one can form the product group by taking pairsof elements (g′, g′′) ∈ G′ ×G′′. However, when the two groups act on the samespace, but elements of G′ and G′′ don’t commute, a different sort of productgroup is needed. As an example, consider the case of pairs (a2, R2) of elements

171

a2 ∈ R3 and R2 ∈ SO(3), acting on R3 by translation and rotation

v→ (a2, R2) · v = a2 +R2v

If we then act on the result with (a1, R1) we get

(a1, R1)(a2, R2) · v = (a1, R1) · (a2 +R2v) = a1 +R1a2 +R1R2v

Note that this is not what we would get if we took the product group law onR3 × SO(3), since then the action of (a1, R1)(a2, R2) on R3 would be

v→ a1 + a2 +R1R2v

To get the correct group action on R3, we need to take R3 × SO(3) not withthe product group law, but instead with the group law

(a1, R1) · (a2, R2) = (a1 +R1a2, R1R2)

This group law differs from the standard product law, using R1a2, which is theresult of R1 ∈ SO(3) acting on a2 ∈ R3. We will denote the set R3 × SO(3)with this group law by

R3 o SO(3)

This is the group of transformations of R3 preserving the standard inner prod-uct.

The same construction works in arbitrary dimensions, where one has

Definition (Euclidean group). The Euclidean group E(d) (sometimes writtenISO(d) for “inhomogeneous” rotation group) in dimension d is the product ofthe translation and rotation groups of Rd as a set, with multiplication law

(a1, R1) · (a2, R2) = (a1 +R1a2, R1R2)

(where aj ∈ Rd, Rj ∈ SO(d)) and can be denoted by

Rd o SO(d)

E(d) can also be written as a matrix group, taking it to be the subgroupof GL(d + 1,R) of matrices of the form (R is a d by d orthogonal matrix, a ad-dimensional column vector) (

R a0 1

)One gets the multiplication law for E(d) from matrix multiplication since(

R1 a1

0 1

)(R2 a2

0 1

)=

(R1R2 a1 +R1a2

0 1

)The Lie algebra of E(d) will be given by taking the sum of Rd (the Lie

algebra of Rd) and so(d), with a Lie bracket that takes into account the action

172

of SO(d) on Rd. So elements are pairs (a, X) with a ∈ Rd and X an anti-symmetric d by d matrix, and can be written in matrix form as(

X a0 0

)Note that the use of the same symbol a in the notation for an element (a, 0) inthe Lie algebra of E(3) and for an element (a,1) in the group E(3) is justifiedby the fact that, thinking of these as matrices, they are isomorphic under theexponential map.

The Lie bracket is given by the matrix commutator, so

[

(X1 a1

0 0

),

(X2 a2

0 0

)] =

([X1, X2] X1a2 −X2a1

0 0

)(15.1)

15.2 Semi-direct products N oK and their rep-resentations, the case of N commutative

The Euclidean group example of the previous section can be generalized to thefollowing

Definition (Semi-direct product, N commutative). Given a group K, a com-mutative group N , and an action φ of K on N by automorphisms, the semi-direct product N oK is the set of pairs (n, k) ∈ N ×K with group law

(n1, k1)(n2, k2) = (n1 + φk1(n2), k1k2)

Here we have written the commutative group law for N additively, and theaction of k ∈ K on N is by an automorphism

φk : N → N

The Euclidean group E(d) is an example with N = Rd,K = SO(d). Fora ∈ Rd, R ∈ SO(d) one has

φRa = Ra

In chapter 37 we will see another important example, the Poincare group whichgeneralizes E(3) to include a time dimension, treating space and time accordingto the principles of special relativity. One of the problems will outline how toshow that the Heisenberg group H2d+1 is another example: it can be written asa semi-direct product of two commutative subgroups

H2d+1 = Rd+1 o Rd

The reason we have been using the symbol N for the first group in thesemidirect product G = N oK is that N ⊂ G (identifying N with elements ofthe form (n, 1)) is an example of a normal subgroup:

173

Definition (Normal subgroup). N is a normal subgroup of G iff

n ∈ N =⇒ gng−1 ∈ N, ∀n ∈ N, g ∈ G

A standard notation for this isN G

In our semi-direct product case, the condition gng−1 ∈ N is only non-trivialfor g ∈ K ⊂ G (here K is identified with the subgroup of G of elements of theform (0, k)), so one needs

(n, 1) ∈ N =⇒ (0, k)(n, 1)(0, k−1) ∈ N ⊂ G, ∀n ∈ N, k ∈ K

This is true since

(0, k)(n, 1)(0, k−1) = (φk(n), k)(0, k−1) = (φk(n), kk−1) = (φk(n), 1) ∈ N ⊂ G

which also shows that one can recover the map φ needed to define G fromknowing the group law in G and the subgroups N and K.

The symbol N o K is supposed to be a mixture of the × and symbols(note that some authors define it to point in the other direction). It has thedefect that it does not specify which φ is being used to construct the semi-directproduct, and there may be multiple possibilities.

Turning to the question of how to find representations of N o K, one canproceed according to the following outline, starting by considering the represen-tations of N .

For a commutative group likeN , irreducible representations are one-dimensional,so correspond to characters. As a result, the set of representations of N acquiresits own group structure, also commutative, and one can define

Definition (Character group). For N a commutative group, let N be the set ofcharacters of N , i.e. functions

χ : N → C

that satisfy the homomorphism property

χ(n1n2) = χ(n1)χ(n2)

The elements of N form a group, with multiplication

(χ1χ2)(n) = χ1(n)χ2(n)

We only will actually need the case N = Rd, where we have already seenthat the differentiable irreducible representations are one-dimensional and givenby

χp(q) = eip·q

So the character group in this case is N = Rd, with elements labeled by thevector p.

174

Recall that whenever we have an action of a group G on a space M , we getan action of G on functions on M by

g · f(x) = f(g−1 · x)

In the case of a semi-direct product, K acts on the set N so we get an action of Kon the functions on N , and thus on N . We can use this to study representationsof the semi-direct product by proceeding roughly as follows.

Given a representation (π, V ) of the semi-direct product N o K, we canconsider this as a representation of the subgroup N , and decompose it intoirreducibles. This decomposition can be thought of as

V = “⊕α∈N

”Vα

where Vα is the subspace on which N acts by the character α ∈ N , i.e.

v ∈ Vα =⇒ π(n)v = α(n)v

If V and N were finite dimensional this would be a finite sum. We are interestedthough in cases where V is infinite dimensional, but do not want to try andenter into the difficult analytical machinery necessary to properly deal with thisinfinite-dimensionality, so this sum should be considered just a motivation forthe construction that follows.

If one α ∈ N occurs in this sum, so will all the other α ∈ N that one gets byacting on α with elements of K. The set of these is called the K-orbit Oα of αin N under the action of K. This suggests that we try and construct irreduciblerepresentations of N oK by taking functions on K-orbits in N , with distinctorbits giving distinct irreducible representations.

In the case of Euclidean groups E(d), N is Rd and non-zero SO(d) orbitswill be spheres. In chapter 17 we will work out in detail what happens for d = 2and d = 3. Chapter 37 will examine the case of Poincare group, and showingthat the Schrodinger representation can be understood as a representation onfunctions on Rd ⊂ N for the Heisenberg group case will be left as a problem.

Looking at functions on K-orbits Oα does not give all possibilities for irre-ducible representations. To get the rest one needs to proceed as follows.

For each α ∈ N , one can define a subgroup Kα ⊂ K consisting of groupelements whose action on N leaves α invariant:

Definition (Stabilizer group or little group). The subgroup Kα ⊂ K of elementsk ∈ K such that

k · α = α

for a given α ∈ N is called the stabilizer subgroup (by mathematicians) or littlesubgroup (by physicists).

To get all irreducible representations of a semi-direct product, one needs toconsider not just functions onOα, but a new “twisted” notion of function, taking

175

values at α in a vector space that is a representation of Kα. Such more generalirreducibles will be labeled not just by the orbit Oα but also by a choice of irre-ducible representation W of Kα. To properly define these “twisted” functionson the orbits involves constructing something called an “induced representa-tion”. The following is rather abstract and somewhat peripheral to the mainexamples we are considering at this point.

Digression (Induced representations). For very general groups G,H, with Ha subgroup of G, to any representation (ρ,W ) of H we can associate a repre-sentation of G as follows

Definition (Induced representation). For (ρ,W ) a representation of H, theinduced representation of G is

(π, IndGH(W ))

where

IndGH(W ) = f : G→W, f(gh) = ρ(h−1)f(g)

for h ∈ H. The action of G on this function space is given by

π(g)f(g0) = f(g−1g0)

Note that once one knows the value of a function in IndGH(W ) at a pointg ∈ G, it is determined at all other points in the coset [gH]. So one can think ofa function in IndGH(W ) as a sort of “twisted” W -valued function on the quotientspace G/H. Only if the representation of H on W is trivial are these the usualW -valued functions on G/H.

In our definition of IndGH(W ) we have not specified which class of functionsto consider. For compact Lie groups G,H this turns out not to matter much,but for non-compact examples it becomes a serious issue. In addition, in suchcases if one wants the induced representation of G to be unitary when one startswith a unitary representation of H one has to modify slightly the definition ofIndGH(W ). Starting with an irreducible representation W , IndGH(W ) may ormay not be an irreducible representation.

For a semi-direct product N o K of the sort discussed in this section, ir-reducible representations will be labeled by pairs (Oα,W ), where W is an irre-ducible representation of Kα. The irreducible representation can be constructedby extending W to a representation of N × Kα by letting N act on W by thecharacter α, and then taking the induced representation

IndNoKNoKαW

Elements of this induced representation space can be thought of as “twisted” W -valued functions on Oα, giving more general representations than the ones onstandard functions on Oα.

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15.3 The Jacobi group and general semi-directproducts

In chapter 12 we found that taking degree zero and one polynomials in phasespace coordinates qj , pj with the Poisson bracket as Lie bracket gives the Liealgebra of the Heisenberg group H2d+1. Homogeneous polynomials of degree twogive the Lie algebra of the group Sp(2d,R). The action of Sp(2d,R) preservesthe symplectic form Ω which determines the multiplication law in H2d+1 (seechapter 13). As a result, elements of Sp(2d,R) act by automorphisms of H2d+1

and this will allow us to construct an example of a semidirect product N oKwhere N = H2d+1 is not a commutative group (here K = Sp(2d,R)).

The general definition of a semi-direct product, for N not necessarily com-mutative, is

Definition (Semi-direct product). A semi-direct product

G = N oK

of two groups N and K is a group such that

1. As a setG = N ×K

and any element g ∈ G can uniquely be written as g = nk, with n ∈ Nand k ∈ K (as elements of G, n is (n, 1), k is (1, k)).

2. N is a normal subgroup of G, i.e.

n ∈ N =⇒ knk−1 ∈ N, ∀n ∈ N, k ∈ K

3. There is a homomorphism

φ : k ∈ K → φk ∈ Aut(N)

(Aut(N) is the group of automorphisms of N) such that the group law inN oK is given by

(n1k1)(n2k2) = n1k1n2k2 = n1φk1(n2)k1k2

This last condition gives the group law knowing φ. If one knows the grouplaw, one can recover φ as

φk(n) = knk−1

The one example with non-commutative N that we are interested in is

Definition (Jacobi group). The Jacobi group in d dimensions is the semi-directproduct group

GJ(d) = H2d+1 o Sp(2d,R)

where for k ∈ Sp(2d,R), φk is the automorphism of H2d+1 = R2d⊕R that actson the R2d preserving the symplectic form Ω that defines the group multiplica-tion.

177

The Lie algebra gJ(d) of the Jacobi group GJ(d) is just precisely the Liealgebra of polynomials of degree less than three on the phase space R2d, asstudied in chapter 13. The normal subgroup N = H3 ⊂ GJ(1) has Lie algebrah2d+1 with basis 1, qj , pj and only non-zero Lie bracket

qj , pk = δjk

The subgroupK = Sp(2d,R) has a Lie algebra with basis elements qjpk, qjqk, pjpk.The non-trivial semi-direct product structure here comes about from an ac-

tion of Sp(2d,R) on H2d+1 by automorphisms. Infinitesimally, this action isgiven by the Poisson bracket of a degree two polynomial with a linear poly-nomial, as studied in section 13.3. There equation 13.9 gives the action of anelement L ∈ sp(2d,R) on linear functions on M and thus on h2d+1. One canexponentiate to get an action of the Lie group K = Sp(2d,R). This actionis a Lie algebra automorphism since it preserves Ω. Using exponential coordi-nates (see equation 11.1) to relate H2d+1 and h2d+1 gives the action of K as anautomorphism of the Heisenberg group.


Semi-direct products and their representations are not commonly discussed ineither physics or mathematics textbooks, with the exception of the case of thePoincare group of special relativity, that will be discussed in chapter 37. Sometextbooks that do cover the subject include section 3.8 of [55], Chapter 5 of[60], [9], and [63]. The general theory was developed by Mackey during the late1940s and 1950s, and his lecture notes on representation theory [37] are a goodsource for the details of this.

178

Chapter 16

Symmetries andIntertwining Operators

The Schrodinger representation of the Heisenberg Lie algebra of linear functionson phase space is the fundamental construction in quantum mechanics, and someaspects of quantum physics (e.g. the momentum operator) can be understoodpurely in terms of this representation on the state space H. We have seen thatthe polynomials of degree two on phase space also form a Lie algebra, sp(2d,R),and that a representation of this Lie algebra on the same state space can be foundby taking products of pairs of Heisenberg Lie algebra representation operators.

This phenomenon can be understood at the group level in terms of the Ja-cobi group, the semi-direct product of the Heisenberg and symplectic groups.The Schrodinger construction provides a unitary representation of the Heisen-berg group, but this will carry extra structure arising from the fact that thesymplectic group acts on the Heisenberg group by automorphisms (since it actson phase space, preserving the Poisson bracket). Each element of the symplecticgroup takes a given construction of the Schrodinger representation to a unitar-ily equivalent one, providing an operator called an “intertwining operator”, andthese intertwining operators will give (up to a phase factor), a representation ofthe symplectic group. Differentiating, we get a representation of sp(2d,R) (upto addition of a scalar), which is the same representation found by quantizingquadratic functions on phase space.

In later chapters we will see that many of the operators providing the funda-mental observables of quantum mechanics come from exactly this construction,corresponding to the representation of some sub-Lie algebra of sp(2d,R) on thestate space.

179

16.1 Intertwining operators and the metaplecticrepresentation

For a general semi-direct product N oK with non-commutative N , the repre-sentation theory can be quite complicated. For the Jacobi group case though, itturns out that things simplify dramatically because of the Stone-von Neumanntheorem which says that, up to unitary equivalence, we only have one irreduciblerepresentation of N = H2d+1.

In the general case, recall that for each k ∈ K the definition of the semi-direct product gives an automorphism φk : N → N . Given a representation πof N , for each k we can define a new representation πk of N by first acting withφk:

πk(n) = π(φk(n))

In the special case of the Jacobi group, if we do this and define a new representionof the Heisenberg group by

ΓS,k(n) = ΓS(φk(n))

the Stone-von Neumann theorem assures us that these must be unitarily equiv-alent, so there must exist unitary operators Uk satisfying

ΓS,k = UkΓSU−1k = ΓS(φk(n)) (16.1)

Operators like this that relate two representations are called “intertwiningoperators”.

Definition (Intertwining operator). If (π1, V1), (π2, V2) are two representationsof a group G, an intertwining operator between these two representations is anoperator U such that

π2(g)U = Uπ1(g) ∀g ∈ G

In our case V1 = V2 is the Schrodinger representation state space H and Uk :H → H is an intertwining operator between ΓS and ΓS,k for each k ∈ Sp(2d,R).Since

ΓS,k1k2 = Uk1k2ΓSU−1k1k2

the Uk should satisfy the group homomorphism property

Uk1k2= Uk1

Uk2

and give us a representation of the group Sp(2d,R) on H. This is what weexpect on general principles: a symmetry of the classical phase space afterquantization becomes a unitary representation on the quantum state space.

The problem with this argument is that the Uk are not uniquely defined.Schur’s lemma tells us that since the representation on H is irreducible, theoperators commuting with the representation operators can only be complexscalars, but these complex scalars give a phase ambiguity in how we define the

180

Uk. So the Uk only have to give a representation of Sp(2d,R) on H up to aphase, i.e.

Uk1k2 = Uk1Uk2eiθ(k1,k2)

for some real function θ of pairs of group elements.The more detailed explicit information we have about the representation ΓS

from its construction in terms of Q and P operators turns out to show thatthis phase ambiguity is really just a sign ambiguity. This is what is responsiblefor the necessity of introducing a double cover of Sp(2d,R) called Mp(2d,R)if we want a true representation. Exactly how this works is best seen usinga different construction of the representation, one that appears in the theoryof the quantum harmonic oscillator, so we will postpone the details of this tochapter 21.

For infinite dimensional phase spaces, the ambiguity may not just be a signambiguity, a situation that is described as a quantum “anomaly” in the symme-try. This means that one is in a situation where a classical symmetry does notsurvive intact as a symmetry of the quantum state space (i.e. as a unitary rep-resentation on the state space), but only as a representation up to a phase. Theoperators that should give the Lie algebra representation will have extra scalarterms that violate the Lie algebra homomorphism property. This phenomenonwill be examined in more detail in chapter 31.

16.2 Constructing the intertwining operators

While the Stone-von Neumann theorem tells us that Uk with the above prop-erties must exist, it does not tell us how to construct them. We have seenthough that quantization of homogeneous quadratic polynomials on the phasespace R2d provides a representation of the Lie algebra sp(2d,R), which is theLie algebra version of what we want, In this section we’ll outline the standardmethod for constructing the unitary operators Uk in a quantum system thatcorrespond to some group action on classical phase space that preserves thePoisson bracket.

For each k0 ∈ Sp(2d,R), k0 acts on the dual classical phase spaceM = R2d

by linear transformations preserving the Poisson bracket Ω (see section 13.3).Recall from section 11.3 that one can identify R2d+1 = M ⊕ R with eitherthe Heisenberg Lie algebra h2d+1 or the Heisenberg Lie group H2d+1. Theexponential map relating the group and Lie algebra is just the identity (inexponential coordinates on the group). Acting by k0 onM and trivially on theR factor gives a map

φk0 : n ∈ H2d+1 → φk0(n) ∈ H2d+1

that is an automorphism (preserves the group law) since it leaves invariant thebilinear from Ω that determines the group law. Since the exponential map isthe identity, one can equally well take φk0

to act on h2d+1, in which case it willbe an automorphism (preserves Lie brackets) of the Lie algebra structure.

181

The φk0 take one Schrodinger representation to a unitarily equivalent one,and there be a representation of the double cover Mp(2d,R) by unitary op-erators Uk on the state space H of the Schrodinger representation, which willsatisfy the intertwining relation

UkΓS(n)U−1k = ΓS(φk0

(n))

Here k is one of the two elements in Mp(2d,R) corresponding to k0 ∈ Sp(2d,R).On the Lie algebra version of the Schrodinger representation we have the

same automorphism φk0, and, for X ∈ h2d+1, the corresponding intertwining

relation isUkΓ′S(X)U−1

k = Γ′S(φk0(X)) (16.2)

We can considerk = etL

for L ∈ mp(2d,R) = sp(2d,R) and take the derivative at t = 0 (using 5.1) toget

[U ′L,Γ′S(X)] = Γ′S(L ·X) (16.3)

Here

L ·X =d

dtφetL(X)|t=0

is the infinitesimal action of the automorphism φk0on the Heisenberg Lie alge-

bra, and U ′L is the derivative of Uk. It is a skew-adjoint operator on H, givinga unitary Lie algebra representation of mp(2d,R) = sp(2d,R). Note that thisintertwining property leaves undetermined a potential scalar addition to the U ′Lor, equivalently, a phase factor in the Uk.

In equation 16.3, a basis for the X will be the constant function 1 and thephase space coordinates qj , pj . In this basis L (see 13.7) will act trivially on 1and by

L ·(

qp

)= LT

(qp

)on the coordinate function basis ofM (see 13.9 and its proof). We want to findthe operators U ′L, which will (up to a constant) be a quadratic combination ofthe operators Qj = iΓ′S(qj), Pj = iΓ′S(pj). To find the right such combination,we can first solve the problem instead at the classical level, using equation 13.9to determine the quadratic function µL on phase space corresponding to thematrix L ∈ sp(2d,R). This quadratic function will satisfy the Poisson bracketrelation

µL,(

qp

) = LT

(qp

)(16.4)

This is the Lie bracket relation in the Lie algebra h(2d+1)osl(2d,R) describingthe action of sl(2d,R) on h(2d+1). Quantization is a Lie algebra homomorphismfrom this to linear operators, so the U ′L that we want should be the quantizationof µL. It will satisfy

[U ′L,

(QP

)] = LT

(QP

)Exponentiating this U ′L will give us our Uk, and thus the operators we want.

182

16.3 Some examples

This has been a rather abstract discussion, but it will provide operators in thequantum theory corresponding to groups G ⊂ GJ(d) of classical phase spacesymmetries. In this section we’ll work out a couple of the simplest possibleexamples in great detail. One motivation for this is that these examples makeclear the conventions being chosen, another is that they determine much ofthe basic structure of what the operators corresponding to symmetries looklike, even in the quite sophisticated infinite-dimensional quantum field theoryexamples we will come to later.

16.3.1 The SO(2) action on the d = 1 phase space

In the case d = 1 one has elements g0 ∈ SO(2) ⊂ Sp(2,R) acting on cqq+ cpp ∈M by (

cqcp

)→ g0

(cqcp

)=


)(cqcp

)so

g0 = eθL

where

L =

(0 1−1 0

)To find the intertwining operators, we first find the quadratic function µL

in q, p that satisfies

µL,(qp

) = LT

(qp

)=

(−pq

)By equation 13.2 this is

µL =1

2

(q p

)(1 00 1

)(qp

)=

1

2(q2 + p2)

Quantizing µL and multiplying by −i, one has a unitary Lie algebra repre-sentation U ′ of so(2) with

U ′L = − i2

(Q2 + P 2)

satisfying

[U ′L,

(QP

)] =

(−PQ

)(16.5)

and intertwining operators

Ug0= ±eθU

′L = ±e−i θ2 (Q2+P 2)

which we will see later give a representation of SO(2) up to a sign, and providea true representation

Ug = e−iθ2 (Q2+P 2)

183

on a double cover of SO(2). In either case, we have

e−iθ2 (Q2+P 2)

(QP

)eiθ2 (Q2+P 2) =


)(QP

)(16.6)

16.3.2 The SO(2) action by rotations of the plane for d = 2

In the case d = 2 there is a another example of an SO(2) group which is asubgroup of the symplectic group, here Sp(4,R). This is the group of rotationsof the configuration space R2, with a simultaneous rotation of the momentumspace, leaving invariant the Poisson bracket. The group SO(2) acts on cq1q1 +cq2q2 + cp1

p1 + cp2p2 ∈M by

cq1cq2cp1

cp2

→ g0

cq1cq2cp1

cp2

=

cos θ − sin θ 0 0sin θ cos θ 0 0

0 0 cos θ − sin θ0 0 sin θ cos θ

cq1cq2cp1

cp2

so g0 is an exponential of θL ∈ sp(4,R), where

L =

0 −1 0 01 0 0 00 0 0 −10 0 1 0

L acts on phase space coordinate functions by

q1

q2

p1

p2

→ LT

q1

q2

p1

p2

=

q2

−q1

p2

−p1

By equation 13.9, with

A =

(0 −11 0

), B = C = 0

the quadratic function µL that satisfies

µL,

q1

q2

p1

p2

= LT

q1

q2

p1

p2

=

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

q1

q2

p1

p2

=

q2

−q1

p2

−p1

is

µL = −q ·(

0 −11 0

)p = q1p2 − q2p1

(which is just the formula given that the angular momentum corresponding torotation about an axis perpendicular to this plane is l = q1p2 − q2p1).

184

Quantizing give a representation of the Lie algebra so(2) with

U ′L = −i(Q1P2 −Q2P1)

satisfying

[U ′L,

(Q1

Q2

)] =

(Q2

−Q1

), [U ′L,

(P1

P2

)] =

(P2

−P1

)and a representation of SO(2)

Ueiθ = e−iθ(Q1P2−Q2P1)

with conjugation by Ueiθ rotating linear combinations of the Q1, Q2 (or theP1, P2) each by an angle θ.

Ueiθ (cq1Q1 + cq2Q2)U−1eiθ

= c′q1Q1 + c′q2Q2

where (c′q1c′q2

)=


)(cq1cq2

)Replacing the matrix L by any real 2 by 2 matrix

A =

(a11 a12

a21 a22

)we get an action of the group GL(2,R) ⊂ Sp(4,R) onM, and after quantizationa Lie algebra representation

U ′A = i(Q1 Q2

)(a11 a12

a21 a22

)(P1

P2

)which will satisfy

[U ′A,

(Q1

Q2

)] = −A

(Q1

Q2

), [U ′A,

(P1

P2

)] = AT

(P1

P2

)Note that the action of A on the momentum operators is the dual of the actionon the position operators. Only in the case of an orthogonal action (the SO(2)earlier) are these the same, with AT = −A.


The conventional derivation of the operators U ′L found in most physics text-books uses Lagrangian methods and Noether’s theorem. The purely Hamilto-nian method used here treats configuration and momentum variables on thesame footing, useful especially to understand group actions that mix them. Foranother treatment of these operators along the lines of this chapter, see section14 of [25].

The issue of the phase factor in the intertwining operators and the metaplec-tic double cover will be discussed in later chapters using a different realizationof the Heisenberg Lie algebra representation. For a discussion of this in termsof the Schrodinger representation, see part I of [35].

185

186

Chapter 17

The Quantum Free Particleas a Representation of theEuclidean Group

In this chapter we will explicitly construct unitary representations of the Eu-clidean groups E(2) and E(3) of spatial symmetries in two and three dimensions.These groups commute with the Hamiltonian of the free particle, and their ir-reducible representations will be given just by the quantum state space of afree particle in either two ot three spatial dimensions. The operators Pj willprovide the infinitesimal action of translations on the state space, while angularmomentum operators Lk will provide the infinitesimal rotation action (therewill be only one of these in two dimensions, three in three dimensions).

The Hamiltonian of the free particle is proportional to the operator |P|2.This is a quadratic operator that commutes with the action of all the elementsof the Lie algebra of the Euclidean group, and so is a Casimir operator play-ing an analogous role to that of the SO(3) Casimir operator |L|2 of chapter8.4. Irreducible representations will be labeled by the eigenvalue of this oper-ator, which in this case will be proportional to the energy. In the Schrodingerrepresentation where the Pj are differentiation operators, this will be a second-order differential operator, and the eigenvalue equation will be a second-orderdifferential equation (the time-independent Schrodinger equation).

17.1 Representations of E(2)

We’ll begin with the case of two spatial dimensions, partly for simplicity, partlybecause physical systems that are translationally invariant in one direction canoften be treated as effectively two dimensional. A basis for the Lie algebra ofE(2) is given by the functions

l = q1p2 − q2p1, p1, p2

187

on the d = 2 phase space M = R4. The non-zero Lie bracket relations are givenby the Poisson brackets

l, p1 = p2, l, p2 = −p1

with an isomorphism with the representation of this Lie algebra in terms of 3by 3 matrices given by

l↔

0 −1 01 0 00 0 0

, p1 ↔

0 0 10 0 00 0 0

, p2 ↔

0 0 00 0 10 0 0

The expression of this Lie algebra in terms of linear and quadratic functions

on the phase space means that it is realized as a sub-Lie algebra of the JacobiLie algebra gJ(2). Quantization via the Schrodinger representation ΓS thenprovides a unitary representation of the Lie algebra of E(2) on the state spaceH of functions of the position variables q1, q2, in terms of operators

Γ′S(p1) = −iP1 = − ∂

∂q1, Γ′S(p2) = −iP1 = − ∂

∂q2(17.1)

and

Γ′S(l) = −iL = −i(Q1P2 −Q2P1) = −(q1∂

∂q2− q2

∂

∂q1) (17.2)

One can recognize 17.2 as the quadratic operator derived for the SO(2) rotationaction in chapter 16.3.2.

The Hamiltonian operator for the free particle is

H =1

2m(P 2

1 + P 22 ) = − 1

2m(∂2

∂q21

+∂2

∂q22

)

and solutions to the Schrodinger equation can be found by solving the eigenvalueequation

Hψ(q1, q2) = − 1

2m(∂2

∂q21

+∂2

∂q22

)ψ(q1, q2) = Eψ(q1, q2)

The operators L,P1, P2 commute with H and so provide a representation of theLie algebra of E(2) on the space of wave-functions of energy E.

This construction of a unitary representation of E(2) gives an example of thegeneral story described in chapter 15 for a semi-direct product with commutativeN . Here N = R2 is the translation group on R2, and elements of N are thefunctions

χp = eip·q

labeled by vectors p = (p1, p2). Rotations R in the group K = SO(2) act onsuch functions by

eip·q 7→ eip·R−1q = ei(Rp)·q

(since R is an orthogonal matrix, R−1 = RT ). Under this action of SO(2), thereare two different kinds of orbits of a label p:

188

• p = 0. This is the point orbit, and corresponds to a representation ofE(2) on which the translation subgroup N acts trivially. Any SO(2)representation can occur, so the irreducible representations of E(2) we getthis way are just the standard one-dimensional complex representationsof SO(2) = U(1). One can think of these as irreducible representations ofE(2) corresponding to irreducible representations of the stabilizer groupof p = 0, which is all of SO(2). From the point of view of solutions to theSchrodinger equation, one just gets the trivial representation of SO(2), onconstant functions ψ. To get the other representations, one would have togeneralize to the case of “twisted” functions described in chapter 15.

• |p|2 = 2mE > 0. This is the circle of radius√

2mE in momentum space,with SO(2) acting by rotation of the circle.

To each point on the second sort of orbit we associate the wave-function or state

ψp(q) = eip·q = |p〉

which is a solution of the time-independent Schrodinger equation with energy

E =|p|2

2m> 0

Recall that such |p〉 give a sort of continuous basis of H, even though theseare not square-integrable functions. The formalism for working with them usesdistributions and the orthonormality relation

〈p|p′〉 = δ(p− p′)

An arbitrary ψ(q) ∈ H can be written as a continuous linear combination of

the |p〉, i.e. as an inverse Fourier transform of a function ψ(p) on momentumspace as

ψ(q) =1

2π

∫∫eip·qψ(p)d2p

In momentum space the time-independent Schrodinger equation becomes

(|p|2

2m− E)ψ(p) = 0

so we get a solution for any choice of ψ(p) that is non-zero only on the circle|p|2 = 2mE (we won’t try to characterize which class of such functions toconsider, which would determine which class of functions solving the Schrodingerequation we end up with after Fourier transform).

Going to polar coordinates p = (p cos θ, p sin θ), the space of solutions to

the time-independent Schrodinger equation at energy E is given by ψ(p) of theform

ψ(p) = ψE(θ)δ(p2 − 2mE)

189

To put this delta-function in a more useful form, note that for p ≈√

2mE onehas

p2 − 2mE ≈ 1

2√

2mE(p−

√2mE)

so one has the equality of distributions

δ(p2 − 2mE) = δ(2√

2mE(p−√

2mE)) =1

2√

2mEδ(p−

√2mE)

It is this space of functions ψE(θ) of functions on the circle of radius√

2mEthat will provide an infinite-dimensional representation of the group E(2), onethat turns out to be irreducible, although we will not show that here. Theposition space wave-function corresponding to ψE(θ) will be

ψ(q) =1

2π

∫∫eip·qψE(θ)δ(p2 − 2mE)pdpdθ

=1

2π

∫∫eip·qψE(θ)

1

2√

2mEδ(p−

√2mE)pdpdθ

=1

4π

∫ 2π

0

ei√

2mE(q1 cos θ+q2 sin θ)ψE(θ)dθ

Functions ψE(θ) with simple behavior in θ will correspond to wave-functions

with more complicated behavior in position space. For instance, taking ψE(θ) =e−inθ one finds that the wave-function along the q2 direction is given by

ψ(0, q) =1

4π

∫ 2π

0

ei√

2mE(q sin θ)e−inθdθ

=1

2Jn(√

2mEq)

where Jn is the n’th Bessel function.Equations 17.1 and 17.2 give the representation of the Lie algebra of E(2)

on wave-functions ψ(q). The representation of this Lie algebra on the ψE(θ) is

just given by the Fourier transform, and we’ll denote this Γ′S . Using the formulafor the Fourier transform we find that

Γ′S(p1) = − ∂

∂q1= −ip1 = −i

√2mE cos θ

Γ′S(p2) = − ∂

∂q2= −ip2 = −i

√2mE sin θ

are multiplication operators and, taking the Fourier transform of 17.2 gives thedifferentiation operator

Γ′S(l) =− (p1∂

∂p2− p2

∂

∂p1)

=− ∂

∂θ

190

(use integration by parts to show qj = i ∂∂pj

and thus the first equality, then the

chain rule for functions f(p1(θ), p2(θ)) for the second).

At the group level, E(2) has elements (a, R(φ)) which can be written as aproduct (a, R(φ)) = (a,1)(0, R(φ)) or, in terms of matricescosφ − sinφ a1

sinφ cosφ a2

0 0 1

=

1 0 a1

0 1 a2

0 0 1

cosφ − sinφ 0sinφ cosφ 0

0 0 1

The group has a unitary representation

(a, R(φ))→ u(v, R(φ))

on the position space wave-functions ψ(q), given by the induced action on func-tions from the action of E(2) on position space R2

u(a, R(φ))ψ(q) =ψ((a, R(φ))−1 · q)

= = ψ((−R(−φ)a, R(−φ)) · q)

=ψ(R(−φ)(q− a))

This is just the Schrodinger representation ΓS of the Jacobi group GJ(2), re-stricted to the subgroup E(2) of transformations of phase space that are transla-tions in q and rotations in both q and p vectors, preserving their inner product(and thus the symplectic form). One can see this by considering the actionof translations as the exponential of the Lie algebra representation operatorsΓ′S(pj) = −iPj

u(a,1)ψ(q) = e−i(a1P1+a2P2)ψ(q) = ψ(q− a)

and the action of rotations as the exponential of the Γ′S(l) = −iL

u(0, R(φ))ψ(q) = e−iφLψ(q) = ψ(R(−φ)q)

One has a Fourier-transformed version u of this representation, with rota-tions acting now on momentum space and on the ψE by

u(0, R(φ))ψE(θ) = ψE(θ − ψ)

Translations act here by multiplication operators

u(a,1)ψE(θ) = e−i(a·p)ψE(θ) = e−i√

2mE(a1 cos θ+a2 sin θ)ψE(θ)

In the case of the R action by time translations, we found that we had both aSchrodinger picture where unitary operators U(t) = e−itH time-evolved states,and a Heisenberg picture where states stayed constant, but operators evolvedby conjugation by U(t). Here we can also have the unitary transformations

191

u(a, R(φ)) act on states, or on operators by conjugation. Acting on the oper-ators Γ′S(q) = −iQ,Γ′S(p) = −iP, this conjugation action is by intertwiningoperators (see 16.2) that are just the u(a, R(φ)) . One has

u(a, R(φ))Γ′S(q,p)u(a, R(φ))−1 = Γ′S((a, R(φ)) · (q,p))

where the Euclidean group acts on phase space by

(q,p)→ (a, R(φ)) · (q,p) = (a +R(φ)q, R(φ)p)

17.2 The quantum particle and E(3) representa-tions

In the physical case of three spatial dimensions, the state space of the theory of aquantum free particle is again a Euclidean group representation, with the samerelationship to the Schrodinger representation as in two spatial dimensions. Themain difference is that the rotation group is now three dimensional and non-commutative, so instead of the single Lie algebra basis element l we have threeof them, satisfying Poisson bracket relations that are the Lie algebra relationsof so(3)

l1, l2 = l3, l2, l3 = l1, l3, l1 = l2

The pj give the other three basis elements of the Lie algebra of E(3). Theycommute amongst themselves and the action of rotations on vectors providesthe rest of the non-trivial Poisson bracket relations

l1, p2 = p3, l1, p3 = −p2

l2, p1 = −p3, l2, p3 = p1

l3, p1 = p2, l3, p2 = −p1

An isomorphism of this Lie algebra with a Lie algebra of matrices is givenby

l1 ↔

0 0 0 00 0 −1 00 1 0 00 0 0 0

l2 ↔

0 0 1 00 0 0 0−1 0 0 00 0 0 0

l3 ↔

0 −1 0 01 0 0 00 0 0 00 0 0 0

p1 ↔

0 0 0 10 0 0 00 0 0 00 0 0 0

p2 ↔

0 0 0 00 0 0 10 0 0 00 0 0 0

p3 ↔

0 0 0 00 0 0 00 0 0 10 0 0 0

The lj are quadratic functions in the qj , pj , given by the classical mechanical

expression for the angular momentum

l = q× p

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or, in components

l1 = q2p3 − q3p2, l2 = q3p1 − q1p3, l3 = q1p2 − q2p1

The Euclidean group E(3) is a subgroup of the Jacobi group GJ(3) in thesame way as in two dimensions, and the Schrodinger representation ΓS providesa representation of E(3) with Lie algebra version

Γ′S(l1) = −iL1 = −i(Q2P3 −Q3P2) = −(q2∂

∂q3− q3

∂

∂q2)

Γ′S(l2) = −iL2 = −i(Q3P1 −Q1P3) = −(q3∂

∂q1− q1

∂

∂q3)

Γ′S(l3) = −iL3 = −i(Q1P2 −Q2P1) = −(q1∂

∂q2− q2

∂

∂q1)

Γ′S(pj) = −iPj = − ∂

∂qj

These are just the infinitesimal versions of the action of E(3) on functionsinduced from its action on position space R3. Given an element g = (a, R) ∈E(3) ⊂ GJ(3) we have a unitary transformation on wave-functions

u(a, R)ψ(q) = ΓS(g)ψ(q) = ψ(g−1 · q) = ψ(R−1(q− a))

These group elements will be a product of a translation and a rotation, andthe unitary transformations u are exponentials of the Lie algebra actions above,with

u(a,1)ψ(q) = e−i(a1P1+a2P2+a3P3)ψ(q) = ψ(q− a)

for a translation by a, and

u(0, R(φ, ej))ψ(q) = e−iφLjψ(q) = ψ(R(−φ, ej)q)

for R(φ, ej) a rotation about the j-axis by angle φ.This representation of E(3) on wave-functions is reducible, since in terms of

momentum eigenstates, rotations will only take eigenstates with one value of themomentum to those with another value of the same length-squared. We can getan irreducible representation by using the Casimir operator P 2

1 +P 22 +P 2

3 , whichcommutes with all elements in the Lie algebra of E(3). The Casimir operatorwill act on an irreducible representation as a scalar, and the representation willbe characterized by that scalar. Up to a constant the Casimir operator is justthe Hamiltonian

H =1

2m(P 2

1 + P 22 + P 2

3 )

and so the constant characterizing an irreducible will just be the energy E,and our irreducible representation will be on the space of solutions of the time-independent Schrodinger equation

1

2m(P 2

1 + P 22 + P 2

3 )ψ(q) = − 1

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)ψ(q) = Eψ(q)

193

Using the Fourier transform

ψ(q) =1

(2π)32

∫R3

eip·qψ(p)d3p

the time-independent Schrodinger equation becomes

(|p|2

2m− E)ψ(p) = 0

and we have distributional solutions

ψ(p) = ψE(p)δ(|p|2 − 2mE)

characterized by functions ψE(p) defined on the sphere |p|2 = 2mE. Suchcomplex-valued functions on the sphere of radius

√2mE provide a Fourier-

transformed version u of the irreducible representation of E(3). Here the actionof the group E(3) is by

u(a,1)ψE(p) = e−i(a·p)ψE(p)

for translations, byu(0, R)ψE(p) = ψE(R−1p)

for rotations, and by

u(a, R)ψE(p) = u(a,1)u(0, R)ψE(p) = e−ia·R−1pψE(R−1p)

for a general element.

17.3 Other representations of E(3)

In the E(2) case we could get all irreducible representations from considering

the degenerate case of the point orbit in momentum space N = R2 and the irre-ducible representations of SO(2), together with the non-trivial case of functionson circular orbits of radius r, where r2 is the eigenvalue of the Casimir operator.For any momentum vector on such a circular orbit, the stabilizer subgroup ofSO(2) action leaving it invariant is trivial.

In three dimensions, we again have representations labeled by rotation grouporbits in momentum space N = R3, which now are spheres of radius r. For thedegenerate case of r = 0, we again get an E(3) representation for each irreduciblerepresentation of the rotation group SO(3) (labeled by a non-negative integer,the spin), but these are finite-dimensional representations with the translationsacting trivially. We get infinite-dimensional representations for each sphericalorbit, with the Casimir operator taking value r2 > 0.

The new phenomenon in the E(3) case is that a non-zero momentum vectorp has a non-trivial stabilizer subgroup for the SO(3) action by rotations. Thisstabilizer subroup is the SO(2) subgroup of rotations about the axis p. Recall

194

from our discussion in chapter 15 that we expect to get irreducibles for eachpair of an orbit and an irreducible representation of the stabilizer group. Sinceirreducible representations of SO(2) are labeled by integers n, we expect theseE(3) irreducibles to be labeled by pairs (r, n) where r is the radius of the orbitin momentum space, and n is an integer.

In the E(3) case there are actually two Casimir operators, |P|2 and L · P.One can check that this second operator commutes with everything in the E(3)Lie algebra, just like the first, and it takes the value 0 on the representations onwave-functions that we have considered so far. This is because for our plane-wave solutions of momentum p, L ·P is the infinitesimal generator of rotationsabout p, but these plane waves are invariant under such rotations. To getnon-zero values for L · P one has to somehow generalize the notion of wave-function, with one possibility the use of the “induced representations” discussedin chapter 15. These more general cases will be characterized by the eigenvalueof this second Casimir operator L · P, which on a momentum eigenstate willbe some integer times the length of the momentum. This integer is called the“helicity”, and this number can be interpreted as some sort of “internal” angularmomentum of the quantum particle, one that only has a well-defined componentalong the direction of motion of the particle.

We will not consider such theories in general, but in chapter 29 will considerone special case: the non-relativistic spin-half particle, which will have helicityvalues ± 1

2 , the fractional values indicating the use of the double cover of SO(3).In this construction we will use two-component wave-functions, with solutionstaking values in a one-dimensional subspace that varies with p, providing anappropriate sort of “twisted” function in that context.


The angular momentum operators are a standard topic in every quantum me-chanics textbook, see for example chapter 12 of [53]. The characterization here offree-particle wave-functions at fixed energy as giving irreducible representationsof the Euclidean group is not so conventional, but it is just the non-relativisticversion of the conventional description of relativistic quantum particles in termsof representations of the Poincare group (see chapter 37).

195

196

Chapter 18

Central Potentials and theHydrogen Atom

When the Hamiltonian function is invariant under rotations, we then expecteigenspaces of the corresponding Hamiltonian operator to carry representationsof SO(3). These spaces of eigenfunctions of a given energy break up into irre-ducible representations of SO(3), and we have seen that these are labeled bya integer l = 0, 1, 2, . . .. One can use this to find properties of the solutions ofthe Schrodinger equation whenever one has a rotation-invariant potential en-ergy, and we will work out what happens for the case of the Coulomb potentialdescribing the hydrogen atom. We will see that this specific case is exactlysolvable because it has a second not-so-obvious SO(3) symmetry, in addition tothe one coming from rotations in R3.

18.1 Quantum particle in a central potential

In classical physics, to describe not free particles, but particles experiencingsome sort of force, one just needs to add a “potential energy” term to thekinetic energy term in the expression for the energy (the Hamiltonian function).In one dimension, for potential energies that just depend on position, one has

h =p2

2m+ V (q)

for some function V (q). In the physical case of three dimensions, this will be

h =1

2m(p2

1 + p22 + p2

3) + V (q1, q2, q3)

Quantizing and using the Schrodinger representation, the Hamiltonian op-

197

erator for a particle moving in a potential V (q1, q2, q3) will be

H =1

2m(P 2

1 + P 22 + P 3

3 ) + V (Q1, Q2, Q3)

=−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

) + V (q1, q2, q3)

=−~2

2m∆ + V (q1, q2, q3)

We will be interested in so-called “central potentials”, potential functions thatare functions only of q2

1 + q22 + q2

3 , and thus only depend upon r, the radialdistance to the origin. For such V , both terms in the Hamiltonian will beSO(3) invariant, and eigenspaces of H will be representations of SO(3).

Using the expressions for the angular momentum operators in spherical coor-dinates derived in chapter 8, one can show that the Laplacian has the followingexpression in spherical coordinates

∆ =∂2

∂r2+

2

r

∂

∂r− 1

r2L2

where L2 is the Casimir operator in the representation ρ, which we have shownhas eigenvalues l(l+1) on irreducible representations of dimension 2l+1 (integralspin l). So, restricted to such an irreducible representation, we have

∆ =∂2

∂r2+

2

r

∂

∂r− l(l + 1)

r2

To solve the Schrodinger equation, we need to find the eigenfunctions of H.These will be irreducible representations of SO(3) acting on H, which we haveseen can be explicitly expressed in terms of the spherical harmonic functionsY ml (θ, φ) in the angular variables. So, to find eigenfunctions of the Hamiltonian

H = − ~2

2m∆ + V (r)

we need to find functions glE(r) depending on l = 0, 1, 2, . . . and the energyeigenvalue E satisfying

(−~2

2m(d2

dr2+

2

r

d

dr− l(l + 1)

r2) + V (r))glE(r) = EglE(r)

Given such a glE(r) we will have

HglE(r)Y ml (θ, φ) = EglE(r)Y ml (θ, φ)

and theψ(r, θ, φ) = glE(r)Y ml (θ, φ)

will span a 2l+ 1 dimensional (since m = −l,−l+ 1, . . . , l−1, l) space of energyeigenfunctions for H of eigenvalue E.

198

For a general potential function V (r), exact solutions for the eigenvalues Eand corresponding functions glE(r) cannot be found in closed form. One specialcase where we can find such solutions is for the three-dimensional harmonic os-cillator, where V (r) = 1

2mω2r2. These are much more easily found though using

the creation and annihilation operator techniques to be discussed in chapter 19.

The other well-known and physically very important case is the case of a 1r

potential, called the Coulomb potential. This describes a light charged particlemoving in the potential due to the electric field of a much heavier chargedparticle, a situation that corresponds closely to that of a hydrogen atom. Inthis case we have

V = −e2

r

where e is the charge of the electron, so we are looking for solutions to

(−~2

2m(d2

dr2+

2

r

d

dr− l(l + 1)

r2)− e2

r)glE(r) = EglE(r)

Since having

d2

dr2(rg) = Erg

is equivalent to

(d2

dr2+

2

r)g = Eg

for any function g, glE(r) will satisfy

(−~2

2m(d2

dr2− l(l + 1)

r2)− e2

r)rglE(r) = ErglE(r)

The solutions to this equation can be found through a rather elaborate pro-cess. For E ≥ 0 there are non-normalizable solutions that describe scatteringphenomena that we won’t study here. For E < 0 solutions correspond to aninteger n = 1, 2, 3, . . ., with n ≥ l + 1. So, for each n we get n solutions, withl = 0, 1, 2, . . . , n− 1, all with the same energy

En = − me4

2~2n2

A plot of the different energy eigenstates looks like this:

199

The degeneracy in the energy values leads one to suspect that there is someextra group action in the problem commuting with the Hamiltonian. If so, theeigenspaces of energy eigenfunctions will come in irreducible representations ofsome larger group than SO(3). If when one restricts to the SO(3) subgroup,the representation is reducible, giving n copies of our SO(3) representation ofspin l, that would explain the pattern observed here. In the next section wewill see that this is the case, and there use representation theory to derive theabove formula for En.

We won’t go through the process of showing how to explicitly find the func-tions glEn(r) but just quote the result. Setting

a0 =~2

me2

(this has dimensions of length and is known as the “Bohr radius”), and defining

200

gnl(r) = glEn(r) the solutions are of the form

gnl(r) ∝ e−rna0 (

2r

na0)lL2l+1

n+l (2r

na0)

where the L2l+1n+l are certain polynomials known as associated Laguerre polyno-

mials.So, finally, we have found energy eigenfunctions

ψnlm(r, θ, φ) = gnl(r)Yml (θ, φ)

for

n = 1, 2, . . .

l = 0, 1, . . . , n− 1

m = −l,−l + 1, . . . , l − 1, l

The first few of these, properly normalized, are

ψ100 =1√πa3

0

e−ra0

(called the 1S state, “S” meaning l = 0)

ψ200 =1√

8πa30

(1− r

2a0)e−

r2a0

(called the 2S state), and the three dimensional l = 1 (called 2P , “P” meaningl = 1) states with basis elements

ψ211 = − 1

8√πa3

0

r

a0e−

r2a0 sin θeiφ

ψ211 = − 1

4√

2πa30

r

a0e−

r2a0 cos θ

ψ21−1 =1

8√πa3

0

r

a0e−

r2a0 sin θe−iφ

18.2 so(4) symmetry and the Coulomb potential

The Coulomb potential problem is very special in that it has an additionalsymmetry, of a non-obvious kind. This symmetry appears even in the classi-cal problem, where it is responsible for the relatively simple solution one canfind to the essentially identical Kepler problem. This is the problem of findingthe classical trajectories for bodies orbiting around a central object exerting agravitational force, which also has a 1

r potential. Kepler’s second law for such

201

motion comes from conservation of angular momentum, which corresponds tothe Poisson bracket relation

lj , h = 0

Here we’ll take the Coulomb version of the Hamiltonian that we need for thehydrogen atom problem

h =1

2m|p|2 − e2

r

One can read the relation lj , h = 0 in two ways: it says both that the hamil-tonian h is invariant under the action of the group (SO(3)) whose infinitesimalgenerators are lj , and that the components of the angular momentum (lj) areinvariant under the action of the group (R of time translations) whose infinites-imal generator is h.

Kepler’s first and third laws have a different origin, coming from the existenceof a new conserved quantity for this special choice of Hamiltonian. This quantityis, like the angular momentum, a vector, often called the Lenz (or sometimesRunge-Lenz, or even Laplace-Runge-Lenz) vector.

Definition (Lenz vector). The Lenz vector is the vector-valued function on thephase space R6 given by

w =1

m(p× l)− e2 q

|q|2

Simple manipulations of the cross-product show that one has

l ·w = 0

We won’t here explicitly calculate the various Poisson brackets involving thecomponents wj of w, since this is a long and unilluminating calculation, butwill just quote the results, which are

•wj , h = 0

This says that, like the angular momentum, the vector wa is a conservedquantity under time evolution of the system, and its components generatesymmetries of the classical system.

•wj , lk = εjklwl

These relations say that the generators of the SO(3) symmetry act on wjthe way one would expect, since wj is a vector.

•wj , wk = εjklll(

−2h

m)

This is the most surprising relation, and it has no simple geometricalexplanation (although one can change variables in the problem to try andgive it one). It expresses a highly non-trivial relationship between the twosets of symmetries generated by the vectors l,w and the Hamiltonian h.

202

The wj are cubic in the q and p variables, so one would expect that theGroenewold-van Hove no-go theorem would tell one that there is no consistentway to quantize this system by finding operators Wj corresponding to the wjthat would satisfy the commutation relations corresponding to these Poissonbrackets. It turns out though that this can be done, although not for functionsdefined over the entire phase-space. One gets around the no-go theorem bydoing something that only works when the Hamiltonian h is positive (we’ll betaking a square root of −h).

The choice of operators Wj that works is

W =1

2m(P× L− L×P)− e2 Q

|Q|2

where the last term is the operator of multiplication by e2qj/|q|2. By elaborateand unenlightening computations the Wj can be shown to satisfy the commu-tation relations corresponding to the Poisson bracket relations of the wj :

[Wj , H] = 0

[Wj , Lk] = i~εjklWl

[Wj ,Wk] = i~εjklLl(−2

mH)

as well as

L ·W = W · L = 0

The first of these shows that energy eigenstates will be preserved not just bythe angular momentum operators Lj , but by a new set of non-trivial operators,the Wj , so will be representations of a larger Lie algebra thatn so(3).

In addition, one has the following relation between W 2, H and the Casimiroperator L2

W 2 = e41 +2

mH(L2 + ~21)

and it is this which will allow us to find the eigenvalues of H, since we knowthose for L2, and can find those of W 2 by changing variables to identify a secondso(3) Lie algebra.

To do this, first change normalization by defining

K =

√−m2E

W

where E is the eigenvalue of the Hamiltonian that we are trying to solve for.Note that it is at this point that we violate the conditions of the no-go theorem,since we must have E < 0 to get a K with the right properties, and this restrictsthe validity of our calculations to a subset of phase space. For E > 0 one canproceed in a similar way, but the Lie algebra one gets is different (so(3, 1) insteadof so(4)).

203

One then has the following relation between operators

2H(K2 + L2 + ~21) = −me41

and the following commutation relations

[Lj , Lk] = i~εjklLl

[Kj , Lk] = i~εjklKl

[Kj ,Kk] = i~εjklLl

Defining

M =1

2(L + K), N =

1

2(L−K)

one has[Mj ,Mk] = i~εjklMl

[Nj , Nk] = i~εjklNl

[Mj , Nk] = 0

This shows that we have two commuting copies of so(3) acting on states, spannedrespectively by the Mj and Nj , with two corresponding Casimir operators M2

and N2.Using the fact that

L ·K = K · L = 0

one finds thatM2 = N2

Recall from our discussion of rotations in three dimensions that representa-tions of so(3) = su(2) correspond to representations of Spin(3) = SU(2), thedouble cover of SO(3) and the irreducible ones have dimension 2l + 1, with lhalf-integral. Only for l integral does one get representations of SO(3), and itis these that occur in the SO(3) representation on functions on R3. For four di-mensions, we found that Spin(4), the double cover of SO(4), is SU(2)×SU(2),and one thus has spin(4) = so(4) = su(2)×su(2) = so(3)×so(3). This is exactlythe Lie algebra we have found here, so one can think of the Coulomb problem ashaving an so(4) symmetry. The representations that will occur can include thehalf-integral ones, since neither so(3) is the so(3) of physical rotations in 3-space(those are generated by L = M + N, which will have integral eigenvalues of l).

The relation between the Hamiltonian and the Casimir operators M2 andN2 is

2H(K2 + L2 + ~21) = 2H(2M2 + 2N2 + ~21) = 2H(4M2 + ~21) = −me41

On irreducible representations of so(3) of spin µ, we will have

M2 = µ(µ+ 1)1

204

for some half-integral µ, so we get the following equation for the energy eigen-values

E = − −me4

2~2(4µ(µ+ 1) + 1)= − −me4

2~2(2µ+ 1)2

Letting n = 2µ + 1, for µ = 0, 12 , 1, . . . we get n = 1, 2, 3, . . . and precisely the

same equation for the eigenvalues described earlier

En = − me4

2~2n2

It is not hard to show that the irreducible representations of a product likeso(3) × so(3) are just tensor products of irreducibles, and in this case the twofactors of the product are identical due to the equality of the Casimirs M2 = N2.The dimension of the so(3)×so(3) irreducibles is thus (2µ+1)2 = n2, explainingthe multiplicity of states one finds at energy eigenvalue En.

18.3 The hydrogen atom

The Coulomb potential problem provides a good description of the quantumphysics of the hydrogen atom, but it is missing an important feature of thatsystem, the fact that electrons are spin 1

2 systems. To describe this, one reallyneeds to take as space of states two-component wave-functions

|ψ〉 =

(ψ1(q)ψ2(q)

)in a way that we will examine in detail in chapter 29. The Hamiltonian operatorfor the hydrogen atom is the same on both components, so the only effect of theadditional wave-function component is to double the number of energy eigen-states at each energy. Electrons are fermions, so anti-symmetry of multi-particlewave functions implies the Pauli principle that states can only be occupied bya single particle. As a result, one finds that when adding electrons to an atomdescribed by the Coulomb potential problem, the first two fill up the lowestCoulomb energy eigenstate (the ψ100 or 1S state at n = 1), the next eight fillup the n = 2 states ( two each for ψ200, ψ211, ψ210, ψ21−1), etc. This goes a longways towards explaining the structure of the periodic table of elements.

When one puts a hydrogen atom in a constant magnetic field B, the Hamil-tonian acquires a term

2e

mcB · σ

of exactly the sort we began our study of quantum mechanics with for a simpletwo-state system. It causes a shift in energy eigenvalues proportional to ±|B|for the two different components of the wave-function, and the observation ofthis energy splitting makes clear the necessity of treating the electron using thetwo-component formalism.

205


This is a standard topic in all quantum mechanics books. For example, seechapters 12 and 13 of [53]. The so(4) calculation is not in [53], but is in some ofthe other such textbooks, a good example is chapter 7 of [4]. For an extensivediscussion of the symmetries of the 1

r potential problem, see [24].

206

Chapter 19

The Harmonic Oscillator

In this chapter we’ll begin the study of the most important exactly solvablephysical system, the harmonic oscillator. Later chapters will discuss extensionsof the methods developed here to the case of fermionic oscillators, as well as freequantum field theories, which are harmonic oscillator systems with an infinitenumber of degrees of freedom.

For a finite number of degrees of freedom, the Stone-von Neumann theo-rem tells us that there is essentially just one way to non-trivially represent the(exponentiated) Heisenberg commutation relations as operators on a quantummechanical state space. We have seen two unitarily equivalent constructionsof these operators: the Schrodinger representation in terms of functions on ei-ther coordinate space or momentum space. It turns out that there is anotherclass of quite different constructions of these operators, one that depends uponintroducing complex coordinates on phase space and then using properties ofholomorphic functions. We’ll refer to this as the Bargmann-Fock representation,although quite a few mathematicians have had their name attached to it for onegood reason or another (some of the other names one sees are Friedrichs, Segal,Shale, Weil, as well as the descriptive terms holomorphic and oscillator).

Physically the importance of this representation is that it diagonalizes theHamiltonian operator for a fundamental sort of quantum system: the harmonicoscillator. In the Bargmann-Fock representation the energy eigenstates of sucha system are the simplest states and energy eigenvalues are just integers. Theseintegers label the irreducible representations of the U(1) symmetry generatedby the Hamiltonian, and they can be interpreted as counting the number of“quanta” in the system. It is the ubiquity of this example that justifies the“quantum” in “quantum mechanics”. The operators on the state space can besimply understood in terms of basic operators called annihilation and creationoperators which increase or decrease by one the number of quanta.

207

19.1 The harmonic oscillator with one degree offreedom

An even simpler case of a particle in a potential than the Coulomb potential ofthe last chapter is the case of V (q) quadratic in q. This is also the lowest-orderapproximation when one studies motion near a local minimum of an arbitraryV (q), expanding V (q) in a power series around this point. We’ll write this as

h =p2

2m+

1

2mω2q2

with coefficients chosen so as to make ω the angular frequency of periodic motionof the classical trajectories. These satisfy Hamilton’s equations

p = −∂V∂q

= mω2q, q =p

m

so

q = −ω2q

which will have solutions with periodic motion of angular frequency ω. Thesesolutions can be written as

q(t) = c+eiωt + c−e

−iωt

for c+, c− ∈ C where, since q(t) must be real, we have c− = c+. The space ofsolutions of the equation of motion is thus two-dimensional, and abstractly onecan think of this as the phase space of the system.

More conventionally, one can parametrize the phase space by initial valuesthat determine the classical trajectories, for instance by the position q(0) andmomentum p(0) at an initial time t(0). Since

p(t) = mq = mc+iωeiωt −mc−iωe−iωt = imω(c+e

iωt − c+e−iωt)

we have

q(0) = c+ + c− = 2<(c+), p(0) = imω(c+ − c−) = 2mω=(c+)

so

c+ =1

2q(0) + i

1

2mωp(0)

The classical phase space trajectories are

q(t) = (1

2q(0) + i

1

2mωp(0))eiωt + (

1

2q(0)− i 1

2mωp(0))e−iωt

p(t) = (imω

2q(0)− 1

2p(0))eiωt + (

−imω2

q(0) +1

2p(0))e−iωt

208

Instead of using two real coordinates to describe points in the phase space(and having to introduce a reality condition when using complex exponentials),one can instead use a single complex coordinate

z(t) =1√2

(q(t) +i

mωp(t))

Then the equation of motion is a first-order rather than second-order differentialequation

z = −iωzwith solutions

z(t) = z(0)e−iωt

The classical trajectories are then realized as complex functions of t, and paramet-rized by the complex number

z(0) =1√2

(q(0) +i

mωp(0))

Since the Hamiltonian is just quadratic in the p and q, we have seen that wecan construct the corresponding quantum operator uniquely using the Schroding-er representation. For H = L2(R) we have a Hamiltonian operator

H =P 2

2m+

1

2mω2Q2 = − ~2

2m

d2

dq2+

1

2mω2q2

To find solutions of the Schrodinger equation, as with the free particle, oneproceeds by first solving for eigenvectors of H with eigenvalue E, which meansfinding solutions to

HψE = (− ~2

2m

d2

dq2+

1

2mω2q2)ψE = EψE

Solutions to the Schrodinger equation will then be linear combinations of thefunctions

ψE(q)e−i~Et

Standard but somewhat intricate methods for solving differential equationslike this show that one gets solutions for E = En = (n+ 1

2 )~ω, n a non-negativeinteger, and the normalized solution for a given n (which we’ll denote ψn) willbe

ψn(q) = (mω

π~22n(n!)2)

14Hn(

√mω

~q)e−

mω2~ q

2

where Hn is a family of polynomials called the Hermite polynomials. Theψn provide an orthonormal basis for H (one does not need to consider non-normalizable wave-functions as in the free particle case), so any initial wave-function ψ(q, 0) can be written in the form

ψ(q, 0) =

∞∑n=0

cnψn(q)

209

with

cn =

∫ +∞

−∞ψn(q)ψ(q, 0)dq

(note that the ψn are real-valued). At later times, the wavefunction will be

ψ(q, t) =

∞∑n=0

cnψn(q)e−i~Ent =

∞∑n=0

cnψn(q)e−i(n+ 12 )ωt

19.2 Creation and annihilation operators

It turns out that there is a quite easy method to solve for eigenfunctions ofthe harmonic oscillator Hamiltonian, which also leads to a new representationof the Heisenberg group (of course unitarily equivalent to the Schrodinger oneby the Stone-von Neumann theorem). Instead of working with the self-adjointoperators Q and P that satisfy the commutation relation

[Q,P ] = i~1

we define

a =

√mω

2~Q+ i

√1

2mω~P, a† =

√mω

2~Q− i

√1

2mω~P

which satisfy the commutation relation

[a, a†] = 1

To simplify calculations, from now on we will set ~ = m = ω = 1. Thiscorresponds to a specific choice of units for energy, distance and time, and thegeneral case of arbitrary constants can be recovered by rescaling the results ofour calculations. So, now

a =1√2

(Q+ iP ), a† =1√2

(Q− iP )

and

Q =1√2

(a+ a†), P =1

i√

2(a− a†)

and the Hamiltonian operator is

H =1

2(Q2 + P 2) =

1

2(1

2(a+ a†)2 − 1

2(a− a†)2)

=1

2(aa† + a†a)

=a†a+1

2

The fact that the eigenvalues of H will be (n+ 12 ) (in units of ~ω) corresponds

to

210

Theorem. If a has an eigenvector |0〉 with eigenvalue 0, the number operatorN = a†a has eigenvalues n = 0, 1, 2, . . . on H = L2(R).

Proof. If |c〉 is a normalized eigenvector of N with eigenvalue c, one has

c = 〈c|a†a|c〉 = |a|c〉|2 ≥ 0

One also has

Na|c〉 = a†aa|c〉 = (aa† − 1)a|c〉 = a(N − 1)|c〉 = (c− 1)a|c〉

so a|c〉 will have eigenvalue c− 1 for N .Since

|a|c〉|2 = 〈c|a†a|c〉 = 〈c|N |c〉 = c〈c|c〉

if |c〉 has norm 1 then1√ca|c〉 = |c− 1〉

will be normalized.In order to have non-negative eigenvalues for N , the sequence one gets by

repeatedly applying a to a N eigenvector must terminate. The only way for thisto happen is for there to be an eigenvector |0〉 such that

a|0〉 = 0

We cannot have non-integral eigenvalues c since repeatedly applying a wouldultimately lead to states with negative values of c.

Similarly, one can show that

a†|c〉 =√c+ 1|c+ 1〉

so by repeatedly applying a† to |0〉 we get states with eigenvalues 0, 1, 2, · · · forN .

The eigenfunctions of H can thus be found by first solving

a|0〉 = 0

for |0〉 (the lowest energy or “vacuum” state) which will have energy eigenvalue12 , then acting by a† n-times on |0〉 to get states with energy eigenvalue n+ 1

2 .The equation for |0〉 is thus

a|0〉 =1√2

(Q+ iP )ψ0(q) =1√2

(q +d

dq)ψ0(q) = 0

One can check that solutions to this are all of the form

ψ0(q) = Ce−q2

2

211

Choosing C to make this of norm 1 gives

ψ0(q) =1

π14

e−q2

2

and the rest of the energy eigenfunctions can be found by computing

|n〉 =a†√n· · · a

†√

2

a†√1|0〉 =

1

π14 2

n2

√n!

(q − d

dq)ne−

q2

2

In the physical interpretation of this quantum system, the state |n〉, withenergy ~ω(n + 1

2 ) is thought of as a state describing n “quanta”. The state|0〉 is the “vacuum state” with zero quanta, but still carrying a “zero-point”energy of 1

2~ω. The operators a† and a have somewhat similar properties tothe raising and lowering operators we used for SU(2) but their commutatoris different (just the identity operator), leading to simpler behavior. In thiscase they are called “creation” and “annihilation” operators respectively, dueto the way they change the number of quanta. The relation of such quanta tophysical particles like the photon is that quantization of the electromagnetic fieldinvolves quantization of an infinite collection of oscillators, with the quantumof an oscillator corresponding physically to a photon with a specific momentumand polarization. This leads to a well-known problem of how to handle theinfinite vacuum energy corresponding to adding up 1

2~ω for each oscillator.

The first few eigenfunctions are plotted below. The lowest energy eigenstateis a Gaussian centered at q = 0, with a Fourier transform that is also a Gaussiancentered at p = 0. Classically the lowest energy solution is an oscillator at rest atits equilibrium point (q = p = 0), but for a quantum oscillator one cannot havesuch a state with a well-defined position and momentum. Note that the plotgives the wave functions, which in this case are real and can be negative. Thesquare of this function is what has an intepretation as the probability densityfor measuring a given position.

212

19.3 The Bargmann-Fock representation

Working with the operators a and a† and their commutation relation

[a, a†] = 1

makes it clear that there is a simpler way to represent these operators thanthe Schrodinger representation as operators on position space functions that wehave been using, while the Stone-von Neumann theorem assures us that this willbe unitarily equivalent to the Schrodinger representation. This representationappears in the literature under a large number of different names, depending onthe context, all of which refer to the same representation:

Definition (Bargmann-Fock or oscillator or holomorphic or Segal-Shale-Weilrepresentation). The Bargmann-Fock (etc.) representation is given by taking

H = C[w]

(this is the vector space of polynomials of a complex variable w), and operatorsacting on this space

a =d

dw, a† = w

213

The inner product on H is

〈ψ1|ψ2〉 =1

π

∫C

ψ1(w)ψ2(w)e−|w|2

dudv (19.1)

where w = u+ iv.

One has

[a, a†]wn =d

dw(wwn)− w d

dwwn = (n+ 1− n)wn = wn

so this commutator is the identity operator on polynomials

[a, a†] = 1

and

Theorem. The Bargmann-Fock representation has the following properties

• The elementswn√n!

of H for n = 0, 1, 2, . . . are orthornormal.

• The operators a and a† are adjoints with respect to the given inner producton H.

• The basiswn√n!

of H for n = 0, 1, 2, . . . is complete.

• The representation of the Heisenberg algebra on H is irreducible.

Proof. The proofs of the above statements are not difficult, in outline they are

• For orthonormality one can just compute the integrals∫C

wmwne−|w|2

dudv

in polar coordinates.

• To show that w and ddw are adjoint operators, use integration by parts.

• For completeness, assume 〈n|ψ〉 = 0 for all n. The expression for the |n〉as Hermite polynomials times a Gaussian implies that∫

F (q)e−q2

2 ψ(q)dq = 0

214

for all polynomials F (q). Computing the Fourier transform of ψ(q)e−q2

2

gives ∫e−ikqe−

q2

2 ψ(q)dq =

∫ ∞∑j=0

(−ikq)j

j!e−

q2

2 ψ(q)dq = 0

So ψ(q)e−q2

2 has Fourier transform 0 and must be 0 itself. Alternatively,one can invoke the spectral theorem for the self-adjoint operator H, whichguarantees that its eigenvectors form a complete and orthonormal set.

• The operators a and a† take one from any energy eigenstate |n〉 to anyother, so linear combinations in H form an irreducible representation. Bycompleteness there are no states |ψ〉 orthogonal to all the |n〉. Alterna-tively, one can argue that if there was such a |ψ〉, by the argument thatshowed that the eigenvalues of N are natural numbers, the same argu-ment would imply that |ψ〉 would be in a chain of states terminating witha state |0〉′ such that a|0〉′ = 0, with |0〉′ 6= |0〉. But we saw that therewas only one solution for a|0〉 = 0. In general, given a representation ofthe Heisenberg algebra, one can check for irreducibility by seeing if thereis a unique solution to a|0〉 = 0.

Since in this representation the number operator N = a†a satisfies

Nwn = wd

dwwn = nwn

the monomials in w diagonalize the number and energy operators, so one has

|n〉 =wn√n!

for the normalized energy eigenstate of energy ~ω(n+ 12 ).

19.4 Multiple degrees of freedom

Up until now we have been working with the simple case of one physical degree offreedom, i.e. one pair (Q,P ) of position and momentum operators satisfying theHeisenberg relation [Q,P ] = i1, or one pair of adjoint operators a, a† satisfying[a, a†] = 1. We can easily extend this to any number d of degrees of freedom bytaking tensor products of our state space Hd=1, and d copies of our operators,acting on each factor of the tensor product. Our new state space will be

H = Hd=1 ⊗ · · · ⊗ Hd=1︸︷︷︸d times

and we will have operators

Qj , Pj , j = 1, . . . d

215

satisfying[Qj , Pk] = iδjk1, [Qj , Qk] = [Pj , Pk] = 0

where Qj and Pj just act on the j’th term of the tensor product in the usualway.

We can now define annihilation and creation operators in the general case:

Definition (Annihilation and creation operators). The 2d operators

aj =1√2

(Qj + iPj), a†j =1√2

(Qj − iPj), j = 1, . . . , d

are called annihilation (the aj) and creation (the adaggerj) operators.

One can easily check that these satisfy:

Definition (Canonical commutation relations). The canonical commutation re-lations (often abbreviated CCR) are

[aj , a†k] = δjk1, [aj , ak] = [a†j , a

†k] = 0

Using the fact that tensor products of function spaces correspond to func-tions on the product space, in the Schrodinger representation we have

H = L2(Rd)

and in the Bargmann-Fock representation

H = C[w1, . . . , wd]

the polynomials in D complex variables.The harmonic oscillator Hamiltonian for d degrees of freedom will be

H =1

2

d∑j=1

(P 2j +Q2

j ) =

d∑j=1

(a†jaj +1

2)

where one should keep in mind that one can rescale each degree of freedomseparately, allowing different parameters ωj for the different degrees of freedom.The energy and number operator eigenstates will be written

|n1, . . . , nd〉

wherea†jaj |n1, . . . , nd〉 = Nj |n1, . . . , nd〉 = nj |n1, . . . , nd〉

Either the Pj , Qk or the aj , a†k together with the identity operator will give

a representation of the Heisenberg Lie algebra h2d+1 on H, and by exponentia-tion a representation of the Heisenberg group H2d+1. Quadratic combinationsof these operators will give a representation of sp(2d,R), the Lie algebra ofSp(2d,R). In the next two chapters we will study these and other aspects ofthe quantum harmonic oscillator as a unitary representation.

216


All quantum mechanics books should have a similar discussion of the harmonicoscillator, with a good example the detailed one in chapter 7 of Shankar [53].

217

218

Chapter 20

The Harmonic Oscillator asa Representation of theHeisenberg Group

The quantum harmonic oscillator explicitly constructed in the previous chapterprovides new insight into the representation theory of the Heisenberg and meta-plectic groups, using the existence of not just the Schrodinger representationΓS , but the unitarily equivalent Bargmann-Fock version. In this chapter we’llexamine various aspects of the Heisenberg group H2d+1 part of this story thatthe formalism of annihilation and creation operators illuminates, going beyondthe understanding of this representation one gets from the physics of the freeparticle and the Schrodinger representation.

The Schrodinger representation ΓS of H2d+1 uses a specific choice of extrastructure on classical phase space, a decomposition of its coordinates into po-sitions qj and momenta pj . For the unitarily equivalent Bargmann-Fock repre-sentation (which we’ll denote ΓBF ), a different sort of extra structure is needed,a decomposition into complex coordinates and their complex conjugates. Thisis called a “complex structure”: we must first complexify our phase space R2d,then make a specific choice of complex coordinates and their complex conju-gates, corresponding to how we choose to define annihilation and creation op-erators. This choice has physical significance, it is equivalent to a specificationof the lowest energy state |0〉 ∈ H. Equivalently, it corresponds to a choice ofHamiltonian operator H.

The Heisenberg group action on H = C[w1, . . . , wd] given by ΓBF does notcommute with the Hamiltonian H, so it is not what is normally described asa “symmetry”, which would take states to other states with the same energy.While it doesn’t commute with the Hamiltonian, it does have physically inter-esting aspects, since it takes the state |0〉 to a distinguished set of states knownas “coherent states”. These states are labeled by points of the phase space R2d

and provide the closest analog possible in the quantum system of classical states

219

(i.e. those with a well-defined value of position and momentum variables).

20.1 Complex structures and quantization

Quantization of phase space M = R2d using the Schrodinger representationgives a representation Γ′S which takes the qj and pj coordinate functions onphase space to operators Qj and Pj on HS = L2(Rd). To deal with the har-monic oscillator, it is far more convenient to define annihilation and creationoperators aj , a

†j and work with the Bargmann-Fock representation Γ′BF of these

on polynomials HBF = C[w1, . . . , wd]. In the Schrodinger case we needed tomake a choice, that of taking states to be functions of the qj , or of the pj . Adifferent kind of choice needs to be made in the Bargmann-Fock case: certaincomplex linear combinations of the Qj , Pj are taken as annihilation operators,and complex conjugate ones are taken to be creation operators. Once we decidewhich are the annihilation operators, that specifies the state |0〉 as the statethey all annihilate.

To understand what the possible consistent such choices are, we need tobegin by introducing the following definition:

Definition (Complex structure). A complex structure on R2d is a linear oper-ator

J : R2d → R2d

such thatJ2 = −1

The point of this is that, abstractly, to make a real vector space a complexvector space, we need to know how to multiply vectors by complex numbers.We already know how to multiply them by real numbers, so we just need tospecify what happens when we multiply by the imaginary unit i. The choice ofa complex structure J provides exactly that piece of information.

Given such a pair (V = R2d, J), we can now break up complex linear com-binations of vectors in V into those on which J acts as i and those on which itacts as −i (since J2 = −1, its eigenvalues must be ±i). We have

V ⊗C = V +J ⊕ V

−J

where V +J is the +i eigenspace of the operator J on V ⊗C and V −J is the −i

eigenspace. Complex conjugation takes elements of V +J to V −J and vice-versa.

To get the standard identification of R2 (with coordinates x, y) and C (withcoordinate z), one chooses a complex structure J0 such that

J0x = −y, J0y = x

Then, complexifying R2 so that we can take complex linear combinations of xand y, we have

J0(x+ iy) = i(x+ iy), J0(x− iy) = −i(x− iy)

220

andR2 ⊗C = C⊕C

Of the two copies of C we get, one is V +J0

, with coordinate z = x+ iy, the other

is V −J0, with coordinate z = x− iy. The generalization of this standard choice of

J to the case of R2d is straight-forward: using coordinates xj , yj we can define

J0xj = −yj , J0yj = xj

and get complex coordinates zj = xj + iyj on V +J0

= Cd.

The standard definition of annihilation and creation operators aj and a†j ascertain complex linear combinations of the Qj and Pj involves just this sortof choice of an identification using the standard complex structure. We canconstruct a generalization of the Bargmann-Fock representation by choosingany complex structure J on the dual phase space M = M∗, which gives adecomposition

M⊗C =M+J ⊕M

−J

Recall that the Heisenberg Lie algebra is

h2d+1 =M⊕R

with Lie bracket determined by the anti-symmetric bilinear form Ω

[(u, c), (v, d)] = (0,Ω(u, v))

Taking complex linear combinations of h2d+1 elements to get a new Lie algebra,the complexification h2d+1⊗C, we just use the same formula for the Lie bracket,with u, v now complex vectors in M⊗C, and c, d complex numbers.

The choice of a complex structure J on M allows one to write elements ofh2d+1 ⊗C as

(u = u+ + u−, c) ∈ h2d+1 ⊗C = (M⊗C)⊕C =M+J ⊕M

−J ⊕C

whereu+ ∈M+

J , u− ∈M−J

We require that the choice of J be compatible with the anti-symmetric formΩ, satisfying

Ω(Ju, Jv) = Ω(u, v) (20.1)

Then, if u+, v+ ∈M+J , u

−, v− ∈M−J we have

Ω(u+, v+) = Ω(Ju+, Jv+) = Ω(iu+, iv+) = −Ω(u+, v+) = 0

and by the same argumentΩ(u−, v−) = 0

soΩ(u+ + u−, v+ + v−) = Ω(u+, v−) + Ω(u−, v+)

221

These relations imply that if we quantize two elements of M+J they will

commute with each other, and similarly for elements ofM−J . The only non-zerocommutation relations will come from the case of commuting a quantization ofsome u+ ∈M+

J with a quantization of some u− ∈M−J . This is a generalizationof the canonical commutation relations, where under quantization u+ becomesan annihilation operator, u− a creation operator.

To define a generalization of the Bargmann-Fock representation for each J ,we need to find a unitary Lie algebra representation Γ′BF,J of h2d+1 such that,when we extend by complex linearity to linear operators

Γ′BF,J(u, c), (u, c) ∈ (M⊗C)⊕C = h2d+1 ⊗C

we have

Γ′BF,J(u, c) = Γ′BF,J(u+, 0) + Γ′BF,J(u−, 0) + Γ′BF,J(0, c)

with Γ′BF,J(u+, 0) given by an annihilation operator, Γ′BF,J(u−, 0) a creationoperator, and Γ′BF,J(0, c) = −ic1. The first two components can be explicitlyrealized as differentiation and multiplication operators on a state spaceHBF,J ofcomplex polynomials in d variables, the third is just multiplication by a scalar.As a technical aside, an additional positivity condition on J is required hereto make the analog of the standard Bargmann-Fock inner product well-defined,giving a unitary representation.

The standard Bargmann-Fock representation Γ′BF corresponds to a standardchoice of complex structure J0 on M, one such that on basis elements

J0qj = −pj , J0pj = qj

On arbitrary elements ofM one has (treating the d = 1 case, which generalizeseasily)

J0(cqq + cpp) = cpq − cqp

so J0 in matrix form is

J0

(cqcp

)=

(0 1−1 0

)(cqcp

)=

(cp−cq

)It is compatible with Ω since

Ω(J0(cqq + cpp), J0(c′qq + c′pp)) =(

(0 1−1 0

)(cqcp

))T(

0 1−1 0

)(

(0 1−1 0

)(c′qc′p

))

=(cq cp

)(0 −11 0

)(0 1−1 0

)(0 1−1 0

)(c′qc′p

)=(cq cp

)( 0 1−1 0

)(c′qc′p

)=Ω(cqq + cpp, c

′qq + c′pp)

222

Basis elements of M+J0

are then

zj =1√2

(qj + ipj)

since one has

J0zj =1√2

(−pj + iqj) = izj

Basis elements of M−J0are the complex conjugates

zj =1√2

(qj − ipj)

The definition of Poisson brackets extends to functions of complex coordi-nates with the conjugate variable to zj becoming izj since

zj , zk = 1√2

(qj + ipj),1√2

(qj − ipj) = −iδjk

orzj , izk = δjk

These Poisson bracket relations are just the Lie bracket relations on the Liealgebra h2d+1 ⊗C, since as we have seen, a choice of complex structure J0 hasbeen made allowing us to write

h2d+1 ⊗C =M+J0⊕M−J0

⊕C

and the only non-zero Lie brackets in this Lie algebra are the ones betweenelements of M+

J0and M−J0

given by the Poisson bracket relation above.For another form of these Poisson bracket relations, write arbitrary elements

u, v ∈M⊗ C in terms of basis elements zj , zj as

u =∑j

(cjzj + c′jzj), v =∑j

(djzj + d′jzj)

Thenu, v = −i

∑j

(cjd′j − djc′j) = Ω(u, v) (20.2)

Quantization takes

zj → aj , zj → a†j , 1→ 1

and these operators can be realized on the state space C[w1, . . . , wd]. The

aj , a†j are neither self-adjoint nor skew-adjoint, but if we follow our standard

convention of multiplying by −i to get the Lie algebra representation, we have

Γ′BF (zj , 0) = −iaj = −i ∂

∂wj, Γ′BF (zj , 0) = −ia†j = −iwj , Γ′BF (0, 1) = −i1

223

To check that this is a Lie algebra homomorphism, we compute

[Γ′BF (zj , 0),Γ′BF (zk, 0)] =[−iaj ,−ia†k] = −[aj , a†k]

=δjk(−i)(−i1) = δjkΓ′BF (0,−i)

which is correct since, for elements of the complexified Heisenberg algebrah2d+1 ⊗C

[(zj , 0), (zk, 0)] = (0,Ω(zj , zk)) = (0, zj , zk) = (0,−iδjk)

20.2 The Bargmann transform

The Stone von-Neumann theorem implies the existence of

Definition. Bargmann transformThere is a unitary map called the Bargmann transform

B : HBF → HS

between the Bargmann-Fock and Schrodinger representations, with operatorssatisfying the relation

Γ′BF (X) = B−1Γ′S(X)B

for X ∈ h2d+1.

In practice, knowing B explicitly is often not needed, since one can use therepresentation independent relation

aj =1√2

(Qj + iPj)

to express operators either purely in terms of aj and a†j , which have a simpleexpression

aj =∂

∂wj, a†j = wj

in the Bargmann-Fock representation, or purely in terms of Qj and Pj whichhave a simple expression

Qj = qj , Pj = −i ∂∂qj

in the Schrodinger representation.To give an idea of what the Bargmann transform looks like explicitly, we’ll

just give the formula for the d = 1 case here, without proof. If ψ(q) is a statein HS = L2(R), then

(Bψ)(z) =1

π14

∫ +∞

−∞e−z

2− q2

2 +2qzψ(q)dq

224

One can check this equation for the case of the lowest energy state in theSchrodinger representation, where |0〉 has coordinate space representation

ψ(q) =1

π14

e−q2

2

and

Bψ(z) =1

π14

∫ +∞

−∞e−z

2− q2

2 +2qz 1

π14

e−q2

2 dq

=1

π12

∫ +∞

−∞e−z

2−q2+2qzdq

=1

π12

∫ +∞

−∞e−(q−z)2

dq

=1

π12

∫ +∞

−∞e−q

2

dq

=1

20.3 Coherent states and the Heisenberg groupaction

Since the Hamiltonian for the harmonic oscillator does not commute with theoperators aj or a†j which give the representation of the Lie algebra h2d+1 onthe state space HBF , the Heisenberg Lie group and its Lie algebra are notsymmetries of the system in the conventional sense. Energy eigenstates donot break up into irreducible representations of the group but rather the entirestate space makes up such an irreducible representation. The state space for theharmonic oscillator does however have a distinguished state, the lowest energystate |0〉, and one can ask what happens to this state under the Heisenberggroup action. We’ll study this question for the simplest case of d = 1.

Considering the basis of operators for the Lie algebra representation 1, a, a†,we see that the first acts as a constant on |0〉, generating a phase tranformationof the state, while the second annihilates |0〉, so generates group transformationsthat leave the state invariant. It is only the third operator a†, that takes |0〉 todistinct states, and one could consider the family of states

eαa†|0〉

for α ∈ C. The transformations eαa†

are not unitary since αa† is not skew ad-joint. It is better to fix this by replacing αa† with the skew-adjoint combinationαa† − αa, defining

Definition (Coherent states). The coherent states in H are the states

|α〉 = eαa†−αa|0〉

where α ∈ C.

225

Since eαa†−αa is unitary, the |α〉 will be a family of distinct normalized states

in H, with α = 0 corresponding to the lowest energy state |0〉. These are, upto phase transformation, precisely the states one gets by acting on |0〉 witharbitrary elements of the Heisenberg group H3.

Using the Baker-Campbell-Hausdorff formula gives

|α〉 = eαa†−αa|0〉 = eαa

†e−αae−

|α|22 |0〉

and since a|0〉 = 0 one has

|α〉 = e−|α|2

2 eαa†|0〉 = e−

|α|22

∞∑n=0

αn√n!|n〉

Since a|n〉 =√n|n− 1〉 one finds

a|α〉 = e−|α|2

2

∞∑n=0

αn√(n− 1)!

|n− 1〉 = α|α〉

and this property is equivalently used as a definition of coherent states.Note that coherent states are superpositions of different states |n〉, so are

not eigenvectors of the number operator N . They are eigenvectors of

a =1√2

(Q+ iP )

with eigenvalue α so one can try and think of α as a complex number whosereal part gives the position and imaginary part the momentum. This does notlead to a violation of the Heisenberg uncertainly principle since this is not aself-adjoint operator, and thus not an observable. Such states are however veryuseful for describing certain sorts of physical phenomena, for instance the stateof a laser beam, where (for each momentum component of the electromagneticfield) one does not have a definite number of photons, but does have a definiteamplitude and phase.

One thing coherent states do provide is an alternate complete set of normone vectors in H, so any state can be written in terms of them. However, thesestates are not orthogonal (they are eigenvectors of a non-self-adjoint operator sothe spectral theorem for self-adjoint operators does not apply). One can easilycompute that

|〈β|α〉|2 = e−|α−β|2

Digression (Spin coherent states). One can perform a similar constructionreplacing the group H3 by the group SU(2), and the state |0〉 by a highest weightvector of an irreducible representation (πn, V

n = Cn+1) of spin n2 . Writing |n2 〉

for a highest weight vector, we have

π′n(S3)|n2〉 =

n

2, π′n(S+)|n

2〉 = 0

226

and we can create a family of spin coherent states by acting on |n2 〉 by elementsof SU(2). If we identify states in this family that differ just by a phase, thestates are parametrized by a sphere.

By analogy with the Heisenberg group coherent states, with π′n(S+) playingthe role of the annihilation operator a and π′n(S−) playing the role of the creationoperator a†, we can define a skew-adjoint transformation

1

2θeiφπ′n(S−)− 1

2θe−iφπ′n(S+)

and exponentiate to get a family of unitary transformations parametrized by(θ, φ). Acting on the highest weight state we get a definition of the family ofspin coherent states as

|θ, φ〉 = e12 θe

iφπ′n(S−)− 12 θe−iφπ′n(S+)|n

2〉

One can show that the SU(2) group element used here corresponds, in terms ofits action on vectors, to a rotation by an angle θ about the axis (sinφ,− cosφ, 0),so one can associate the state |θ, φ〉 to the unit vector along the z-axis, rotatedby this transformation.


Some references for more details about the Bargmann transform and the Bargmann-Fock representation as a representation of h2d+1 are [19] and [8]. Coherent statesand spin coherent states are discussed in chapter 21 of [53].

227

228

Chapter 21

The Harmonic Oscillatorand the MetaplecticRepresentation, d = 1

In the last chapter we examined those aspects of the harmonic oscillator quan-tum system and the Bargmann-Fock representation that correspond to quan-tization of phase space functions of order less than or equal to one, giving aunitary representation ΓBF of the Heisenberg group H2d+1. We’ll now turnto what happens when one also includes quadratic functions, which will give arepresentation of Mp(2d,R) on the harmonic oscillator state space, extendingΓBF to a representation of the full Jacobi group GJ(d). In this chapter we willsee what happens in some detail for the d = 1 case, where the symplectic groupis just SL(2,R).

A subtle aspect of the Bargmann-Fock construction is that it depends on aspecific choice of complex structure J , a choice which corresponds to a choiceof Hamiltonian operator H and of lowest energy state |0〉, as well as a specificsubgroup U(1) ⊂ SL(2,R). The nature of the double-cover needed for ΓBFto be a true representation (not just a representation up to sign) is best seenby considering the action of this U(1) on the harmonic oscillator state space.The Lie algebra of this U(1) acts on energy eigenstates with an extra 1

2 term,well-known to physicists as the non-zero energy of the vacuum state, and thisshows the need for the double-cover.

21.1 The metaplectic representation for d = 1

In the last chapter we saw that choosing a complex structure on the dual phasespace M = R2d allows one to break up complex linear combinations of theQj , Pj into annihilation and creation operators, giving the Bargmann-Fock rep-resentation Γ′BF of h2d+1 on H = C[w1, . . . , wd]. This then provides a simple

229

description of the quantum harmonic oscillator system in d degrees of freedom.Recall from our discussion of the Schrodinger representation Γ′S in section 16

that we can extend this representation from h2d+1 to include quadratic combi-nations of the qj , pj , getting a unitary representation of the semi-direct producth2d+1 o sp(2d,R). Restricting attention to the sp(2d,R) factor, we get themetaplectic representation, and it is this that we will construct explicitly usingΓ′BF instead of Γ′S in this chapter, beginning in this section with the case d = 1,where sp(2,R) = sl(2,R). Unless otherwise noted, we will be using the stan-dard choice of complex structure to define Γ′BF , later discussing what happensfor other choices of J .

One can readily compute the Poisson brackets of quadratic combinations ofz and z using the basic relation z, z = −i and the Leibniz rule, finding thefollowing for the non-zero cases

zz, z2 = 2iz2, zz, z2 = −2iz2, z2, z2 = 4izz

In the case of the Schrodinger representation, our quadratic combinations of pand q were real, and we could identify the Lie algebra they generated with theLie algebra sl(2,R) of traceless 2 by 2 real matrices with basis

E =

(0 10 0

), F =

(0 01 0

), G =

(1 00 −1

)Since we have complexified, our quadratic combinations of z and z are in the

complexification of sl(2,R), the Lie algebra sl(2,C) of traceless 2 by 2 complexmatrices. We can take as a basis of sl(2,C) over the complex numbers

Z = E − F, X± =1

2(G± i(E + F ))

which satisfy

[Z,X+] = 2iX+, [Z,X−] = −2iX−, [X+, X−] = −iZ

and then use as our isomorphism between quadratics in z, z and sl(2,C)

z2

2↔ X+,

z2

2↔ X−, zz ↔ Z

The element

zz =1

2(q2 + p2)↔ Z =

(0 1−1 0

)exponentiates to give a SO(2) = U(1) subgroup of SL(2,R) with elements ofthe form

eθZ =


)Note that h = 1

2 (p2 + q2) = zz is the classical Hamiltonian function for theharmonic oscillator.

230

We can now quantize quadratics in z and z using annihilation and creationoperators acting on H = C[w]. There is no operator ordering ambiguity for

z2 → a2 =d2

dw2, z2 → (a†)2 = w2

For the case of zz (which is real), in order to get the sl(2,R) commutationrelations to come out right (in particular, the Poisson bracket z2, z2 = 4izz),we must take the symmetric combination

zz → 1

2(aa† + a†a) = a†a+

1

2= w

d

dw+

1

2

(which of course is just the standard Hamiltonian for the quantum harmonicoscillator).

Multiplying as usual by−i one can now define an extension of the Bargmann-Fock representation to an sl(2,C) representation by taking

Γ′BF (X+) = − i2a2, Γ′BF (X−) = − i

2(a†)2, Γ′BF (Z) = −i1

2(a†a+ aa†)

One can check that we have made the right choice of Γ′BF (Z) to get an sl(2,C)representation by computing

[Γ′BF (X+),Γ′BF (X−)] =[− i2a2,− i

2(a†)2] = −1

2(aa† + a†a)

=− iΓ′BF (Z)

As a representation of the real sub-Lie algebra sl(2,R) one has (using thefact that G,E + F,E − F is a real basis of sl(2,R)):

Definition (Metaplectic representation of sl(2,R)). The representation Γ′BFon H = C[w] given by

Γ′BF (G) = Γ′BF (X+ +X−) = − i2

((a†)2 + a2)

Γ′BF (E + F ) = Γ′BF (−i(X+ −X−)) = −1

2((a†)2 − a2)

Γ′BF (E − F ) = Γ′BF (Z) = −i12

(a†a+ aa†)

is a representation of sl(2,R) called the metaplectic representation.

Note that one explicitly see from these expressions that this is a unitaryrepresentation, since all the operators are skew-adjoint (using the fact that aand a† are each other’s adjoints).

This representation Γ′BF will be unitarily equivalent to the Schrodinger ver-sion Γ′S found earlier when quantizing q2, p2, pq as operators on H = L2(R).It is however much easier to work with since it can be studied as the state

231

space of the quantum harmonic oscillator, with the Lie algebra acting simplyby quadratics in the annihilation and creation operators.

One thing that can now easily be seen is that this representation Γ′BF doesnot integrate to give a representation of the group SL(2,R). If the Lie algebrarepresentation Γ′BF comes from a Lie group representation ΓBF of SL(2,R), wehave

ΓBF (eθZ) = eθΓ′BF (Z)

where

Γ′BF (Z) = −i(a†a+1

2) = −i(N +

1

2)

so

π(eθZ)|n〉 = e−iθ(n+ 12 )|n〉

Taking θ = 2π, this gives an inconsistency

ΓBF (1)|n〉 = −|n〉

which has its origin in the physical phenomenon that the energy of the lowestenergy eigenstate |0〉 is 1

2 rather than 0, so not an integer.

This is precisely the same sort of problem we found when studying thespinor representation of the Lie algebra so(3). Just as in that case, the problemindicates that we need to consider not the group SL(2,R), but a double cover,the metaplectic group Mp(2,R). The behavior here is quite a bit more subtlethan in the Spin(3) double cover case, where Spin(3) was just the group SU(2),and topologically the only non-trivial cover of SO(3) was the Spin(3) one sinceπ1(S)(3) = Z2. Here one has π1(SL(2,Z)) = Z, and each extra time one goesaround the U(1) subgroup we are looking at one gets a topologically differentnon-contractible loop in the group. As a result, SL(2,R) has lots of non-trivialcovering groups, of which only one interests us, the double cover Mp(2,R). In

particular, there is an infinite-sheeted universal cover ˜SL(2,R), but that playsno role here.

Digression. This group Mp(2,R) is quite unusual in that it is a finite-dim-ensional Lie group, but does not have any sort of description as a group offinite-dimensional matrices. This is related to the fact that its only interestingirreducible representation is the infinite-dimensional one we are studying. Thelack of any significant irreducible finite-dimensional representations correspondsto its not having a matrix description, which would give such a representation.Note that the lack of a matrix description means that this is a case where thedefinition we gave of a Lie algebra in terms of the matrix exponential does notapply. The more general geometric definition of the Lie algebra of a group interms of the tangent space at the identity of the group does apply, although to dothis one really needs a construction of the double cover Mp(2,R), which is quitenon-trivial. This is not actually a problem for purely Lie algebra calculations,since the Lie algebras of Mp(2,R) and SL(2,R) can be identified.

232

Another aspect of the metaplectic representation that is relatively easy tosee in the Bargmann-Fock construction is that the state space H = C[w] is notan irreducible representation, but is the sum of two irreducible representations

H = Heven ⊕Hodd

where Heven consists of the even-degree polynomials, Hodd the odd degree poly-nomials. Since the generators of the Lie algebra representation are degree twocombinations of annihilation and creation operators, they will not change theparity of a polynomial, taking even to even and odd to odd. The separate irre-ducibility of these two pieces is due to the fact that (when n = m(2)), one canget from state |n〉 to any another |m〉 by repeated application of the Lie algebrarepresentation operators.

21.2 Normal-ordering and the choice of complexstructure

Recall that our construction of the metaplectic representation depends on usinga choice of a basis q, p of M = R2 to get an irreducible unitary representationof the Heisenberg algebra, and the group SL(2,R) acts on M in a way thattakes one such construction to another unitarily equivalent one. Elements ofthis group are given by matrices (

α βγ δ

)satisfying αδ − βγ = 1. The Bargmann-Fock construction depends on a choiceof complex structure J to provide the splitting M⊗ C = M+ ⊕M− into ±ieigenspaces of J (and thus into annihilation and creation operators). We haveso far used the standard complex structure

J0 =

(0 1−1 0

)and one can ask how the SL(2,R) action interacts with this choice. Only asubgroup of SL(2,R) will respect the splitting provided by J0, it will be thesubgroup of matrices commuting with J0. Acting with such matrices on M(and extending by complex linearity to an action on M⊗C), vectors that areJ0 eigenvectors with eigenvalue i will transform into other J0 eigenvectors witheigenvalue i, and similarly for the −i eigenspace. This commutativity conditionis explicitly (

α βγ δ

)(0 1−1 0

)=

(0 1−1 0

)(α βγ δ

)so (

−β α−δ γ

)=

(γ δ−α −β

)233

which implies β = −γ and α = δ. The elements of SL(2,R) that we want willbe of the form (

α β−β α

)with unit determinant, so α2+β2 = 1. This subgroup is the same SO(2) = U(1)subgroup of matrices of the form(

cos θ sin θ− sin θ cos θ

)studied in section 16.3.1.

Recall that here SL(2,R) acts by automorphisms on the Lie algebra h3, andthe infinitesimal version of this action is given by the Poisson brackets betweenthe order-two polynomials of sl(2,R) and the order-one or zero polynomials ofh3. These are

pq, p = p, pq, q = −q

p2

2, q = −p, q

2

2, p = q

As we saw in section 16.3.1, we have here an SO(2) ⊂ SL(2,R) action onM, preserving Ω, and thus a moment map

L ∈ so(2)→ µL =1

2(q2 + p2)

Once we intoduce a complex structure and variables z, z we have

µL = zz

and the Poisson bracket relations above break up into two sorts, the first ofwhich is

zz, z = iz, zz, z = −iz (21.1)

These are examples of the general equation (16.4)

µL, X = L ·X

where now X ∈ M ⊗ C and L acts with eigenvalue i on the M+ which hasbasis z, and with eigenvalue −i on M− which has basis z.

Quantizing µL = zz as − i2 (aa† + a†a) (which we saw in section 21.1 gave a

representation of sl(2,R)), we have the actions on operators

[− i2

(aa† + a†a), a] = ia, [− i2

(aa† + a†a), a†] = −ia† (21.2)

which are the infinitesimal versions of

UθaU−1θ = eiθa, Uθa

†U−1θ = e−iθa† (21.3)

234

whereUθ = e−i

θ2 (aa†+a†a)

We see that, on operators, conjugation by the action of the U(1) subgroupof SL(2,R) does not mix creation and annihilation operators. On harmonic

oscillator states the Uθ act as

Uθ|n〉 = e−i(n+ 12 )θ|n〉

providing a representation of U(1) on the harmonic oscillator state space up toa sign, or equivalently, of a double cover of this U(1).

The second sort of Poisson bracket relation one has are these

z2, z = −2iz, z2, z = 0, z2, z = 2iz, z2, z = 0

which after quantization become

[a2, a†] = 2a, [a2, a] = 0, [(a†)2, a] = −2a†, [(a†)2, a†] = 0

Exponentiating the skew-adjoint operators

(a†)2 + a2, −i((a†)2 − a2)

will give the action of the metaplectic representation in the other two direc-tions than the U(1) direction corresponding to zz. Such transformations actnon-trivially on the chosen complex structure J0, taking it to a different oneJ ′, and giving a new set of annihilation a′ and creation operators (a†)′ thatmixes the original ones. This sort of transformation is known to physicists asa “Bogoliubov transformation” and is useful for describing systems where thelowest-energy state has an indefinite numbe of quanta. The lowest energy statewill be the state |0〉J′ satisfying

a′|0〉J′ = 0

(this determines the state up to phase). The different J ’s (or equivalently, differ-ent possible |0〉J up to phase) will be parametrized by the space SL(2,R)/U(1).This is a space well-known to mathematicians as having a two-dimensional hy-perbolic geometry, with two specific models the Poincare upper half plane modeland the Poincare disk model. Acting with both SL(2,R) and the Heisenberggroup on |0〉 we get a generalization of our construction of coherent states inthe previous chapter, with these new coherent states parametrized not just byan element of M, but also by an element of SL(2,R)/U(1).

Whenever one has a product involving both z and z that one would liketo quantize, the non-trivial commutation relation of a and a† means that onehas different inequivalent possibilities, depending on the order one chooses forthe a and a†. In this case, we chose to quantize zz using the symmetric choice12 (aa† + a†a) = a†a+ 1

2 because it then satisfied the commutation relations forsl(2,R). We could instead have chosen to use a†a, which is an example of a“normal-ordered” product.

235

Definition. Normal-ordered productGiven any product P of the a and a† operators, the normal ordered product

of P , written : P : is given by re-ordering the product so that all factors a† areon the left, all factors a on the right, for example

: a2a†a(a†)3 := (a†)4a3

The advantage of working with the normal-ordered choice a†a instead of12 (aa†+a†a) is that it acts trivially on |0〉 and has integer eigenvalues on all thestates. There is then no need to invoke a double-covering. The disadvantageis that one gets a representation of u(1) that does not extend to sl(2,R). Onealso needs to keep in mind that the definition of normal-ordering depends uponthe choice of J , or equivalently, the choice of lowest energy state |0〉J . A betternotation would be something like : P :J rather than just : P :.

The choice of J is a necessary part of the construction of the Bargmann-Fock representation, and it shows up in the following distinguished features ofthe representation:

• The choice of the space M+J ⊂M⊗C. For the standard choice J0 of J ,

M+J0

will be a copy of C, with basis z = 1√2(q + ip). If we act on M by

an SL(2,R) transformation(qp

)→(q′

p′

)=

(α βγ δ

)(qp

)and use the standard J0 in the new coordinates, then z′ = 1√

2(q′ + ip′)

will be a basis ofM+J . After quantization, we have a different annihilation

operator a′.

• The choice of normal-ordering prescription. SL(2,R) transformationsthat mix annihilation and creation operators change the definition of thenormal ordering symbol : :.

• The choice (up to a scalar factor) of a distinguished state inH, the vacuumstate |0〉J . This is the state annihilated by a′, so in the Schrodingerrepresentation it satisfies

(q′ + ip′)ψ(q) =((αq + βp) + i(γq + δp))ψ(q)

=((α+ iγ)q + (−iβ + δ)d

dq)ψ(q) = 0

with solutionψ(q) ∝ e−

12 (−iβ+δ

α+iγ )q2

• The choice of Hamiltonian, for which |0〉 is the lowest energy state. Witha different J , the new classical hamiltonian will be h = z′z′, the newquantum Hamiltonian

H =1

2(a′(a′)† + (a′)†a′)

236

In each of these cases, if the SL(2,R) transformation commutes with the stan-dard J (and so lies in a U(1) subgroup), one can check that the choice involveddoes not change.


The metaplectic representation is not usually mentioned in the physics litera-ture, and the discussions in the mathematical literature tend to be aimed atan advanced audience. Two good examples of such detailed discussions can befound in [19] and chapters 1 and 11 of [60].

Remarkably, the metaplectic representation plays a significant role in num-ber theory, in which context the fundamental theorem of quadratic reciprocitybecomes a description of the nature of the non-trivial metaplectic cover of thesymplectic group. This is also closely related to the theory of theta functions.This relationship was first studied by Andre Weil in the early sixties [65]. Forsome (rather advanced and technical) expositions of this material, see [7], [35],and [39].

237

238

Chapter 22

The Harmonic Oscillator asa Representation of U(d)

As a representation of Mp(2d,R), ΓBF is unitarily equivalent (by the Bargmanntransform) to ΓS , the Schrodinger representation version studied earlier. TheBargmann-Fock version though makes some aspects of the representation easierto study; in particular it makes clear the nature of the double-cover that appears.A subtle aspect of the Bargmann-Fock construction is that it depends on aspecific choice of complex structure J , a choice which corresponds to a choiceof Hamiltonian operator H and of lowest energy state |0〉. This same choicepicks out a subgroup U(d) ⊂ Sp(2d,R) of transformations which commutewith J , and the harmonic oscillator state space gives one a representation ofa double-cover of this group. We will see that a re-ordering (called “normal-ordering”) of the products of annihilation and creation operators turns thisinto a representation of U(d) itself. In this way, a U(d) action on the finite-dimensional phase space gives intertwining operators that provide an infinitedimensional representation of U(d) on the harmonic oscillator state space H.This method for turning symmetries of the classical phase space into unitaryrepresentations of the symmetry group on a quantum state space is elaboratedin great detail here not just because of its application to these simple quantumsystems, but because it will turn out to be fundamental in our later study ofquantum field theories.

We will se in detail how in the case d = 2 the harmonic oscillator state spaceprovides a construction of all SU(2) ⊂ U(2) irreducible representations. Thecase d = 3 corresponds to the physical example of a quadratic central potentialin three dimensions, with the rotation group acting on the state space as asubgroup SO(3) ⊂ U(3).

239

22.1 The metaplectic representation for d de-grees of freedom

Generalizing to the case of d degrees of freedom, we can consider the polynomialsof order two in coordinates qj , pj on phase space M = R2d. We saw in chapter13.3 that these are a basis for the Lie algebra sp(2d,R), using the Poissonbracket as Lie bracket. The quantization of qj and pj as operators Qj , Pj inthe Schrodinger representation gives the representation Γ′S of this Lie algebraon H = L2(Rd). To get the Bargmann-Fock representation Γ′BF , recall that weneed to introduce annihilation and creation operators, which can be done withthe standard choice

aj =1√2

(Qj + iPj), a†j =1√2

(Qj − iPj), j = 1, . . . d

and then realize these operators as differentiation and multiplication operators

aj =∂

∂wj, a†j = wj

on the state space

H = C[w1, · · · , wd]

of polynomials in d complex variables. The Bargmann-Fock inner product (seeequation 19.1) can easily be generalized from the d = 1 case to this one.

Here we are quantizing the dual phase space M = R2d by choosing thestandard complex structure J on R2d and using it to decompose

M⊗C =M+ ⊕M−

where zj , j = 1, · · · , d are basis elements inM+ while zj , j = 1, · · · , d are basiselements inM−. The choice of J gives a decomposition of the complexified Liealgebra sp(2d,C) into three sub-algebras as follows:

• A Lie subalgebra with basis elements zjzk (as usual, the Lie bracket is thePoisson bracket). There are 1

2 (d2 +d) distinct such basis elements. This isa commutative Lie subalgebra, since the Poisson bracket of any two basiselements is zero.

• A Lie subalgebra with basis elements zjzk. Again, it has dimension 12 (d2+

d) and is a commutative Lie subalgebra.

• A Lie subalgebra with basis elements zjzk, which has dimension d2. Com-puting Poisson brackets one finds

zjzk, zlzm =zjzk, zlzm+ zkzj , zlzm=− izjzmδkl + izlzkδjm (22.1)

240

This last subalgebra is the one we will mostly be interested in since we willsee that quantization of elements of this subalgebra produces the operators ofmost interest in the applications we will examine. Taking all complex linearcombinations, this subalgebra can be identified with the Lie algebra gl(d,C) ofall d by d complex matrices, since if Ejk is the matrix with 1 at the j-th rowand k-th column, zeros elsewhere, one has

[Ejk, Elm] = Ejmδkl − Elkδjm

and these provide a basis of gl(d,C). Identifying bases by

izjzk ↔ Ejk

gives the isomorphism of Lie algebras. This gl(d,C) is the complexification ofu(d), the Lie algebra of the unitary group U(d). Elements of u(d) will corre-spond to skew-adjoint matrices so real linear combinations of the real quadraticfunctions on M.

zjzk + zjzk, i(zjzk − zjzk)

The moment map here is

A ∈ u(d)→ µA = i∑j,k

zjAjkzk

and we have

Theorem 22.1. One has the Poisson bracket relation

µA, µA′ = µ[A,A′]

so the moment map is a Lie algebra homomorphism.

One also has (for column vectors z with components z1, . . . , zd)

µA, z = −Az, µA, z = AT z (22.2)

Proof. Using 22.1 one has

µA, µA′ = −∑j,k,l,m

zjAjkzk, zlA′lmzm

= −∑j,k,l,m

AjkA′lmzjzk, zlzm

= i∑j,k,l,m

AjkA′lm(zjzmδkl − zlzkδjm)

= i∑j,k

zj [A,A′]jkzk = µ[A,A′]

241

To show 22.2, compute

µA, zl =i∑j,k

zjAjkzk, zl = i∑j,k

Ajkzj , zlzk

=−∑k

Alkzk

and

µA, zl =i∑j,k

zjAjkzk, zl = i∑j,k

zjAjkzk, zl

=∑j

zjAjl

Note that in chapter 13 we found the moment map µL = −q·Ap for elementsL ∈ sp(2d,R) of the block-diagonal form(

A 00 −AT

)where A is a real d by d matrix and so in gl(d,R). That block decompositioncorresponded to the decomposition of M into q and p. Here we have com-plexified, and are working with respect to a different decomposition, that ofM⊗C = M+ ⊕M−. The A are complex, skew-adjoint, and in a different Liesubalgebra, u(d) ⊂ sp(2d,R).

The standard Hamiltonian

h =

d∑j=1

zjzj

lies in this sub-algebra (it is the case A = −i1), and one can show that itsPoisson brackets with the rest of the sub-algebra are zero. It gives a basiselement of the one-dimensional u(1) subalgebra that commutes with the rest ofthe u(d) subalgebra.

In section 21.2 for the case d = 1 we found that there was a U(1) ⊂ SL(2,R)group acting on M preserving Ω, and commuting with J . Complexifying Mthis U(1) acted separately on M+ and M−, and there was a moment map µLgiving the function zz on M . Here we have a U(d) ⊂ Sp(2d,R), again actingon M preserving Ω, and commuting with J , so also acting separately on M+

and M− after complexification.Turning to the quantization problem, for any j, k one has

zjzk → −iajak, zjzk → −ia†ja†k

so there is no ambiguity in the quantization of the two subalgebras given bypairs of the z coordinates or pairs of the z coordinates. If j 6= k one can take

zjzk → −iaja†k = −ia†kaj

242

and there is again no ordering ambiguity. The definition of normal orderinggeneralizes simply, since the order of annihilation and creation operators withdifferent values of j is immaterial. If j = k, as in the d = 1 case there is a choiceto be made. One possibility is to take

zjzj → −i1

2(aja

†j + a†jaj) = −i(a†jaj +

1

2)

which will have the proper sp(2d,R) commutation relations (in particular for

commutators of a2j with (a†j)

2), but require going to a double cover to get a truerepresentation of the group. The other is to use the normal-ordered version

zjzj → −ia†jaj

If one does this, one has shifted the usual quantized operators of the Bargmann-Fock representation by a scalar 1

2 , and after exponentiation the state space Hprovides a representation of U(d), with no need for a double cover. As a u(d)representation however, this does not extend to a representation of sp(2d,C),

since commutation of a2j with (a†j)

2 can land one on the unshifted operators.We saw above that the infinitesimal action of u(d) ⊂ sp(2d,R) preserves the

decomposition ofM⊗C =M+⊕M−, and this will be true after exponentiatingfor U(d) ⊂ Sp(2d,R). We won’t show this here, but U(d) is the maximal sub-group that preserves this decomposition. The analog of the d = 1 parametriza-tion of possible |0〉J by SL(2,R)/U(1) here would be a parametrization of suchstates (or, equivalently, of possible choices of J) by the space Sp(2d,R)/U(d),known as the “Siegel upper half-space”.

Since the normal-ordering doesn’t change the commutation relations obeyedby products of a†j , ak, one can quantize the quadratic expression for µA and get

quadratic combinations of the aj , a†k with the same commutation relations as in

theorem 22.1. Letting

U ′A =∑j,k

a†jAjkak

then

Theorem 22.2. For A a skew-adjoint d by d matrix one has

[U ′A, U′A′ ] = U[A,A′]

SoA ∈ u(d)→ U ′A

is a Lie algebra representation of u(d) on H = C[w1, . . . , wd], the harmonicoscillator state space in d degrees of freedom.

One also has (for column vectors a with components a1, . . . , ad)

[U ′A,a] = −Aa, [U ′A,a†] = ATa† (22.3)

Proof. Essentially the same proof as 22.1.

243

The Lie algebra representation U ′A of u(d) exponentiates to give a represen-tation of U(d) on H = C[w1, . . . , wd] by operators

UeA = eU′A

These satisfy

UeAa(UeA)−1 = e−Aa, UeAa†(UeA)−1 = eAT

a† (22.4)

(the relations 22.3 are the derivative of these). This shows that the UeA areintertwining operators for a U(d) action on annihilation and creation operatorsthat preserves the canonical commutation relations (the relations that say the

aj , a†j give a representation of the complexified Heisenberg Lie algebra). Here

the use of normal-ordered operators means that U ′A is a representation of u(d)that differs by a constant from the metaplectic representation, and UeA differsby a phase-factor. This does not affect the commutation relations with U ′A orthe conjugation action of UeA . Recall that the representation one gets this waydiffers in two ways from the metaplectic representation. It acts on the samespace H, but it is a true representation of U(d), no double-cover is needed. Italso does not extend to a representation of the larger group Sp(2d,R).

The operators U ′A and UeA commute with the Hamiltonian operator. Fromthe physics point of view, this is useful, as it provides a decomposition of en-ergy eigenstates into irreducible representations of U(d). From the mathematicspoint of view, the quantum harmonic oscillator state space provides a construc-tion of a large class of irreducible representations of U(d) (the energy eigenstatesof a given energy).

22.2 Examples in d = 2 and 3

22.2.1 Two degrees of freedom and SU(2)

In the case d = 2, we will re-encounter our earlier construction of SU(2) repre-sentations in terms of homogeneous polynomials, in a new context. This use ofthe energy eigenstates of a two-dimensional harmonic oscillator appears in thephysics literature as the Schwinger boson method for studying representationsof SU(2).

The state space for the d = 2 Bargmann-Fock representation is

H = C[w1, w2]

the polynomials in two complex variables w1, w2. Recall from our SU(2) dis-cussion that it was useful to organize these polynomials into finite dimensionalsets of homogeneous polynomials of degree n for n = 0, 1, 2, . . .

H = H0 ⊕H1 ⊕H2 ⊕ · · ·

There are four annihilation or creation operators

a1 =∂

∂w1, a†1 = w1, a2 =

∂

∂w2, a†2 = w2

244

acting on H. These are the quantizations of complexified phase space coordi-nates z1, z2, z1, z2 with quantization just the Bargmann-Fock Lie algebra repre-sentation of the Lie algebra these span

Γ′BF (1) = −i1, Γ′BF (zj) = −iaj , Γ′BF (zj) = −ia†j

Our original dual phase space was M = R4, with a group Sp(4,R) actingon it, preserving the Poisson bracket. When picking the coordinates z1, z2, wehave made a standard choice of complex structure J , which breaks up as

M⊗C =M+ ⊕M− = C2 ⊕C2

where z1, z2 are coordinates onM+, z1, z2 are coordinates onM−. This choiceof J picks out a distinguished subgroup U(2) ⊂ Sp(4,R).

The quadratic combinations of the creation and annihilation operators giverepresentations on H of three subalgebras of the complexification sp(4,C) ofsp(4,R):

• A three dimensional commutative Lie sub-algebra spanned by z1z2, z21 , z

22 ,

with quantization

Γ′BF (z1z2) = −ia1a2, Γ′BF ((z21) = −ia2

1, Γ′BF (z22) = −ia2

2

• A three dimensional commutative Lie sub-algebra spanned by z1z2, z21, z

22,

with quantization

Γ′BF (z1z2) = −ia†1a†2, Γ′BF (z2

1) = −i(a†1)2, Γ′BF (z22) = −i(a†2)2

• A four dimensional Lie subalgebra isomorphic to gl(2,C) with basis

z1z1, z2z2, z1z2, z1z2

and quantization

Γ′BF (z1z1) = − i2

(a†1a1 + a1a†1), Γ′BF (z2z2) = − i

2(a†2a2 + a2a

†2)

Γ′BF (z1z2) = −ia†1a2, Γ′BF (z1z2) = −ia†2a1

Real linear combinations of

z1z1, z2z2, z1z2 + z2z1, i(z1z2 − z2z1)

span the Lie algebra u(2) ⊂ sp(4,R), and Γ′BF applied to these gives aunitary Lie algebra representation by skew-adjoint operators.

Inside this last subalgebra, there is a distinguished element h = z1z1 + z2z2

that Poisson-commutes with the rest of the subalgebra, which after quantizationgives the quantum Hamiltonian

H =1

2(a1a

†1 + a†1a1 + a2a

†2 + a†2a2) = N1 +

1

2+N2 +

1

2= w1

∂

∂w1+w2

∂

∂w2+ 1

245

This operator will just multiply a homogeneous polynomial by its degree plusone, so it acts just by multiplication by n + 1 on Hn. Exponentiating thisoperator (multiplied by −i) one gets a representation of a U(1) subgroup of themetaplectic cover Mp(4,R). Taking instead the normal-ordered version

: H := a†1a1 + a†2a2 = N1 +N2 = w1∂

∂w1+ w2

∂

∂w2

one gets a representation of the U(1) ⊂ Sp(4,R) that acts with opposite weighton M+ and M−. Neither H nor : H : commutes with operators coming fromquantization of the first two subalgebras. These change the eigenvalue of H or: H : by ±2 so take

Hn → Hn±2

in particular taking |0〉 to either 0 or a state in H2.h is a basis element for the u(1) in u(2) = u(1)⊕ su(2). For the su(2) part a

correspondence to our basis Xj = −iσj2 in terms of 2 by 2 matrices is

X1 ↔1

2(z1z2 + z2z1), X2 ↔

i

2(z2z1 − z1z2), X3 ↔

1

2(z1z1 − z2z2)

This relates two different but isomorphic ways of describing su(2): as 2 by 2matrices with Lie bracket the commutator, or as quadratic polynomials, withLie bracket the Poisson bracket.

Quantizing using normal-ordering of operators give a representation of su(2)on H

Γ′(X1) = − i2

(a†2a1 + a†1a2), Γ′(X2) =1

2(a†2a1 − a†1a2)

Γ′(X3) = − i2

(a†1a1 − a†2a2)

Note that another way to get this result is by the theorem of the last section,computing the quadratic operators as(

a†1 a†2)Xj

(a1

a2

)If one compares this to the representation π′ of su(2) discussed in chapter 8,

one finds thatΓ′(X) = π′(−XT )

so these are the same representations, up to the fact that that (w1, w2) variablesof the polynomials in the state space here transform under SU(2) in a dual man-ner to the variables (z1, z2) used in chapter 8. We see that, up to this changefrom a C2 to its dual, and the normal-ordering (which only affects the u(1) fac-tor, shifting the Hamiltonian by a constant), the Bargmann-Fock representationon polynomials and the SU(2) representation on homogeneous polynomials areclosely related. Unlike the representation on homogeneous polynomials though,the Bargmann-Fock representation does extend as a unitary representation toa larger group (H5 oMp(4,R)).

246

One can show that the inner product on the space of polynomials in twovariables studied in one of the problem sets is exactly the Bargmann-Fock innerproduct, restricted to homogeneous polynomials.

The fact that we have an SU(2) group acting on the state space of the d = 2harmonic oscillator and commuting with the action of the Hamiltonian H meansthat energy eigenstates can be organized as irreducible representations of SU(2).In particular, one sees that the space Hn of energy eigenstates of energy n+ 1will be a single irreducible SU(2) representation, the spin n

2 representation ofdimension n+ 1 (so n+ 1 will be the multiplicity of energy eigenstates of thatenergy).

22.2.2 Three degrees of freedom and SO(3)

The case d = 3 corresponds physically to the so-called isotropic quantumharmonic oscillator system, and it is an example of the sort of central po-tential problem we have already studied (since the potential just depends onr2 = q2

1 + q22 + q2

3). For such problems, we saw that since the classical Hamilto-nian is rotationally invariant, the quantum Hamiltonian will commute with theaction of SO(3) on wavefunctions and energy eigenstates can be decomposedinto irreducible representations of SO(3).

Here the Bargmann-Fock representation gives an action of H7oMp(6,R) onthe state space, with a U(3) subgroup commuting with the Hamiltonian (moreprecisely one has a double cover of U(3), but by normal-ordering one can get anactual U(3)). The eigenvalue of the U(1) corresponding to the Hamiltonian givesthe energy of a state, and states of a given energy will be sums of irreduciblerepresentations of SU(3). This works much like in the d = 2 case, although hereour irreducible representations are the spaces Hn of homogeneous polynomialsof degree n in three variables rather than two. These spaces have dimension12 (n + 1)(n + 2). A difference with the SU(2) case is that one does not get allirreducible representations of SU(3) this way.

The rotation group SO(3) will be a subgroup of this U(3) and one can askhow the SU(3) irreducible Hn decomposes into a sum of irreducibles of thesubgroup (which will be characterized by l = 0, 1, 2, · · · ). One can show thatfor even n we get all even values of l from 0 to n, and for odd n we get all oddvalues of l from 1 to n. A derivation can be found in many quantum mechanicstextbooks, see for example pgs. 351-2 of [53].

The angular momentum operators L1, L2, L3 will be examples of the quadraticcombinations of annihilation and creation operators discussed in the generalcase. For example, one can simply calculate

L3 =Q1P2 −Q2P1

=− i

2((a1 + a†1)(a2 − a†2)− (a2 + a†2)(a1 − a†1))

=− i

2(a†1a2 − a1a

†2 − a

†2a1 + a2a

†1)

=i(a†2a1 − a†1a2)

247

One could also use the fact that in the matrix representation of the Lie alge-bra so(3) acting on vectors, infinitesimal rotations are given by antisymmetricmatrices, and use our theorem from earlier in this section to find the quadraticoperator corresponding to a given antisymmetric matrix. For instance, for in-finitesimal rotations about the 3 axis,

l3 =

0 −1 01 0 00 0 0

and our theorem says this will be represented on H by∑

j,k

a†j(l3)jkak = −a†1a2 + a†2a1

which is just, as expected, −i times the self-adjoint operator L3 found above.


The references from chapter 21([19], [60]) also contain the general case discussedhere. The construction of the intertwining operators U(d) ⊂ Sp(2d,R) usingannihilation and creation operators is a standard topic quantum field theorytextbooks, although there in an infinite rather than finite-dimensional context(and not explicitly in the language used here). We will encounter the quantumfield theory version in later chapters.

248

Chapter 23

The Fermionic Oscillator

The quantum state spaces describing fundamental physical systems that occurin the real world often are “fermionic”: they have a property of anti-symmetryunder the exchange of two states. It turns out that the entire quantum me-chanical formalism we have developed so far has a parallel version in which suchanti-symmetry is built in from the beginning. Besides providing further exam-ples of the connections between representation theory and quantum mechanics,this new formalism encompasses in a surprising way some mathematics of greatimportance in geometry.

In this chapter we’ll consider in detail the basic example of how this works,as a simple variation on techniques we used to study the harmonic oscillator,leaving for later chapters a discussion of the new mathematics and new generalformalism embodied in the example.

23.1 Canonical commutation relations and thebosonic oscillator

Recall that the Hamiltonian for the quantum harmonic oscillator system in ddegrees of freedom (setting ~ = m = ω = 1) is

H =

d∑j=1

1

2(Q2

j + P 2j )

and that it can be diagonalized by introducing operators

aj =1√2

(Qj + iPj), a†j =1√2

(Qj − iPj)

that satisfy the so-called canonical commutation relations (CCR)

[aj , a†k] = δjk1, [aj , ak] = [a†j , a

†k] = 0

249

This is done by noting that the operators

Nj = a†jaj

have as eigenvalues the natural numbers

nj = 0, 1, 2, . . .

and that

H =1

2

d∑j=1

(a†jaj + aja†j) =

d∑j=1

(Nj +1

2)

The eigenvectors of H can be labeled by the nj and one has

H|n1, n2, · · · , nd〉 = En1,n2,··· ,nd |n1, n2, · · · , nd〉

where the energy eigenvalues are given by

En1,n2,··· ,nd =

d∑j=1

(nj +1

2)

The lowest energy eigenstate will be |0, 0, · · · , 0〉, with eigenvalue d2 .

Putting back in factors of ~, ω and noting that one can choose ω to bea different number ωj for each degree of freedom, one finds that the energyeigenvalues are

En1,n2,··· ,nd =

d∑j=1

(nj +1

2)~ωj

We will see later in this course that collections of identical particles can bedescribed using this formalism, when states are symmetric under interchangeof two particles (and are then called “bosons”). The state |n1, n2, · · · , nN 〉 isinterpreted as describing n1 bosons of one momentum, n2 of another momentum,etc. The lowest energy state |0, 0, · · · , 0〉 describes the “vacuum” state with noparticles, but it carries an energy

d∑j=1

1

2~ωj

Because of this ability to describe bosonic particles, we’ll often call the harmonicoscillator the “bosonic” oscillator.

23.2 Canonical anti-commutation relations andthe fermionic oscillator

The simple change in the harmonic oscillator problem that takes one from bosonsto fermions is the replacement of the bosonic annihilation and creation operators

250

(which we’ll now denote aB and aB†) by fermionic annihilation and creation

operators called aF and aF†, which satisfy a variant of the bosonic commutation

relations[aF , a

†F ]+ = 1, [aF , aF ]+ = 0, [a†F , a

†F ]+ = 0

The change from the bosonic case is just the replacement of the commutator

[A,B] ≡ AB −BA

of operators by the anti-commutator

[A,B]+ ≡ AB +BA

The fermionic number operator

NF = a†FaF

now satisfies

N2F = a†FaFa

†FaF = a†F (1− a†FaF )aF = NF − a†F

2a2F = NF

(using the fact that a2F = a†F

2= 0). So one has

N2F −NF = NF (NF − 1) = 0

which implies that the eigenvalues of NF are just 0 and 1. We’ll denote eigen-vectors with such eigenvalues by |0〉 and |1〉. The simplest representation of the

operators aF and a†F on a complex vector space HF will be on C2, and choosingthe basis

|0〉 =

(01

), |1〉 =

(10

)the operators are represented as

aF =

(0 01 0

), a†F =

(0 10 0

), NF =

(1 00 0

)Since

H =1

2(a†FaF + aFa

†F )

is just 12 the identity operator, to get a non-trivial quantum system, instead we

make a sign change and set

H =1

2(a†FaF − aFa

†F ) = NF −

1

21 =

(12 00 − 1

2

)The energies of the energy eigenstates |0〉 and |1〉 will then be ± 1

2 since

H|0〉 = −1

2|0〉, H|1〉 =

1

2|1〉

251

Note that the quantum system we have constructed here is nothing but ourold friend the qubit. Taking complex linear combinations of the operators

aF , a†F , NF ,1

we get all linear transformations of HF = C2 (so this is an irreducible represen-tation of the algebra of these operators). The relation to the Pauli matrices isjust

a†F =1

2(σ1 + iσ2), aF =

1

2(σ1 − iσ2), H =

1

2σ3

23.3 Multiple degrees of freedom

For the case of d degrees of freedom, one has this variant of the canonicalcommutation relations (CCR) amongst the bosonic annihilation and creation

operators aBj and aB†j :

Definition (Canonical anti-commutation relations). A set of 2d operators aF j , aF†j , j =

1, . . . , d is said to satisfy the canonical anti-commutation relations (CAR) whenone has

[aF j , aF†k]+ = δjk1, [aF j , aF k]+ = 0, [aF

†j , aF

†k]+ = 0

In this case the state space is the tensor product of N copies of the singlefermionic oscillator state space

HF = (C2)⊗d = C2 ⊗C2 ⊗ · · · ⊗C2︸︷︷︸d times

The dimension of HF in this case will be 2d. On this space the operators aF jand aF

†j can be explicitly given by

aF j = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3︸︷︷︸j−1 times

⊗(

0 01 0

)⊗ 1⊗ · · · ⊗ 1

aF†j = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3︸︷︷︸

j−1 times

⊗(

0 10 0

)⊗ 1⊗ · · · ⊗ 1

The factors of σ3 ensure that the canonical anti-commutation relations

[aF j , aF k]+ = [aF†j , aF

†k]+ = [aF j , aF

†k]+ = 0

are satisfied for j 6= k since in these cases one will get a factor in the tensorproduct of

[σ3,

(0 01 0

)]+ = 0 or [σ3,

(0 10 0

)]+ = 0

252

While this sort of tensor product construction is useful for discussing thephysics of multiple qubits, in general it is easier to not work with large tensorproducts, and the Clifford algebra formalism we will describe in the next chapteravoids this.

The number operators will be

NF j = aF†jaF j

These will commute with each other, so can be simultaneously diagonalized,with eigenvalues nj = 0, 1. One can take as an orthonormal basis of HF the 2d

states

|n1, n2, · · · , nd〉

As an example, for the case d = 3 the pattern of states and their energylevels for the bosonic and fermionic cases looks like this

In the bosonic case the lowest energy state is at positive energy and there arean infinite number of states of ever increasing energy. In the fermionic case the

253

lowest energy state is at negative energy, with the pattern of energy eigenvaluesof the finite number of states symmetric about the zero energy level.

No longer setting ~ = ω = 1 and allowing different ω for different degrees offreedom, the Hamiltonian will be

H =1

2

d∑j=1

~ωj(aF †jaF j − aF jaF†j)

and one can write the energy eigenstates as

|n1, n2, · · · , nd〉

for nj = 0, 1. They will have energy eigenvalues

En1,n2,··· ,nd =

d∑j=1

(nj −1

2)~ωj


Most quantum field theory books and a few quantum mechanics books containsome sort of discussion of the fermionic oscillator, see for example Chapter 21.3of [53] or Chapter 5 of [10]. The standard discussion often starts with consideringa form of classical analog using anti-commuting “fermionic” variables and thenquantizes to get the fermionic oscillator. Here we are doing things in the oppositeorder, starting in this chapter with the quantized oscillator, then consideringthe classical analog in a later chapter.

254

Chapter 24

Weyl and Clifford Algebras

We have seen that just changing commutators to anti-commutators takes theharmonic oscillator quantum system to a very different one (the fermionic os-cillator), with this new system having in many ways a parallel structure. Itturns out that this parallelism goes much deeper, with every aspect of the har-monic oscillator story having a fermionic analog. We’ll begin in this chapter bystudying the operators of the corresponding quantum systems.

24.1 The Complex Weyl and Clifford algebras

In mathematics, a “ring” is a set with addition and multiplication laws that areassociative and distributive (but not necessarily commutative), and an “algebra”is a ring that is also a vector space over some field of scalars. The canonicalcommutation and anti-commutation relations define interesting algebras, calledthe Weyl and Clifford algebras respectively. The case of complex numbers asscalars is simplest, so we’ll start with that, before moving on to the real numbercase.

24.1.1 One degree of freedom, bosonic case

Starting with the one degree of freedom case (corresponding to two operatorsQ,P , which is why the notation will have a 2) we can define

Definition (Complex Weyl algebra, one degree of freedom). The complex Weylalgebra in the one degree of freedom case is the algebra Weyl(2,C) generated by

the elements 1, aB , a†B, satisfying the canonical commutation relations:

[aB , a†B ] = 1, [aB , aB ] = [a†B , a

†B ] = 0

In other words, Weyl(2,C) is the algebra one gets by taking arbitrary prod-ucts and complex linear combinations of the generators. By repeated use of thecommutation relation

aBa†B = 1 + a†BaB

255

any element of this algebra can be written as a sum of elements in normal order,of the form

cl,m(a†B)lamB

with all annihilation operators aB on the right, for some complex constants cl,m.As a vector space over C, Weyl(2,C) is infinite-dimensional, with a basis

1, aB , a†B , a

2B , a

†BaB , (a†B)2, a3

B , a†Ba

2B , (a†B)2aB , (a†B)3, . . .

This algebra is isomorphic to a more familiar one. Setting

a†B = w, aB =∂

∂w

one sees that Weyl(2,C) can be identified with the algebra of polynomial coeffi-cient differential operators on functions of a complex variable w. As a complexvector space, the algebra is infinite dimensional, with a basis of elements

wl∂m

∂wm

In our study of quantization and the harmonic oscillator we saw that thesubset of such operators consisting of linear combinations of

1, w,∂

∂w, w2,

∂2

∂w2, w

∂

∂w

is closed under commutators, so it forms a Lie algebra of dimension 6. This Liealgebra includes as subalgebras the Heisenberg Lie algebra (first three elements)and the Lie algebra of SL(2,R) (last three elements). Note that here we areallowing complex linear combinations, so we are getting the complexification ofthe real six-dimensional Lie algebra that appeared in our study of quantization.

Since the aB and a†B are defined in terms of P and Q, one could of coursealso define the Weyl algebra as the one generated by 1, P,Q, with the Heisenbergcommutation relations, taking complex linear combinations of all products ofthese operators.

24.1.2 One degree of freedom, fermionic case

Changing commutators to anti-commutators, one gets a different algebra, theClifford algebra

Definition (Complex Clifford algebra, one degree of freedom). The complexClifford algebra in the one degree of freedom case is the algebra Cliff(2,C) gen-

erated by the elements 1, aF , a†F , subject to the canonical anti-commutation re-

lations

[aF , a†F ]+ = 1, [aF , aF ]+ = [a†F , a

†F ]+ = 0

256

This algebra is a four dimensional algebra over C, with basis

1, aF , a†F , a†FaF

since higher powers of the operators vanish, and one can use the anti-commutationrelation betwee aF and a†F to normal order and put factors of aF on the right.We saw in the last chapter that this algebra is isomorphic with the algebraM(2,C) of 2 by 2 complex matrices, using

1↔(

1 00 1

), aF ↔

(0 01 0

), a†F ↔

(0 10 0

), a†FaF ↔

(1 00 0

)We will see later on that there is also a way of identifying this algebra with“differential operators in fermionic variables”, analogous to what happens inthe bosonic (Weyl algebra) case.

Recall that the bosonic annihilation and creation operators were originallydefined in term of the P and Q operators by

aB =1√2

(Q+ iP ), a†B =1√2

(Q− iP )

Looking for the fermionic analogs of the operators Q and P , we use a slightlydifferent normalization, and set

aF =1

2(γ1 + iγ2), a†F =

1

2(γ1 − iγ2)

so

γ1 = aF + a†F , γ2 =1

i(aF − a†F )

and the CAR imply that the operators γj satisfy the anti-commutation relations

[γ1, γ1]+ = [aF + a†F , aF + a†F ]+ = 2

[γ2, γ2]+ = −[aF − a†F , aF − a†F ]+ = 2

[γ1, γ2]+ =1

i[aF + a†F , aF − a

†F ]+ = 0

From this we see that

• One could alternatively have defined Cliff(2,C) as the algebra generatedby 1, γ1, γ2, subject to the relations

[γj , γk]+ = 2δjk

• Using just the generators 1 and γ1, one gets an algebra Cliff(1,C), gener-ated by 1, γ1, with the relation

γ21 = 1

This is a two-dimensional complex algebra, isomorphic to C⊕C.

257

24.1.3 Multiple degrees of freedom

For a larger number of degrees of freedom, one can generalize the above anddefine Weyl and Clifford algebras as follows:

Definition (Complex Weyl algebras). The complex Weyl algebra for d degrees

of freedom is the algebra Weyl(2d,C) generated by the elements 1, aBj , aB†j,

j = 1, . . . , d satisfying the CCR

[aBj , aB†k] = δjk1, [aBj , aBk] = [aB

†j , aB

†k] = 0

Weyl(2d,C) can be identified with the algebra of polynomial coefficient dif-ferential operators in m complex variables w1, w2, . . . , wd. The subspace ofcomplex linear combinations of the elements

1, wj ,∂

∂wj, wjwk,

∂2

∂wj∂wk, wj

∂

∂wk

is closed under commutators and is isomorphic to the complexification of the Liealgebra h2d+1 o sp(2d,R) built out of the Heisenberg Lie algebra in 2d variablesand the Lie algebra of the symplectic group Sp(2d,R). Recall that this is theLie algebra of polynomials of degree at most 2 on the phase space R2d, with thePoisson bracket as Lie bracket.

One could also define the complex Weyl algebra by taking complex linearcombinations of products of generators 1, Pj , Qj , subject to the Heisenberg com-mutation relations.

For Clifford algebras one has

Definition (Complex Clifford algebras, using annilhilation and creation op-erators). The complex Clifford algebra for d degrees of freedom is the algebra

Cliff(2d,C) generated by 1, aF j , aF†j for j = 1, 2, . . . , d satisfying the CAR

[aF j , aF†k]+ = δjk1, [aF j , aF k]+ = [aF

†j , aF

†k]+ = 0

or, alternatively, one has the following more general definition that also worksin the odd-dimensional case

Definition (Complex Clifford algebras). The complex Clifford algebra in n vari-ables is the algebra Cliff(n,C) generated by 1, γj for j = 1, 2, . . . , n satisfyingthe relations


We won’t try and prove this here, but one can show that, abstractly asalgebras, the complex Clifford algebras are something well-known. Generalizingthe case d = 1 where we saw that Cliff(2,C) was isomorphic to the algebra of 2by 2 complex matrices, one has isomorphisms

Cliff(2d,C)↔M(2d,C)

258

in the even-dimensional case, and in the odd-dimensional case

Cliff(2d+ 1,C)↔M(2d,C)⊕M(2d,C)

Two properties of Cliff(n,C) are

• As a vector space over C, a basis of Cliff(n,C) is the set of elements

1, γj , γjγk, γjγkγl, . . . , γ1γ2γ3 · · · γn−1γn

for indices j, k, l, · · · ∈ 1, 2, . . . , n, with j < k < l < · · · . To show this,consider all products of the generators, and use the commutation relationsfor the γj to identify any such product with an element of this basis. Therelation γ2

j = 1 shows that one can remove repeated occurrences of aγj . The relation γjγk = −γkγj can then be used to put elements of theproduct in the order of a basis element as above.

• As a vector space over C, Cliff(n,C) has dimension 2n. One way to seethis is to consider the product

(1 + γ1)(1 + γ2) · · · (1 + γn)

which will have 2n terms that are exactly those of the basis listed above.

24.2 Real Clifford algebras

We can define real Clifford algebras Cliff(n,R) just as for the complex case, bytaking only real linear combinations:

Definition (Real Clifford algebras). The real Clifford algebra in n variables isthe algebra Cliff(n,R) generated over the real numbers by 1, γj for j = 1, 2, . . . , nsatisfying the relations


For reasons that will be explained in the next chapter, it turns out that amore general definition is useful. We write the number of variables as n = r+s,for r, s non-negative integers, and now vary not just r + s, but also r − s, theso-called “signature”.

Definition (Real Clifford algebras, arbitrary signature). The real Clifford al-gebra in n = r + s variables is the algebra Cliff(r, s,R) over the real numbersgenerated by 1, γj for j = 1, 2, . . . , n satisfying the relations

[γj , γk]+ = ±2δjk1

where we choose the + sign when j = k = 1, . . . , r and the − sign when j = k =r + 1, . . . , n.

259

In other words, as in the complex case different γj anti-commute, but onlythe first r of them satisfy γ2

j = 1, with the other s of them satisfying γ2j = −1.

Working out some of the low-dimensional examples, one finds:

• Cliff(0, 1,R). This has generators 1 and γ1, satisfying

γ21 = −1

Taking real linear combinations of these two generators, the algebra onegets is just the algebra C of complex numbers, with γ1 playing the role ofi =√−1.

• Cliff(0, 2,R). This has generators 1, γ1, γ2 and a basis

1, γ1, γ2, γ1γ2

with

γ21 = −1, γ2

2 = −1, (γ1γ2)2 = γ1γ2γ1γ2 = −γ21γ

22 = −1

This four-dimensional algebra over the real numbers can be identified withthe algebra H of quaternions by taking

γ1 ↔ i, γ2 ↔ j, γ1γ2 ↔ k

• Cliff(1, 1,R). This is the algebra M(2,R) of 2 by 2 matrices, with onepossible identification as follows

1↔(

1 00 1

), γ1 ↔

(0 11 0

), γ2 ↔

(0 −11 0

), γ1γ2 ↔

(−1 00 1

)Note that one can construct this using the aF , a

†F for the complex case

Cliff(2,C)

γ1 = aF + a†F , γ2 = aF − a†Fsince these are represented as real matrices.

It turns out that Cliff(r, s,R) is always one or two copies of matrices of real,complex or quaternionic elements, of dimension a power of 2, but this requiresa rather intricate algebraic argument that we will not enter into here. For thedetails of this and the resulting pattern of algebras one gets, see for instance[34]. One special case where the pattern is relatively simple is when one hasr = s. Then n = 2r is even-dimensional and one finds

Cliff(r, r,R) = M(2r,R)

We will see in the next chapter that just as quadratic elements of the Weylalgebra give a basis of the Lie algebra of the symplectic group, quadratic ele-ments of the Clifford algebra give a basis of the Lie algebra of the orthogonalgroup.

260


A good source for more details about Clifford algebras and spinors is Chapter12 of the representation theory textbook [60]. For the details of what happensfor all Cliff(r, s,R), another good source is Chapter 1 of [34].

261

262

Chapter 25

Clifford Algebras andGeometry

The definitions given in last chapter of Weyl and Clifford algebras were purelyalgebraic, based on a choice of generators. These definitions do though have amore geometrical formulation, with the definition in terms of generators corre-sponding to a specific choice of coordinates. For the Weyl algebra, the geometryinvolved is known as symplectic geometry, and we have already seen that in thebosonic case quantization of a phase space R2d depends on the choice of anon-degenerate anti-symmetric bilinear form Ω which determines the Poissonbrackets and thus the Heisenberg commutation relations. Such a Ω also deter-mines a group Sp(2d,R), which is the group of linear transformations of R2d

preserving Ω. The Clifford algebra also has a coordinate invariant definition,based on a more well-known structure on a vector space Rn, that of a non-degenerate symmetric bilinear form, i.e. an inner product. In this case thegroup that preserves the inner product is an orthogonal group. In the symplec-tic case anti-symmetric forms require an even number of dimensions, but this isnot true for symmetric forms, which also exist in odd dimensions.

25.1 Non-degenerate bilinear forms

In the case of M = R2d, the dual phase space, for two vectors u, u′ ∈M

u = cq1q1 + cp1p1 + · · ·+ cqdqd + cpdpd ∈M

u′ = c′q1q1 + c′p1p1 + · · ·+ c′qdqd + c′pdpd ∈M

the Poisson bracket determines an antisymmetric bilinear form on M, givenexplicitly by

263

Ω(u, u′) =cq1c′p1− cp1

c′q1 + · · · cqdc′pd − cpdc′qd

=(cq1 cp1

. . . cqd cpd)

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

c′q1c′p1

...c′qdc′pd

Matrices g ∈M(2d,R) such that

gT

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

g =

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

make up the group Sp(2d,R) and preserve Ω, satisfying

Ω(gu, gu′) = Ω(u, u′)

This choice of Ω is much less arbitrary than it looks. One can show thatgiven any non-degenerate anti-symmetric bilinear form on R2d a basis can befound with respect to which it will be the Ω given here (for a proof, see [8]).This is also true if one complexifies, taking (q,p) ∈ C2d and using the sameformula for Ω, which is now a bilinear form on C2d. In the real case the groupthat preserves Ω is called Sp(2d,R), in the complex case Sp(2d,C).

To get a fermionic analog of this, it turns out that all we need to do is replace“non-degenerate anti-symmetric bilinear form Ω(·, ·)” with “non-degenerate sym-metric bilinear form 〈·, ·〉”. Such a symmetric bilinear form is actually somethingmuch more familiar from geometry than the anti-symmetric case analog: it isjust a notion of inner product. Two things are different in the symmetric case:

• The underlying vector space does not have to be even dimensional, onecan take M = Rn for any n, including n odd. To get a detailed analog ofthe bosonic case though, we will mostly consider the even case n = 2d.

• For a given dimension n, there is not just one possible choice of 〈·, ·〉 up tochange of basis, but one possible choice for each pair of integers r, s suchthat r + s = n. Given r, s, any choice of 〈·, ·〉 can be put in the form

〈u,u′〉 =u1u′1 + u2u

′2 + · · ·uru′r − ur+1u

′r+1 − · · · − unu′n

=(u1 . . . un

)

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸

r + signs, s - signs

u′1u′2...

u′n−1

u′n

264

For a proof by Gram-Schmidt orthogonalization, see [8].

We can thus extend our definition of the orthogonal group as the group oftransformations g preserving an inner product

〈gu, gu′〉 = 〈u, u′〉

to the case r, s arbitrary by

Definition (Orthogonal group O(r, s,R)). The group O(r, s,R) is the group ofreal r + s by r + s matrices g that satisfy

gT

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸


g =

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸


SO(r, s,R) ⊂ O(r, s,R) is the subgroup of matrices of determinant +1.

If we complexify and take components of vectors in Cn, using the sameformula for 〈·, ·〉, one can change basis by multiplying the s basis elements bya factor of i, and in this new basis all basis vectors ej satisfy 〈ej , ej〉 = 1. Onefinds that on Cn, as in the symplectic case, up to change of basis there is onlyone non-degenerate bilinear form. The group preserving this is called O(n,C).Note that on Cn 〈·, ·〉 is not the Hermitian inner product (which is anti-linearon the first variable), and it is not positive definite.

25.2 Clifford algebras and geometry

As defined by generators in the last chapter, Clifford algebras have no obviousgeometrical significance. It turns out however that they are powerful tools inthe study of the geometry of linear spaces with an inner product, includingespecially the study of linear transformations that preserve the inner product,i.e. rotations. To see the relation between Clifford algebras and geometry,consider first the positive definite case Cliff(n,R). To an arbitrary vector

v = (v1, v2, . . . , vn) ∈ Rn

we associate the Clifford algebra element v = γ(v) where γ is the map

v ∈ Rn → γ(v) = v1γ1 + v2γ2 + · · ·+ vnγn

Using the Clifford algebra relations for the γj , given two vectors v, w theproduct of their associated Clifford algebra elements satisfies

vw + wv = [v1γ1 + v2γ2 + · · ·+ vnγn, w1γ1 + w2γ2 + · · ·+ wnγn]+

= 2(v1w1 + v2w2 + · · ·+ vnwn)

= 2〈v,w〉

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where 〈·, ·〉 is the symmetric bilinear form on Rn corresponding to the standardinner product of vectors. Note that taking v = w one has

v2 = 〈v,v〉 = ||v||2

The Clifford algebra Cliff(n,R) contains Rn as the subspace of linear combi-nations of the generators γj , and one can think of it as a sort of enhancement ofthe vector space Rn that encodes information about the inner product. In thislarger structure one can multiply as well as add vectors, with the multiplicationdetermined by the inner product.

In general one can define a Clifford algebra whenever one has a vector spacewith a symmetric bilinear form:

Definition (Clifford algebra of a symmetric bilinear form). Given a vector spaceV with a symmetric bilinear form 〈·, ·〉, the Clifford algebra Cliff(V, 〈·, ·〉) is thealgebra generated by 1 and elements of V , with the relations

vw + wv = 2〈v, w〉

Note that different people use different conventions here, with

vw + wv = −2〈v, w〉

another common choice. One also sees variants without the factor of 2.For n dimensional vector spaces over C, we have seen that for any non-

degenerate symmetric bilinear form a basis can be found such that 〈·, ·〉 has thestandard form

〈z,w〉 = z1w1 + z2w2 + · · ·+ znwn

As a result, there is just one complex Clifford algebra in dimension n, the onewe defined as Cliff(n,C).

For n dimensional vector spaces over R with a non-degenerate symmetricbilinear forms of type r, s such that r+s = n, the corresponding Clifford algebrasCliff(r, s,R) are the ones defined in terms of generators in the last chapter.

In special relativity, space-time is a real 4-dimensional vector space with anindefinite inner product corresponding to (depending on one’s choice of conven-tion) either the case r = 1, s = 3 or the case s = 1, r = 3. The group of lineartransformations preserving this inner product is called the Lorentz group, andits orientation preserving component is written as SO(3, 1) or SO(1, 3) depend-ing on the choice of convention. Later on in the class we will consider whathappens to quantum mechanics in the relativistic case, and there encounter thecorresponding Clifford algebras Cliff(3, 1,R) or Cliff(1, 3,R). The generatorsγj of such a Clifford algebra are well-known in the subject as the “Dirac γ-matrices”.

For now though, we will restrict attention to the positive definite case, sojust will be considering Cliff(n,R) and seeing how it is used to study the groupO(n) of n-dimensional rotations in Rn.

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25.2.1 Rotations as iterated orthogonal reflections

We’ll consider two different ways of seeing the relationship between the Cliffordalgebra Cliff(n,R) and the group O(n) of rotations in Rn. The first is basedupon the geometrical fact (known as the Cartan-Dieudonne theorem) that onecan get any rotation by doing multiple orthogonal reflections in different hy-perplanes. Orthogonal reflection in the hyperplane perpendicular to a vector wtakes a vector v to the vector

v′ = v − 2〈v,w〉〈w,w〉

w

something that can easily be seen from the following picture

Identifying vectors v,v′,w with the corresponding Clifford algebra elements,the reflection in w transformation is

v → v′ =v − 2〈v,w〉〈w,w〉

w

=v − (vw + wv)w

〈w,w〉

but since

ww

〈w,w〉=〈w,w〉〈w,w〉

= 1

we have (for non-zero vectors w)

w−1 =w

〈w,w〉

and the reflection transformation is just conjugation by w times a minus sign

v → v′ = v − v − wvw−1 = −wvw−1

So, thinking of vectors as lying in the Clifford algebra, the orthogonal trans-formation that is the result of one reflection is just a conjugation (with a minus

267

sign). These lie in the group O(n), but not in the subgroup SO(n), since theychange orientation. The result of two reflections in different hyperplanes or-thogonal to w1,w2 will be

v → v′ = −w2(−w1vw−11 )w−1

2 = (w2w1)v(w2w1)−1

This will be a rotation preserving orientation, so of determinant one and in thegroup SO(n).

This construction not only gives an efficient way of representing rotations(as conjugations in the Clifford algebra), but it also provides a construction ofthe group Spin(n) in arbitrary dimension n. The spin group is generated froman even number of unit vectors in Rn by taking products of the correspondingelements of the Clifford algebra. This construction generalizes to arbitrary nthe one we gave in chapter 6 of Spin(3) in terms of unit length elements of thequaternion algebra H. One can see the characteristic fact that there are twoelements of the Spin(n) group giving the same rotation in SO(n) by noticingthat changing the sign of the Clifford algebra element does not change theconjugation action, where signs cancel.

25.2.2 The Lie algebra of the rotation group and quadraticelements of the Clifford algebra

For a second approach to understanding rotations in arbitrary dimension, onecan use the fact that these are generated by taking products of rotations in thecoordinate planes. A rotation by an angle θ in the j−k coordinate plane (j < k)will be given by

v→ eθεjkv

where εjk is an n by n matrix with only two non-zero entries: jk entry −1 andkj entry +1 (see equation 5.2.1). Restricting attention to the j − k plane, eθεjk

acts as the standard rotation matrix in the plane(vjvk

)→(


)(vjvk

)In the SO(3) case we saw that there were three of these matrices

l1 = ε23, l2 = ε13, l3 = ε12

providing a basis of the Lie algebra so(3). In n dimensions there will be 12 (n2−n)

of them, providing a basis of the Lie algebra so(n).Just as in the case of SO(3) where unit length quaternions were used, we can

use elements of the Clifford algebra to get these same rotation transformations,but as conjugations in the Clifford algebra. To see how this works, consider thequadratic Clifford algebra element γjγk for j 6= k and notice that

(γjγk)2 = γjγkγjγk = −γjγjγkγk = −1

268

so one has

eθ2 γjγk =(1− (θ/2)2

2!+ · · · ) + γjγk(θ/2− (θ/2)3

3!+ · · · )

= cos(θ

2) + γjγk sin(

θ

2)

Conjugating a vector vjγj+vkγk in the j−k plane by this, one finds (workingout the details here will be a problem on the problem set)

e−θ2 γjγk(vjγj + vkγk)e

θ2 γjγk = (vj cos θ − vk sin θ)γj + (vj sin θ + vk cos θ)γk

which is just a rotation by θ in the j − k plane. Such a conjugation will alsoleave invariant the γl for l 6= j, k. Thus one has

e−θ2 γjγkγ(v)e

θ2 γjγk = γ(eθεjkv) (25.1)

and the infinitesimal version

[−1

2γjγk, γ(v)] = γ(εjkv) (25.2)

Note that these relations are closely analogous to equations 16.6 and 16.5, whichin the symplectic case show that a rotation in the Q,P plane is given by con-jugation by the exponential of an operator quadratic in Q,P . We will examinethis analogy in greater detail in chapter 27.

One can also see that, just as in our earlier calculations in three dimensions,one gets a double cover of the group of rotations, with here the elements e

θ2 γjγk

of the Clifford algebra giving a double cover of the group of rotations in the j−kplane. General elements of the spin group can be constructed by multiplyingthese for different angles in different coordinate planes. One sees that the Liealgebra spin(n) can be identified with the Lie algebra so(n) by

εjk ↔ −1

2γjγk

Yet another way to see this would be to compute the commutators of the − 12γjγk

for different values of j, k and show that they satisfy the same commutationrelations as the corresponding matrices εjk.

Recall that in the bosonic case we found that quadratic combinations of theQj , Pk (or of the aBj , aB

†j) gave operators satisfying the commutation relations

of the Lie algebra sp(2n,R). This is the Lie algebra of the group Sp(2n,R),the group preserving the non-degenerate anti-symmetric bilinear form Ω(·, ·) onthe phase space R2n. The fermionic case is precisely analogous, with the roleof the anti-symmetric bilinear form Ω(·, ·) replaced by the symmetric bilinearform 〈·, ·〉 and the Lie algebra sp(2n,R) replaced by so(n) = spin(n).

In the bosonic case the linear functions of the Qj , Pj satisfied the commuta-tion relations of another Lie algebra, the Heisenberg algebra, but in the fermioniccase this is not true for the γj . In the next chapter we will see that one candefine a notion of a “Lie superalgebra” that restores the parallelism.

269


Some more detail about the relationship between geometry and Clifford algebrascan be found in [34], and an exhaustive reference is [45].

270

Chapter 26

Anti-commuting Variablesand Pseudo-classicalMechanics

The analogy between the algebras of operators in the bosonic (Weyl algebra) andfermionic (Clifford algebra) cases can be extended by introducing a fermionicanalog of phase space and the Poisson bracket. This gives a fermionic ana-log of classical mechanics, sometimes called “pseudo-classical mechanics”, thequantization of which gives the Clifford algebra as operators, and spinors asstate spaces. In this chapter we’ll intoduce “anti-commuting variables” ξj thatwill be the fermionic analogs of the variables qj , pj . These objects will becomegenerators of the Clifford algebra under quantization, and will later be used inthe construction of fermionic state spaces, by analogy with the Schrodinger andBargmann-Fock constructions in the bosonic case.

26.1 The Grassmann algebra of polynomials onanti-commuting generators

Given a phase space V = Rn, one gets classical observables by taking polynomialfunctions on V . These are generated by the linear functions qj , pj , j = 1, . . . , dfor n = 2d, which lie in the dual space V ∗. It is also possible to start with thesame space V ∗ of linear functions on V (with n not necessarily even), and picka different notion of multiplication, one that is anti-commutative on elementsof V ∗. Using such a multiplication, we can generate an analog of the algebra ofpolynomials on V , sometimes called the Grassmann algebra:

Definition (Grassmann algebra). The algebra over the real numbers generatedby ξj , j = 1, . . . , n, satisfying the relations

ξjξk + ξkξj = 0

271

is called the Grassmann algebra, and denoted Λ∗(Rn).

Note that these relations imply that generators satisfy ξ2j = 0. Also note that

sometimes the product in the Grassmann algebra is called the “wedge product”and the product of ξj and ξk is denoted ξj ∧ ξk. We will not use a differentsymbol for the product in the Grassmann algebra, relying on the notation forgenerators to keep straight what is a generator of a conventional polynomialalgebra (e.g. qj or pj) and what is a generator of a Grassman algebra (e.g. ξj).

Recall that the algebra of polynomial functions on V could be thought of asS∗(V ∗), the symmetric part of the tensor algebra on V ∗, with multiplication oftwo linear functions l1, l2 in V ∗ given by

l1l2 =1

2(l1 ⊗ l2 + l2 ⊗ l1) ∈ S2(V ∗)

Similarly, the Grassman algebra on V is the anti-symmetric part of the tensoralgebra on V ∗, with the wedge product of two linear functions l1, l2 in V ∗ givenby

l1 ∧ l2 =1

2(l1 ⊗ l2 − l2 ⊗ l1) ∈ Λ2(V ∗)

The Grassmann algebra behaves in many ways like the polynomial algebraon Rn, but it is finite dimensional, with basis

1, ξj , ξjξk, ξjξkξl, · · · , ξ1ξ2 · · · ξn

for indices j < k < l < · · · taking values 1, 2, . . . , n. As with polynomials,monomials are characterized by a degree (number of generators in the product),which takes values from 0 to n. Λk(Rn) is the subspace of Λ∗(Rn) of linearcombinations of monomials of degree k.

Digression (Differential forms). The Grassmann algebra is also known as theexterior algebra, and we will examine its relationship to anti-symmetric elementsof the tensor algebra later on. Readers may have already seen the exterior al-gebra in the context of differential forms on Rn. These are known to physicistsas “anti-symmetric tensor fields”, and given by taking elements of the exte-rior algebra Λ∗(Rn) with coefficients not constants, but functions on Rn. Thisconstruction is important in the theory of manifolds, where at a point x in amanifold M , one has a tangent space TxM and its dual space (TxM)∗. A set oflocal coordinates xj on M gives basis elements of (TxM)∗ denoted by dxj anddifferential forms locally can be written as sums of terms of the form

f(x1, x2, · · · , xn)dxj ∧ · · · ∧ dxk ∧ · · · ∧ dxl

where the indices j, k, l satisfy 1 ≤ j < k < l ≤ n.

A fundamental principle of mathematics is that a good way to understanda space is in terms of the functions on it. One can think of what we have donehere as creating a new kind of space out of Rn, where the algebra of functions

272

on the space is Λ∗(Rn), generated by coordinate functions ξj with respect to abasis of Rn. The enlargement of conventional geometry to include new kindsof spaces such that this makes sense is known as “supergeometry”, but we willnot attempt to pursue this subject here. Spaces with this new kind of geometryhave functions on them, but do not have conventional points: one can’t askwhat the value of an anti-commuting function at a point is.

Remarkably, one can do calculus on such unconventional spaces, introducinganalogs of the derivative and integral for anti-commuting functions. For the casen = 1, an arbitrary function is

F (ξ) = c0 + c1ξ

and one can take∂

∂ξF = c1

For larger values of n, an arbitrary function can be written as

F (ξ1, ξ2, . . . , ξn) = FA + ξjFB

where FA, FB are functions that do not depend on the chosen ξj (one gets FB byusing the anti-commutation relations to move ξj all the way to the left). Thenone can define

∂

∂ξjF = FB

This derivative operator has many of the same properties as the conventionalderivative, although there are unconventional signs one must keep track of. Anunusual property of this derivative that is easy to see is that one has

∂

∂ξj

∂

∂ξj= 0

Taking the derivative of a product one finds this version of the Leibniz rulefor monomials F and G

∂

∂ξj(FG) = (

∂

∂ξjF )G+ (−1)|F |F (

∂

∂ξjG)

where |F | is the degree of the monomial F .A notion of integration (often called the “Berezin integral”) with many of the

usual properties of an integral can also be defined. It has the peculiar featureof being the same operation as differentiation, defined in the n = 1 case by∫

(c0 + c1ξ)dξ = c1

and for larger n by∫F (ξ1, ξ2, · · · , ξn)dξ1dξ2 · · · dξn =

∂

∂ξn

∂

∂ξn−1· · · ∂

∂ξ1F = cn

273

where cn is the coefficient of the basis element ξ1ξ2 · · · ξn in the expression of Fin terms of basis elements.

This notion of integration is a linear operator on functions, and it satisfiesan analog of integration by parts, since if one has

F =∂

∂ξjG

then ∫Fdξj =

∂

∂ξjF =

∂

∂ξj

∂

∂ξjG = 0

using the fact that repeated derivatives give zero.

26.2 Pseudo-classical mechanics and the fermionicPoisson bracket

The basic structure of Hamiltonian classical mechanics depends on an evendimensional phase space R2d with a Poisson bracket ·, · on functions on thisspace. Time evolution of a function f on phase space is determined by

d

dtf = f, h

for some Hamiltonian function h. This says that taking the derivative of anyfunction in the direction of the velocity vector of a classical trajectory is thelinear map

f → f, h

on functions. As we saw in chapter 12, since this linear map is a derivative, thePoisson bracket will have the derivation property, satisfying the Leibniz rule

f1, f2f3 = f2f1, f3+ f1, f2f3

for arbitrary functions f1, f2, f3 on phase space. Using the Leibniz rule andanti-symmetry, one can calculate Poisson brackets for any polynomials, justfrom knowing the Poisson bracket on generators qj , pj (or, equivalently, theanti-symmetric bilinear form Ω(·, ·)), which we chose to be

qj , qk = pj , pk = 0, qj , pk = −pk, qj = δjk

Notice that we have a symmetric multiplication on generators, while the Poissonbracket is anti-symmetric.

To get pseudo-classical mechanics, we think of the Grassmann algebra Λ∗(Rn)as our algebra of classical observables, an algebra we can think of as functionson a “fermionic” phase space Rn (note that in the fermionic case, the phasespace does not need to be even dimensional). We want to find an appropriatenotion of fermionic Poisson bracket operation on this algebra, and it turns out

274

that this can be done. While the standard Poisson bracket is an antisymmetricbilinear form Ω(·, ·) on linear functions, the fermionic Poisson bracket will bebased on a choice of symmetric bilinear form on linear functions, equivalently,a notion of inner product 〈·, ·〉.

Denoting the fermionic Poisson bracket by ·, ·+, for a multiplication anti-commutative on generators one has to adjust signs in the Leibniz rule, and thederivation property analogous to the derivation property of the usual Poissonbracket is

F1F2, F3+ = F1F2, F3+ + (−1)|F2||F3|F1, F3+F2

where |F2| and |F3| are the degrees of F2 and F3. It will also have the symmetryproperty

F1, F2+ = −(−1)|F1||F2|F2, F1+and one can use these properties to compute the fermionic Poisson bracket forarbitrary functions in terms of the relation for generators.

One can think of the ξj as the “coordinate functions” with respect to abasis ei of Rn, and we have seen that the symmetric bilinear forms on Rn areclassified by a choice of positive signs for some basis vectors, negative signs forthe others. So, on generators ξj one can choose

ξj , ξk+ = ±δjk

with a plus sign for j = k = 1, · · · , r and a minus sign for j = k = r+ 1, · · · , n,corresponding to the possible inequivalent choices of non-degenerate symmetricbilinear forms.

Taking the case of a positive-definite inner product for simplicity, one cancalculate explicitly the fermionic Poisson brackets for linear and quadratic com-binations of the generators. One finds

ξjξk, ξl+ = ξjξk, ξl+ − ξj , ξl+ξk = δjkξl − δjlξk

and

ξjξk, ξlξm+ =ξjξk, ξl+ξm + ξlξjξk, ξm+=δklξjξm − δjlξkξm + δkmξlξj − δjmξlξk

The second of these equations shows that the quadratic combinations of thegenerators ξj satisfy the relations of the Lie algebra of the group of rotations inn dimensions (so(n) = spin(n)). The first shows that the ξkξl acts on the ξj asinfinitesimal rotations in the k − l plane.

In the case of the conventional Poisson bracket, the anti-symmetry of thebracket and the fact that it satisfies the Jacobi identity implies that it is aLie bracket determining a Lie algebra (the infinite dimensional Lie algebra offunctions on a phase space R2d). The fermionic Poisson bracket provides anexample of something called a Lie superalgebra. These can be defined for vectorspaces with some usual and some fermionic coordinates:

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Definition (Lie superalgebra). A Lie superalgebra structure on a real or com-plex vector space V is given by a Lie superbracket [·, ·]. This is a bilinear mapon V which on generators X,Y, Z (which may be usual coordinates or fermionicones) satisfies

[X,Y ] = −(−1)|X||Y |[Y,X]

and a super-Jacobi identity

[X, [Y,Z]] = [[X,Y ], Z] + (−1)|X||Y |[Y, [X,Z]]

where |X| takes value 0 for a usual coordinate, 1 for a fermionic coordinate.

As in the bosonic case, on functions of generators of order less than or equalto two, the Fermionic Poisson bracket , ·, ·+ is a Lie superbracket, giving aLie superalgebra of dimension 1 + n+ 1

2 (n2 − n) (since there is one constant, nlinear terms ξj and 1

2 (n2 − n) quadratic terms ξjξk).We will see in chapter 27 that quantization will give a distinguished rep-

resentation of this Lie superalgebra, in a manner quite parallel to that of theSchrodinger or Bargmann-Fock constructions of a representation in the bosoniccase.

The relation between between the quadratic and linear polynomials in thegenerators is parallet to what happens in the bosonic case. Here we have thefermionic analog of the bosonic theorem 13.1:

Theorem 26.1. The Lie algebra so(n,R) is isomorphic to the Lie algebraΛ2(M∗) (with Lie bracket ·, ·+) of order two anti-commuting polynmials onM = Rn, by the isomorphism

L↔ µL

where L ∈ so(n,R) is an antisymmetric n by n real matrix, and

µL =1

2ξ · Lξ =

1

2

∑j,k

Ljkξjξk

The so(n,R) action on anti-commuting coordinate functions is

µL, ξk+ =∑j

Ljkξj

orµL, ξ+ = LT ξ

Proof. On can first prove the second part of the theorem by computing

1

2

∑j,k

ξjLjkξk, ξl+ =1

2

∑j,k

Ljk(ξjξk, ξl+ − ξj , ξl+ξk)

=1

2(∑j

Ljlξj −∑k

Llkξk)

=∑j

Ljlξj (since L = −LT )

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For the first part of the theorem, the map

L→ µL

is a vector space isomorphism of the space of anti-symmetric matrices andΛ2(Rn). To show that it is a Lie algebra isomorphism, one can use an analogousargument to that of the proof of 13.1. Here one considers the action

ξ → µL, ξ

of µL ∈ so(n,R) on an arbitrary

ξ =∑j

cjξj

and uses the super-Jacobi identity relating the fermionic Poisson brackets ofµL, µL′ , ξ.

26.3 Examples of pseudo-classical mechanics

In pseudo-classical mechanics, the dynamics will be determined by choosing ahamiltonian h in Λ∗(Rn). Observables will be other functions F ∈ Λ∗(Rn), andthey will satisfy the analog of Hamilton’s equations

d

dtF = F, h+

We’ll consider two of the simplest possible examples.

26.3.1 The classical spin degree of freedom

Using pseudo-classical mechanics, one can find a “classical” analog of somethingthat is quintessentially quantum: the degree of freedom that appears in the qubitor spin 1/2 system that we have seen repeatedly in this course. Taking R3 withthe standard inner product as fermionic phase space, we have three generatorsξ1, ξ2, ξ3 satisfying the relations

ξj , ξk+ = δjk

and an 8 dimensional space of functions with basis

1, ξ1, ξ2, ξ3, ξ1ξ2, ξ1ξ3, ξ2ξ3, ξ1ξ2ξ3

If we want the Hamiltonian function to be non-trivial and of even degree, it willhave to be a linear combination

h = B12ξ1ξ2 +B13ξ1ξ3 +B23ξ2ξ3

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The equations of motion on generators will be

d

dtξj(t) = ξj , h+

but since h is a quadratic combination of the generators, recall from the earlierdiscussion of quadratic combinations of the ξj that the right hand side is justan infinitesimal rotation. So the solution to the classical equations of motionwill be a time-dependent rotation of the ξj in the plane perpendicular to

B = (B23,−B13, B12)

at a constant speed proportional to |B|.

26.3.2 The classical fermionic oscillator

We have already studied the fermionic oscillator as a quantum system, and onecan ask whether there is a corresponding pseudo-classical system. Such a systemis given by taking an even dimensional fermionic phase space R2d, with a basisof coordinate functions ξ1, · · · ξ2d that generate Λ∗(R2d). On generators thefermionic Poisson bracket relations come from the standard choice of positivedefinite symmetric bilinear form

ξj , ξk+ = δjk

As discussed earlier, for this choice the quadratic products ξjξk act on the gen-erators by infinitesimal rotations in the j−k plane, and satisfy the commutationrelations of so(2d).

To get a classical system that will quantize to the fermionic oscillator onemakes the choice

h = i

d∑j=1

ξ2j−1ξj

As in the bosonic case, we can make the standard choice of complex structureJ on R2d and get a decomposition

R2d ⊗C = Cd ⊕Cd

into eigenspaces of J of eigenvalue ±i. This is done by defining

θj =1√2

(ξ2j−1 + iξ2j), θj =1√2

(ξ2j−1 − iξ2j)

for j = 1, . . . , d. These satisfy the fermionic Poisson bracket relations

θj , θk+ = θj , θk+ = 0, θj , θk+ = δjk

In the bosonic case we could use any complex structure J that satisfiedΩ(Ju, Ju′) = Ω(u, u′). For the Fermionic case we need instead 〈Ju, Ju′〉 =

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〈u, u′〉. The subgroup of SO(2d) transformations that preserve 〈·, ·〉 and alsocommute with J will be a U(d) subgroup, just like in the bosonic case where aU(d) ⊂ Sp(2d,R) commuted with J . Possible choices of J will be parametrizedby the space SO(2d)/U(d).

In terms of the θj , the Hamiltonian is

h =

d∑j=1

θjθj

Using the derivation property of ·, ·+ one finds

h, θj+ =

d∑k=1

θkθk, θj+ =

d∑k=1

(θkθk, θj+ − θk, θj+θk) = −θj

and, similarly,h, θj+ = θj

so one sees that h is just the generator of U(1) ⊂ U(d) phase rotations on thevariables θj .


For more details on pseudo-classical mechanics, a very readable original ref-erence is [6], and there is a detailed discussion in the textbook [59], chapter7.

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280

Chapter 27

Fermionic Quantization andSpinors

In this chapter we’ll begin by investigating the fermionic analog of the notionof quantization, which takes functions of anti-commuting variables on a phasespace with symmetric bilinear form 〈·, ·〉 and gives an algebra of operators withgenerators satisfying the relations of the corresponding Clifford algebra. Wewill then consider analogs of the constructions used in the bosonic case whichthere gave us an essentially unique representation Γ′ of the corresponding Weylalgebra on a space of states.

We know that for a fermionic oscillator with d degrees of freedom, the alge-bra of operators will be Cliff(2d,C), the algebra generated by annihilation and

creation operators aF j , aF†j . These operators will act on HF , a complex vec-

tor space of dimension 2d, and this will be our fermionic analog of the bosonicΓ′. Since the spin group consists of invertible elements of the Clifford algebra,it has a representation on HF . This is known as the “spinor representation”,and it can be constructed by analogy with the Bargmann-Fock construction inthe bosonic case. We’ll also consider the analog in the fermionic case of theSchrodinger representation, which turns out to have a problem with unitarity,but finds a use in physics as “ghost” degrees of freedom.

27.1 Quantization of pseudo-classical systems

In the bosonic case, quantization was based on finding a representation of theHeisenberg Lie algebra of linear functions on phase space, or more explicitly,for basis elements qj , pj of this Lie algebra finding operators Qj , Pj satisfyingthe Heisenberg commutation relations. In the fermionic case, the analog ofthe Heisenberg Lie algebra is not a Lie algebra, but a Lie superalgebra, withbasis elements 1, ξj , j = 1, . . . , n and a Lie superbracket given by the fermionic

281

Poisson bracket, which on basis elements is

ξj , ξk+ = ±δjk, ξj , 1+ = 0, 1, 1+ = 0

A representation of this Lie superalgebra (and thus a quantization of the pseudo-classical system) will be given by finding a linear map Γ+ that takes basiselements ξj to operators Γ+(ξj) satisfying the anti-commutation relations

[Γ+(ξj),Γ+(ξk)]+ = ±δjk1

As in the bosonic case, quantization takes (for ~ = 1) the function 1 to theidentity operator.

These relations are, up to a factor of√

2, exactly the Clifford algebra rela-tions, since if

γj =√

2Γ+(ξj)

then[γj , γk]+ = ±2δjk

We have seen that the real Clifford algebras Cliff(p, q,R) are isomorphic to eitherone or two copies of the matrix algebras M(2l,R),M(2l,C), or M(2l,H), withthe power l depending on p, q. The irreducible representations of such a matrixalgebra are just the column vectors of dimension 2l, and there will be eitherone or two such irreducible representations for Cliff(p, q,R) depending on thenumber of copies of the matrix algebra.

Note that here we are not introducing the factors of i into the definition ofquantization that in the bosonic case were necessary to get a unitary represen-tation of the Lie group corresponding to the real Heisenberg Lie algebra h2d+1.In the bosonic case we worked with all complex linear combinations of powersof the Qj , Pj (the complex Weyl algebra Weyl(2d,C)), and thus had to identifythe specific complex linear combinations of these that gave unitary represen-tations of the Lie algebra h2d+1 o sp(2d,R). Here we are not complexifyingfor now, but working with the real Clifford algebra Cliff(p, q,R), and it is theirreducible representations of this algebra that provide an analog of the uniqueinteresting irreducible representation of h2d+1. In the Clifford algebra case, therepresentations of interest are not Lie algebra representations and may be onreal vector spaces. There is no analog of the unitarity property of the h2d+1

representation.In the bosonic case we found that Sp(2d,R) acted on the bosonic dual phase

space, preserving the antisymmetric bilinear form Ω that determined the Lie al-gebra h2d+1, so it acted on this Lie algebra by automorphisms. We saw (seechapter 16) that intertwining operators there gave us a representation of thedouble cover of Sp(2d,R) (the metaplectic representation), with the Lie alge-bra representation given by the quantization of quadratic functions of the qj , pjphase space coordinates. There is a closely analogous story in the fermionic case,where SO(p, q,R) acts on the fermionic phase space, preserving the symmetricbilinear form that determines the Clifford algebra relations. Here one can con-struct the spinor representation of spin group double covering SO(p, q,R) using

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intertwining operators, with the Lie algebra representation given by quadraticcombinations of the quantizations of the fermionic coordinates ξj .

The fermionic analog of 16.2 is

UkΓ+(ξ)U−1k = Γ+(φk0(ξ)) (27.1)

Here k0 ∈ SO(p, q,R), ξ ∈M∗ = Rn (n = p+ q), φk0is the action of k0 on M∗

and the Uk for k ∈ Spin(p, q) are the intertwining operators we are looking for(they will give the spinor representation). The fermionic analog of 16.3 is

[U ′L,Γ+(ξ)] = Γ+(L · ξ)

wherek0 = etL, L ∈ so(p, q,R)

and L acts on M∗ as an infinitesimal orthogonal transformation. In terms ofbasis vectors of M∗

ξ =

ξ1...ξn

this says

[U ′L,Γ+(ξ)] = Γ+(LT ξ)

Just as in the bosonic case, the U ′L can be found by looking first at thepseudo-classical case, where one has theorem 26.1 which says

µL, ξ+ = LT ξ

where

µL =1

2ξ · Lξ =

1

2

∑j,k

Ljkξjξk

Extending the fermionic quantization that takes

ξj → Γ+(ξj) =1√2γj

to include quadratics in ξj by

ξjξk → Γ+(ξjξk) =1

2γjγk

one finds

U ′L = Γ+(µL) =1

4

∑j,k

Ljkγjγk

For the case of a rotation in the j − k plane, where with L = εjk we recoverformulas 25.1 and 25.2 from chapter 25, with

[−1

2γjγk, γ(v)] = γ(εjkv)

the infinitesimal action of a rotation on the γ matrices, and

γ(v)→ e−θ2 γjγkγ(v)e

θ2 γjγk = γ(eθεjkv)

the group version.

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27.2 The Schrodinger representation for fermions:ghosts

We would like to construct representations of Cliff(p, q,R) and thus fermionicstate spaces by using analogous constructions to the Schrodinger and Bargmann-Fock ones in the bosonic case. The Schrodinger construction took the statespace H to be a space of functions on a subspace of the classical phase spacewhich had the property that the basis coordinate-functions Poisson-commuted.Two examples of this are the position coordinates qj , since qj , qk = 0, or themomentum coordinates pj , since pj , pk = 0. Unfortunately, for symmetricbilinear forms 〈·, ·〉 of definite sign, such as the positive definite case Cliff(n,R),the only subspace the bilinear form is zero on is the zero subspace.

To get an analog of the bosonic situation, one needs to take the case ofsignature d, d. The fermionic phase space will then be 2d dimensional, withd-dimensional subspaces on which 〈·, ·〉 and thus the fermionic Poisson bracketis zero. Quantization will give the Clifford algebra

Cliff(d, d,R) = M(2d,R)

which has just one irreducible representation, R2d . One can complexify this toget a complex state space

HF = C2d

This state space will come with a representation of Spin(d, d,R) from expo-nentiating quadratic combinations of the generators of Cliff(d, d,R). However,this is a non-compact group, and one can show that on general grounds it can-not have unitary finite-dimensional representations, so there must be a problemwith unitarity.

To see what happens explicitly, consider the simplest case d = 1 of one degreeof freedom. In the bosonic case the classical phase space is R2, and quantizationgives operators Q,P which in the Schrodinger representation act on functionsof q, with Q = q and P = −i ∂∂q . In the fermionic case with signature 1, 1, basiscoordinate functions on phase space are ξ1, ξ2, with

ξ1, ξ1+ = 1, ξ2, ξ2+ = −1, ξ1, ξ2+ = −ξ2, ξ1+ = 0

Defining

η =1√2

(ξ1 + ξ2), π =1√2

(ξ1 − ξ2)

we get objects with fermionic Poisson bracket analogous to those of q and p

η, η+ = π, π+ = 0, η, π+ = 1

Quantizing, we get analogs of the Q,P operators

η =1√2

(Γ+(ξ1) + Γ+(ξ2)), π =1√2

(Γ+(ξ1)− Γ+(ξ2))

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which satisfy anti-commutation relations

η2 = π2 = 0, ηπ + πη = 1

and can be realized as operators on the space of functions of one fermionicvariable η as

η = multiplication by η, π =∂

∂η

This state space is two complex dimensional, with an arbitrary state

f(η) = c11 + c2η

with cj complex numbers. The inner product on this is space given by thefermionic integral

(f1(η), f2(η)) =

∫f∗1 (η)f2(η)dη

with

f∗(ξ) = c11 + c2η

With respect to this inner product, one has

(1, 1) = (η, η) = 0, (1, η) = (η, 1) = 1

This inner product is indefinite and can take on negative values, since

(1− η, 1− η) = −2

Having such negative-norm states ruins any standard interpretation of thisas a physical system, since this negative number is supposed to the probability offinding the system in this state. Such quantum systems are called “ghosts”, anddo have applications in the description of various quantum systems, but onlywhen a mechanism exists for the negative-norm states to cancel or otherwise beremoved from the physical state space of the theory.

27.3 Spinors and the Bargmann-Fock construc-tion

While the fermionic analog of the Schrodinger construction does not give aunitary representation of the spin group, it turns out that the fermionic analogof the Bargmann-Fock construction does, on the fermionic oscillator state spacethat we have already studied. This will work for the case of a positive definitesymmetric bilinear form 〈·, ·〉. Note though that n must be even since this willrequire choosing a complex structure on the fermionic phase space Rn.

The corresponding pseudo-classical system will be the classical fermionicoscillator studied in the last chapter. Recall that this uses a choice of complex

285

structure J on the fermionic phase space R2d, with the standard choice givingthe relations

θj =1√2

(ξ2j−1 + iξ2j), θj =1√2

(ξ2j−1 − iξ2j)

for j = 1, . . . , d between real and complex coordinates. Here 〈·, ·〉 is positive-definite, and the ξj are coordinates with respect to an orthonormal basis, so wehave the standard relation ξj , ξk+ = δjk and the θj , θj satisfy

θj , θk+ = θj , θk+ = 0, θj , θk+ = δjk

To quantize this system we need to find operators Γ+(θj) and Γ+(θj) thatsatisfy

[Γ+(θj),Γ+(θk)]+ = [Γ+(θj),Γ

+(θk)]+ = 0

[Γ+(θj),Γ+(θk)]+ = δjk1

but these are just the CAR satisfied by fermionic annihilation and creationoperators. We can choose

Γ+(θj) = aF j , Γ+(θj) = aF†j

and realize these operators as

aF j =∂

∂θj, aF

†j = multiplication by θj

on the state space Λ∗Cd of polynomials in the anti-commuting variables θj .This is a complex vector space of dimension 2d, isomorphic with the state spaceHF of the fermionic oscillator in d degrees of freedom, with the isomorphismgiven by

1↔ |0〉F , θj ↔ aF†j |0〉F , θjθk ↔ aF

†jaF

†k|0〉, · · · , θ1 . . . θd ↔ aF

†1aF

†2 · · · aF

†d|0〉F

where the indices j, k, . . . take values 1, 2, . . . , d and satisfy j < k < · · · .If one defines a Hermitian inner product (·, ·) on HF by taking these basis

elements to be orthonormal, the operators aF j and aF†j will be adjoints with

respect to this inner product. This same inner product can also be definedusing fermionic integration by analogy with the Bargmann-Fock definition inthe bosonic case as

(f1(θ1, · · · , θd), f2(θ1, · · · , θd)) =

∫f1f2e

−∑dj=1 θjθjdθ1 · · · dθddθ1 · · · dθd

where f1 and f2 are complex linear combinations of the powers of the anticom-muting variables θj . For the details of the construction of this inner product,see chapter 7.2 of [59].

We saw in chapter 25.2 that quadratic combinations of the generators γjof the Clifford algebra satisfy the commutation relations of the Lie algebra

286

so(2d) = spin(2d). The fermionic quantization given here provides an explicitrealization of a representation of the Clifford algebra Cliff(2d,R) on the complexvector space HF since we know that the

√2Γ+(ξj) will satisfy the Clifford

algebra relations, and we have

ξ2j−1 =1√2

(θj + θj), ξ2j =1

i√

2(θj − θj)

As a result, the following linear combinations of annihilation and creation op-erators

aF j + aF†j , −i(aF j − aF

†j)

will satisfy the same Clifford algebra relations as γ2j−1, γ2j and taking quadraticcombinations of these operators on HF provides a representation of the Liealgebra spin(2d).

This representation exponentiates to a representation not of the group SO(2d),but of its double-cover Spin(2d). The representation that we have constructedhere on the fermionic oscillator state space HF is called the spinor representa-tion of Spin(2d), and we will sometimes denote HF with this group action asS.

In the bosonic case, HB is an irreducible representation of the Heisenberggroup, but as a representation of Mp(2d,R), it has two irreducible components,corresponding to even and odd polynomials. The fermionic analog is that HFis irreducible under the action of the Clifford algebra Cliff(2d,C). To showthis, note that Cliff(2d,C) is isomorphic to the matrix algebra M(2d,C) and its

action on HF = C2d is isomorphic to the action of matrices on column vectors.While HF is irreducible as a representation of the Clifford algebra, it is the

sum of two irreducible representations of Spin(2d), the so-called “half-spinor”representations. Spin(2d) is generated by quadratic combinations of the Cliffordalgebra generators, so these will preserve the subspaces

S+ = span|0〉F , aF†jaF

†k|0〉F , · · · ⊂ S = HF

andS− = spanaF †j |0〉F , aF

†jaF

†kaF

†l |0〉F , · · · ⊂ S = HF

This is because quadratic combinations of the aF j , aF†j preserve the parity of

the number of creation operators used to get an element of S by action on |0〉F .The construction of the spinor representation here has involved making a spe-

cific choice of relation between the Clifford algebra generators and the fermionicannihilation and creation operators. This is determined by our choice of com-plex structure J , with the way things work closely analogous to the HeisenbergLie algebra case of section 20.1, with the difference that J must be chosen topreserve not an anti-symmetric bilinear form Ω, but the inner product, so onehas

〈J(·), J(·)〉 = 〈·, ·〉Jsplits the complexification of the real dual phase space M∗ = R2d with its

coordinates ξj into a complex vector space Cd and a conjugate complex vector

287

space. Quantization of vectors in the first of these gives annihilation operators,while vectors in the conjugate Cd are taken to creation operators. The choiceof J is reflected in the existence of a distinguished direction in the spinor space,|0〉F ∈ HF which (up to a scalar factor), is determined by the condition that itis annihilated by all linear combinations of annihilation operators.

The choice of J also picks out a subgroup U(d) ⊂ SO(2d) of those orthogonaltransformations that commute with J . Just as in the bosonic case, two differentrepresentations of the Lie algebra u(d) of U(d) are used:

• The restriction of u(d) ⊂ so(2d) of the spinor representation describedabove. This exponentiates to give a representation not of U(d), but of adouble cover of U(d) that is a subgroup of Spin(2d).

• By normal-ordering operators, one shifts the spinor representation of u(d)by a constant and gets a representation that exponentiates to a true rep-resentation of U(d). This representation is reducible, with irreduciblecomponents the Λk(Cd) for k = 0, 1, . . . , d.

In both cases the representation of u(d) is constructed using quadratic combina-tions of annihilation and creation operators involving one annihilation operatorand one creation operator. Non-zero pairs of two annihilation or two creationoperators will give “Bogoliubov transformations”, changing |0〉F

Given any group element

g0 = eA ⊂ U(d)

acting on the fermionic dual phase space preserving J and the inner product, wecan use exactly the same method as in theorems 22.1 and 22.2 to construct itsaction on the fermionic state space by the second of the above representations.For A a skew-adjoint matrix we have a fermionic moment map

A ∈ u(d)→ µA =∑j,k

θjAjkθk

satisfyingµA, µA′+ = µ[A,A′]

andµA,θ+ = −Aθ, µA,θ+ = ATθ

The Lie algebra representation operators are the

U ′A =∑j,k

a†F jAjkaF k

which satisfy[U ′A, U

′A′ ] = U[A,A′]

and[U ′A,aF ] = −AaF , [U ′A,a

†F ] = ATa†F

288

Exponentiating these gives the intertwining operators, which act on the an-nihilation and creation operators as

UeAaF (UeA)−1 = e−AaF , UeAa†F (UeA)−1 = eAT

a†F

For the simplest example, consider the U(1) ⊂ U(d) ⊂ SO(2d) that acts by

θj → eiφθj , θj → e−iφθj

corresponding to A = −iφ1. The moment map will be

µA = −iφh

where

h =

d∑j=1

θjθj

is the Hamiltonian for the classical fermionic oscillator. Quantizing h by takingthe Hamiltonian operator

H =1

2

d∑j=1

(aF†jaF j − aF jaF

†j)

will give a Lie algebra representation with half-integral eigenvalues (± 12 ), so

exponentiating will give a representation of a double cover of U(d) (and ofSpin(d)). Quantizing h instead using normal-ordering

: H :=

d∑j=1

aF†jaF j

will give a true representation of U(d) and

U ′A = −iφd∑j=1

aF†jaF j

satisfying

[U ′A,aF ] = iφaF , [U ′A,a†F ] = iφa†F

Exponentiating, the action on annihilation and creation operators is

e−iφ∑dj=1 aF

†jaF jaF e

iφ∑dj=1 aF

†jaF j = eiφaF

e−iφ∑dj=1 aF

†jaF ja†F e

iφ∑dj=1 aF

†jaF j = eiφa†F

289

27.4 Parallels between bosonic and fermionic

To summarize much of the material we have covered, it may be useful to con-sider the following table, which explicitly gives the correspondence between theparallel constructions we have studied in the bosonic and fermionic cases.

Bosonic Fermionic

Dual phase space M = R2d Dual phase space M = R2d

Non-degenerate anti-symmetricbilinear form Ω(·, ·) on M

Non-degenerate symmetricbilinear form 〈·, ·〉 on M

Poisson bracket ·, · = Ω(·, ·)on functions on M = R2d

Poisson bracket ·, ·+ = 〈·, ·〉on anti-commuting functions on R2d

Lie algebra of polynomials ofdegree 0, 1, 2

Lie superalgebra of anti-commutingpolynomials of degree 0, 1, 2

Coordinates qj , pj , basis of M Coordinates ξj , basis of M

Quadratics in qj , pj , basis for sp(2d,R) Quadratics in ξj , basis for so(2d)

Weyl algebra Weyl(2d,C) Clifford algebra Cliff(2d,C)

Momentum, position operators Pj , Qj Clifford algebra generators γj

Sp(2d,R) preserves Ω(·, ·) SO(2d,R) preserves 〈·, ·〉

Quadratics in Pj , Qj providerepresentation of sp(2d,R)

Quadratics in γj providerepresentation of so(2d)

J : J2 = −1, Ω(Ju, Jv) = Ω(u, v) J : J2 = −1, 〈Ju, Jv〉 = 〈u, v〉

M⊗C =M+ ⊕M− M⊗C =M+ ⊕M−

U(d) ⊂ Sp(2d,R) commutes with J U(d) ⊂ SO(2d,R) commutes with J

Mp(2d,R) double-cover of Sp(2d,R) Spin(2d) double-cover of SO(2d)

Metaplectic representation Spinor representation

aj , a†j satisfying CCR ajF , aj

†F satisfying CAR

aj |0〉 = 0, |0〉 depends on J ajF |0〉 = 0, |0〉 depends on J


For more about pseudo-classical mechanics and quantization, see [59] Chapter 7.The fermionic quantization map, Clifford algebras, and the spinor representationare discussed in detail in [38]. For another discussion of the spinor representationfrom a similar point of view to the one here, see [60] Chapter 12.

290

Chapter 28

Supersymmetry, SomeSimple Examples

If one considers fermionic and bosonic quantum system that each separatelyhave operators coming from Lie algebra or superalgebra representations on theirstate spaces, when one combines the systems by taking the tensor product, theseoperators will continue to act on the combined system. In certain special casesnew operators with remarkable properties will appear that mix the fermionic andbosonic systems. These are generically known as “supersymmetries”. In suchsupersymmetrical systems, important conventional mathematical objects oftenappear in a new light. This has been one of the most fruitful areas of interactionbetween mathematics and physics in recent decades, beginning with EdwardWitten’s highly influential 1982 paper Supersymmetry and Morse theory [69]. Ahuge array of different supersymmetric quantum field theories have been studiedover the years, although the goal of using these to develop a successful unifiedtheory remains still out of reach. In this chapter we’ll examine in detail someof the simplest such quantum systems, examples of “superymmetric quantummechanics”. These have the characteristic feature that the Hamiltonian operatoris a square.

28.1 The supersymmetric oscillator

In the previous chapters we discussed in detail

• The bosonic harmonic oscillator in N degrees of freedom, with state spaceHB generated by applying N creation operators aBj to a lowest energystate |0〉B . The Hamiltonian is

H =1

2~ω

N∑j=1

(aB†jaBj + aBjaB

†j) =

N∑j=1

(NBj +1

2)~ω

291

where NBj is the number operator for the j’th degree of freedom, witheigenvalues nBj = 0, 1, 2, · · · .

• The fermionic harmonic oscillator in N degrees of freedom, with statespace HF generated by applying N creation operators aF j to a lowestenergy state |0〉F . The Hamiltonian is

H =1

2~ω

N∑j=1

(aF†jaF j − aF jaF

†j) =

N∑j=1

(NF j −1

2)~ω

where NF j is the number operator for the j’th degree of freedom, witheigenvalues nF j = 0, 1.

Putting these two systems together we get a new quantum system with statespace

H = HB ⊗HF

and Hamiltonian

H =

N∑j=1

(NBj +NF j)~ω

Notice that the lowest energy state |0〉 for the combined system has energy 0,due to cancellation between the bosonic and fermionic degrees of freedom.

For now, taking for simplicity the case N = 1 of one degree of freedom, theHamiltonian is

H = (NB +NF )~ω

with eigenvectors |nB , nF 〉 satisfying

H|nB , nF 〉 = (nB + nF )~ω

Notice that while there is a unique lowest energy state |0, 0〉 of zero energy, allnon-zero energy states come in pairs, with two states

|n, 0〉 and |n− 1, 1〉

both having energy n~ω.

This kind of degeneracy of energy eigenvalues usually indicates the existenceof some new symmetry operators commuting with the Hamiltonian operator.We are looking for operators that will take |n, 0〉 to |n − 1, 1〉 and vice-versa,and the obvious choice is the two operators

Q+ = aBa†F , Q− = a†BaF

which are not self adjoint, but are each other’s adjoints ((Q−)† = Q+).

The pattern of energy eigenstates looks like

292

Computing anti-commutators using the CCR and CAR for the bosonic andfermionic operators (and the fact that the bosonic operators commute with thefermionic ones since they act on different factors of the tensor product), onefinds that

Q2+ = Q2

− = 0

and

(Q+ +Q−)2 = [Q+, Q−]+ = H

One could instead work with self-adjoint combinations

Q1 = Q+ +Q−, Q2 =1

i(Q+ −Q−)

which satisfy

[Q1, Q2]+ = 0, Q21 = Q2

2 = H

Notice that the Hamiltonian H is a square of the self-adjoint operator Q+ +Q−, and this fact alone tells us that the energy eigenvalues will be non-negative.It also tells us that energy eigenstates of non-zero energy will come in pairs

|ψ〉, (Q+ +Q−)|ψ〉

293

with the same energy. To find states of zero energy, instead of trying to solvethe equation H|0〉 = 0 for |0〉, one can look for solutions to

Q1|0〉 = 0 or Q2|0〉 = 0

These operators don’t correspond to a Lie algebra representation as H does,but do come from a Lie superalgebra representation, so are described as gen-erators of a “supersymmetry” transformation. In more general theories withoperators like this with the same relation to the Hamiltonian, one may or maynot have solutions to

Q1|0〉 = 0 or Q2|0〉 = 0

If such solutions exist, the lowest energy state has zero energy and is describedas invariant under the supersymmetry. If no such solutions exist, the lowestenergy state will have a non-zero, positive energy, and satisfy

Q1|0〉 6= 0 or Q2|0〉 6= 0

In this case one says that the supersymmetry is “spontaneously broken”, sincethe lowest energy state is not invariant under supersymmetry.

There is an example of a physical quantum mechanical system that has ex-actly the behavior of this supersymmetric oscillator. A charged particle confinedto a plane, coupled to a magnetic field perpendicular to the plane, can be de-scribed by a Hamiltonian that can be put in the bosonic oscillator form (toshow this, we need to know how to couple quantum systems to electromagneticfields, which we will come to later in the course). The equally spaced energylevels are known as “Landau levels”. If the particle has spin one-half, there willbe an additional term in the Hamiltonian coupling the spin and the magneticfield, exactly the one we have seen in our study of the two-state system. Thisadditional term is precisely the Hamiltonian of a fermionic oscillator. For thecase of gyromagnetic ratio g = 2, the coefficients match up so that we haveexactly the supersymmetric oscillator described above, with exactly the patternof energy levels seen there.

28.2 Supersymmetric quantum mechanics witha superpotential

The supersymmetric oscillator system can be generalized to a much wider classof potentials, while still preserving the supersymmetry of the system. In thissection we’ll introduce a so-called “superpotential” W (q), with the harmonicoscillator the special case

W (q) =q2

2For simplicity, we will here choose constants ~ = ω = 1

Recall that our bosonic annihilation and creation operators were defined by

aB =1√2

(Q+ iP ), a†B =1√2

(Q− iP )

294

We will change this definition to introduce an arbitrary superpotential W (q):

aB =1√2

(W ′(Q) + iP ), a†B =1√2

(W ′(Q)− iP )

but keep our definition of the operators

Q+ = aBa†F , Q− = a†BaF

These satisfyQ2

+ = Q2− = 0

for the same reason as in the oscillator case: repeated factors of aF or a†F vanish.Taking as the Hamiltonian the same square as before, we find

H =(Q+ +Q−)2

=1

2(W ′(Q) + iP )(W ′(Q)− iP )a†FaF +

1

2(W ′(Q)− iP )(W ′(Q) + iP )aFa

†F

=1

2(W ′(Q)2 + P 2)(a†FaF + aFa

†F ) +

1

2(i[P,W ′(Q)])(a†FaF − aFa

†F )

=1

2(W ′(Q)2 + P 2) +

1

2(i[P,W ′(Q)])σ3

But iP is the operator corresponding to infinitesimal translations in Q, so wehave

i[P,W ′(Q)] = W ′′(Q)

and

H =1

2(W ′(Q)2 + P 2) +

1

2W ′′(Q)σ3

which gives a large class of quantum systems, all with state space

H = HB ⊗HF = L2(R)⊗C2

(using the Schrodinger representation for the bosonic factor).The energy eigenvalues will be non-negative, and energy eigenvectors with

positive energy will occur in pairs

|ψ〉, (Q+ +Q−)|ψ〉

.There may or may not be a state with zero energy, depending on whether

or not one can find a solution to the equation

(Q+ +Q−)|0〉 = Q1|0〉 = 0

If such a solution does exist, thinking in terms of super Lie algebras, one callsQ1 the generator of the action of a supersymmetry on the state space, anddescribes the ground state |0〉 as invariant under supersymmetry. If no such

295

solution exists, one has a theory with a Hamiltonian that is invariant under su-persymmetry, but with a ground state that isn’t. In this situation one describesthe supersymmetry as “spontaneously broken”. The question of whether a givensupersymmetric theory has its supersymmetry spontaneously broken or not isone that has become of great interest in the case of much more sophisticated su-persymmetric quantum field theories. There, hopes (so far unrealized) of makingcontact with the real world rely on finding theories where the supersymmetryis spontaneously broken.

In this simple quantum mechanical system, one can try and explicitly solvethe equation Q1|ψ〉 = 0. States can be written as two-component complexfunctions

|ψ〉 =

(ψ+(q)ψ−(q)

)and the equation to be solved is

(Q+ +Q−)|ψ〉 =1√2

((W ′(Q) + iP )a†F + (W ′(Q)− iP )aF )

(ψ+(q)ψ−(q)

)=

1√2

((W ′(Q) +d

dq)

(0 10 0

)+ (W ′(Q)− d

dq)

(0 01 0

))

(ψ+(q)ψ−(q)

)=

1√2

(W ′(Q)

(0 11 0

)+

d

dq

(0 1−1 0

))

(ψ+(q)ψ−(q)

)=

1√2

(0 1−1 0

)(d

dq−W ′(Q)σ3)

(ψ+(q)ψ−(q)

)= 0

which has general solution(ψ+(q)ψ−(q)

)= eW (q)σ3

(c+c−

)=

(c+e

W (q)

c−e−W (q

)for complex constants c+, c−. Such solutions will only be normalizable if

c+ = 0, limq→±∞

W (q) = +∞

orc− = 0, lim

q→±∞W (q) = −∞

If, for example, W (q) is an odd polynomial, one will not be able to satisfy eitherof these conditions, so there will be no solution, and the supersymmetry will bespontaneously broken.

28.3 Supersymmetric quantum mechanics anddifferential forms

If one considers supersymmetric quantum mechanics in the case of d degrees offreedom and in the Schrodinger representation, one has

H = L2(Rd)⊗ Λ∗(Rd)

296

the tensor product of complex-valued functions on Rd (acted on by the Weylalgebra Weyl(2d,C)) and anti-commuting functions on Rd (acted on by theClifford algebra Cliff(2d,C)). There are two operators Q+ and Q−, adjointsof each other and of square zero. If one has studied differential forms, thisshould look familiar. This space H is well-known to mathematicians, as thecomplex valued differential forms on Rd, often written Ω∗(Rd), where here the∗ denotes an index taking values from 0 (the 0-forms, or functions) to d (thed-forms). In the theory of differential forms, it is well known that one has anoperator d on Ω∗(Rd) with square zero, called the de Rham differential. Usingthe inner product on Rd, one can put a Hermitian inner product on Ω∗(Rd)by integration, and then d has an adjoint δ, also of square zero. The Laplacianoperator on differential forms is

= (d+ δ)2

The supersymmetric quantum system we have been considering correspondsprecisely to this, once one conjugates d, δ as follows

Q+ = e−V (q)deV (q), Q− = eV (q)δe−V (q)

In mathematics, the interest in differential forms mainly comes from thefact that one can construct them not just on Rd, but on a general differentiablemanifold M , with a corresponding construction of d, δ, operators. In Hodgetheory, one studies solutions of

ψ = 0

(these are called “harmonic forms”) and finds that the dimension of the spaceof such solutions can be used to get topological invariants of the manifold M .Witten’s famous 1982 paper on supersymmetry and Morse theory [69] first ex-posed these connections, using them both to give new ways of thinking aboutthe mathematics involved, as well as applications of topology to the questionof deciding whether supersymmetry was spontaneously broken in various super-symmetric models.


For a reference at the level of these notes, see [22]. For more details aboutsupersymmetric quantum mechanics and its relationship to the Dirac operatorand the index theorem, see the graduate level textbook of Tahktajan [59], andlectures by Orlando Alvarez [1]. These last two sources also describe the formal-ism one can get by putting together the standard Hamiltonian mechanics andits fermionic analog, consistently describing both “classical” system with com-muting and anti-commuting variables and its quantization to get the quantumsystems described in this chapter.

297

298

Chapter 29

The Dirac Operator

In chapter 28 we considered supersymmetric quantum mechanical systems whereboth the bosonic and fermionic variables that get quantized take values in aneven dimensional phase space R2d. There are then two supersymmetry oper-ators Q1 and Q2, so this is sometimes called N = 2 supersymmetry (in analternate normalization, counting complex variables, it is called N = 1). Itturns out however that there are very interesting quantum mechanics systemsthat one can get by quantizing bosonic variables in R2d, but fermionic vari-ables in Rd. The operators appearing in such a theory will be given by thetensor product of the Weyl algebra in 2d variables and the Clifford algebra in dvariables.

In such a theory the operator −|P|2 will have a square root, the Dirac oper-ator /∂. This existence of a square root of the Casimir operator provides a newway to construct irreducible representations of the group of spatial symmetries,using a new sort of quantum free particle, one carrying an internal “spin” de-gree of freedom. Remarkably, fundamental matter particles are well-describedin exactly this way.

29.1 The Dirac operator

Recall from chapter 25 that associated to Rd with a standard inner product,but of a general signature (r, s) (where r + s = d, r is the number of + signs, sthe number of − signs) we have a Clifford algebra Cliff(r, s) with generators γjsatisfying

γjγk = −γkγj , j 6= k

γ2j = +1 forj = 1, · · · , r γ2

j = −1, forj = r + 1, · · · , dTo any vector v ∈ Rd with components vj we can associate a correspondingelement /v in the Clifford algebra by

v ∈ Rd → /v =

d∑j=1

γjvj ∈ Cliff(r, s)

299

Multiplying this Clifford algebra element by itself and using the relations above,we get a scalar, the length-squared of the vector

/v2 = v2

1 + v22 · · ·+ v2

r − v2r+1 − · · · − v2

d = |v|2

This shows that by introducing a Clifford algebra, we can find an interestingnew sort of square-root for expressions like |v|2. In particular, it allows us todefine

Definition (Dirac operator). The Dirac operator is the operator

/∂ =

d∑j=1

γj∂

∂qj

This will be a first-order differential operator with the property that itssquare is the Laplacian

/∂2

=∂2

∂q21

+ · · ·+ ∂2

∂q2r

− ∂2

∂q2r+1

− · · · − ∂2

∂q2d

The Dirac operator /∂ acts not on functions but on functions taking valuesin the spinor vector space S that the Clifford algebra acts on. Picking a matrixrepresentation of the γj , the Dirac operator will be a linear system of first orderequations. In chapter 42 we will study in detail what happens for the case ofr = 3, s = 1 and see how the Dirac operator there provides an appropriatewave-equation with the symmetries of special relativistic space-time.

29.2 The Pauli operator and free spin 12 particles

in d = 3

In dimension d = 3 (r = 3, s = 0), one can choose as generators of the Cliffordalgebra the three Pauli matrices σj . The Dirac operator can then be written as

(σ1∂

∂q1+ σ2

∂

∂q2+ σ3

∂

∂q3)

and it operates on two-component wave-functions(ψ1(q)ψ2(q)

)Using the Dirac operator (often called in this context the “Pauli operator”) wecan write a two-component version of the Schrodinger equation (often called the“Pauli equation” or “Schrodinger-Pauli equation”)

i∂

∂t

(ψ1(q)ψ2(q)

)=−1

2m(σ1

∂

∂q1+ σ2

∂

∂q2+ σ3

∂

∂q3)2

(ψ1(q)ψ2(q)

)(29.1)

300

This free-particle version of the equation is just two copies of the standard free-particle Schrodinger equation, so physically just corresponds to two independentquantum free particles. It becomes much more non-trivial when a coupling toan electromagnetic field is introduced, as will be seen in chapter 40.

The introduction of a two-component wave-function does allow us to findmore interesting irreducible representations of the group E(3), ones that corre-spond to the ± 1

2 cases of eigenvalues of the second Casimir operator J·P|P| rather

than the chapter 17 case where a single wave-function just provided the zeroeigenvalue case.

These representations will as before be on the space of solutions of the time-independent equation, and irreducible for fixed choice of the energy E. Usingthe momentum operator this equation will be

1

2m(σ ·P)2

(ψ1(q)ψ2(q)

)= E

(ψ1(q)ψ2(q)

)In terms of the inverse Fourier transform

ψ1,2(q) =1

(2π)32

∫∫∫eip·qψ1,2(p)d3p

this equation becomes

((σ · p)2 − 2mE)

(ψ1(p)

ψ2(p)

)= (|p|2 − 2mE)

(ψ1(p)

ψ2(p)

)= 0 (29.2)

and as in chapter 17 our solution space is given by functions ψE,1,2(p) on the

sphere of radius√

2mE = |p| in momentum space (although now, two suchfunctions).

Another way to find solutions to this equation is to look for solutions tothe pair of first-order equations involving the three-dimensional Dirac operator.Solutions to

σ · p

(ψ1(p)

ψ2(p)

)= ±√

2mE

(ψ1(p)

ψ2(p)

)will give solutions to 29.2, for either sign. One can rewrite this as

σ · p|p|

(ψ1(p)

ψ2(p)

)= ±

(ψ1(p)

ψ2(p)

)

and we will write solutions to this equation with the + sign as ψE,+(p), those for

the − sign as ψE,−(p). Note that ψE,+(p) and ψE,+(p) are each two-component

complex functions of the momentum, supported on the sphere√

2mE = |p|.In chapter 15 we saw that R ∈ SO(3) acts on single-component momentum

space solutions of the Schrodinger equation by

ψE(p)→ u(0, R)ψE(p) ≡ ψE(R−1p)

301

This takes solutions to solutions since the operator u(0, R) commutes with theCasimir operator |P|2

u(0, R)|P|2 = |P|2u(0, R) ⇐⇒ u(0, R)|P|2u(0, R)−1 = |P|2

This is true since

u(0, R)|P|2u(0, R)−1ψ(p) =u(0, R)|P|2ψ(Rp)

=|R−1P|2ψ(R−1Rp) = |P|2ψ(p)

The operator σ ·P does not commute with the representation u(0, R) because

u(0, R)(σ ·P)u(0, R)−1 = (σ · (RP)) 6= σ ·P

and thus rotations do not act separately on the spaces ψE,+(p) and ψE,−(p).If we want rotations to act separately on these spaces, we need to change

the action of rotations to

ψE,±(p)→ uS(0, R)ψE,±(p) = ΩψE,±(R−1p)

where Ω is one of the two elements of SU(2) corresponding to R ∈ SO(3). Suchan Ω can be constructed using equation 6.2

Ω = Ω(φ,w) = e−iφ2 w·σ

Equation 6.4 shows that Ω is the SU(2) matrix corresponding to a rotation Rby an angle φ about the axis given by a unit vector w.

With this action on solutions we have

uS(0, R)(σ ·P)uS(0, R)−1ψE,±(p) =uS(0, R)(σ ·P)Ω−1ψE,±(Rp)

=Ω(σ ·R−1P)Ω−1ψE,±(R−1Rp)

=(σ ·P)ψE,±(p)

where we have used equation 6.4 to show

Ω(σ ·R−1P)Ω−1 = (σ ·RR−1P) = (σ ·P)

Note that the two representations we get this way are representations notof the rotation group SO(3) but of its double cover Spin(3) = SU(2) (becauseotherwise there is a sign ambiguity since we don’t know whether to choose Ωor −Ω). The translation part of the spatial symmetry group is easily seen tocommute with σ ·P, so we have constructed representations of E(3), or rather,of its double cover

E(3) = R3 o SU(2)

on the two spaces of solutions ψE,±(p). We will see that these two representa-tions are the E(3) representations described in section 17.3, the ones labeled bythe helicity ± 1

2 representations of the stabilizer group SO(2).

302

The translation part of the group acts as in the one-component case, by themultiplication operator

uS(a,1)ψE,±(p) = e−i(a·p)ψE,±(p)

anduS(a,1) = e−ia·P

so the Lie algebra representation is given by the usual P operator. The SU(2)part of the group acts by a commuting product of two different actions

1. The same action on the momentum coordinates as in the one-componentcase, just using R = Φ(Ω), the SO(3) rotation corresponding to the SU(2)group element Ω. For example, for a rotation about the x-axis by angle φwe have

ψE,±(p)→ ψE,±(R(φ, e1)−1p)

Recall that the operator that does this is e−iφL1 where

−iL1 = −i(Q2P3 −Q3P2) = −(q2∂

∂q3− q3

∂

∂q2)

and in general we have operators

−iL = −iQ×P

that provide the Lie algebra version of the representation (recall that atthe Lie algebra level, SO(3) and Spin(3) are isomorphic).

2. The action of the matrix Ω ∈ SU(2) on the two-component wave-functionby

ψE,±(p)→ ΩψE,±(p)

For a rotation by angle φ about the x-axis we have

Ω = e−iφσ12

and the operators that provide the Lie algebra version of the representationare the

−iS = −i12σ

The Lie algebra representation corresponding to the action of both of thesetransformations is given by the operator

−iJ = −i(L + S)

and the standard terminology is to call L the “orbital” angular momentum, Sthe “spin” angular momentum, and J the “total” angular momentum.

The helicity operator that provides the second Casimir operator for this caseis

J ·P|P|

303

and as in the one-component case the L · P part of this acts trivially on oursolutions ψE,±(p). The spin component acts non-trivially and we have

J ·P|P|

ψE,±(p) =1

2

σ · p|p|

ψE,±(p) = ±1

2ψE,±(p)

so we see that our solutions have helicity ± 12 .


The point of view here in terms of representations of E(3) is not very conven-tional, but the material here about spin and the Pauli equation can be found inany quantum mechanics book, see for example chapter 14 of [53].

304

Chapter 30

Lagrangian Methods andthe Path Integral

In this chapter we’ll give a rapid survey of a different starting point for devel-oping quantum mechanics, based on the Lagrangian rather than Hamiltonianclassical formalism. Lagrangian methods have quite different strengths andweaknesses than those of the Hamiltonian formalism, and we’ll try and pointthese out, while referring to standard physics texts for more detail about thesemethods.

The Lagrangian formalism leads naturally to an apparently very differentnotion of quantization, one based upon formulating quantum theory in terms ofinfinite-dimensional integrals known as path integrals. A serious investigationof these would require another and very different volume, so again we’ll have torestrict ourselves to outlining how path integrals work, describing their strengthsand weaknesses, and giving references to standard texts for the details.

30.1 Lagrangian mechanics

In the Lagrangian formalism, instead of a phase space R2d of positions qj andmomenta pj , one considers just the position space Rd. Instead of a Hamiltonianfunction h(q,p), one has

Definition (Lagrangian). The Lagrangian L is a function

L : (q,v) ∈ Rd ×Rd → L(q,v) ∈ R

where v is a tangent vector at q

Given differentiable paths defined by functions

γ : t ∈ [t1, t2]→ Rd

305

which we will write in terms of their position and velocity vectors as

γ(t) = (q(t), q(t))

one can define a functional on the space of such paths

Definition. ActionThe action S for a path γ is

S[γ] =

∫ t2

t1

L(q(t), q(t))dt

The fundamental principle of classical mechanics in the Lagrangian formal-ism is that classical trajectories are given by critical points of the action func-tional. These may correspond to minima of the action (so this is sometimescalled the “principle of least action”), but one gets classical trajectories also forcritical points that are not minima of the action. One can define the appropriatenotion of critical point as follows

Definition. Critical point for SA path γ is a critical point of the functional S[γ] if

δS(γ) ≡ d

dsS(γs)|s=0 = 0

when

γs : [t1, t2]→ Rd

is a smooth family of paths parametrized by an interval s ∈ (−ε, ε), with γ0 = γ.

We’ll now ignore analytical details and adopt the physicist’s interpretationof this as the first-order change in S due to an infinitesimal change δγ =(δq(t), δq(t)) in the path.

When (q(t), q(t)) satisfy a certain differential equation, the path γ will be acritical point and thus a classical trajectory:

Theorem. Euler-Lagrange equationsOne has

δS[γ] = 0

for all variations of γ with endpoints γ(t1) and γ(t2) fixed if

∂L

∂qj(q(t), q(t))− d

dt(∂L

∂qj(q(t), q(t))) = 0

for j = 1, · · · , d. These are called the Euler-Lagrange equations.

Proof. Ignoring analytical details, the Euler-Lagrange equations follow from thefollowing calculations, which we’ll just do for d = 1, with the generalization to

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higher d straightfoward. We are calculating the first-order change in S due toan infinitesimal change δγ = (δq(t), δq(t))

δS[γ] =

∫ t2

t1

δL(q(t), q(t))dt

=

∫ t2

t1

(∂L

∂q(q(t), q(t))δq(t) +

∂L

∂q(q(t), q(t))δq(t))dt

But

δq(t) =d

dtδq(t)

and, using integration by parts

∂L

∂qδq(t) =

d

dt(∂L

∂qδq)− (

d

dt

∂L

∂q)δq

so

δS[γ] =

∫ t2

t1

((∂L

∂q− d

dt

∂L

∂q)δq − d

dt(∂L

∂qδq))dt

=

∫ t2

t1

(∂L

∂q− d

dt

∂L

∂q)δqdt− (

∂L

∂qδq)(t2) + (

∂L

∂qδq)(t1)

If we keep the endpoints fixed so δq(t1) = δq(t2) = 0, then for solutions to

∂L

∂q(q(t), q(t))− d

dt(∂L

∂q(q(t), q(t))) = 0

the integral will be zero for arbitrary variations δq.

As an example, a particle moving in a potential V (q) will be described by aLagrangian

L(q, q) =1

2m

d∑j=1

q2j − V (q)

for which the Euler-Lagrange equations will be

−∂V∂qj

=d

dt(mqj)

This is just Newton’s second law which says that the force coming from thepotential is equal to the mass times the acceleration of the particle.

The derivation of the Euler-Lagrange equations can also be used to studythe implications of Lie group symmetries of a Lagrangian system. When a Liegroup G acts on the space of paths, preserving the action S, it will take classicaltrajectories to classical trajectories, so we have a Lie group action on the spaceof solutions to the equations of motion (the Euler-Lagrange equations). In

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good cases, this space of solutions is just the phase space of the Hamiltonianformalism. On this space of solutions, we have, from the calculation above

δS[γ] = (∂L

∂qδq(X))(t1)− (

∂L

∂qδq(X))(t2)

where now δq(X) is the infinitesimal change in a classical trajectory comingfrom the infinitesimal group action by an element X in the Lie algebra of G.From invariance of the action S under G we must have δS=0, so

(∂L

∂qδq(X))(t2) = (

∂L

∂qδq(X))(t1)

This is an example of a more general result known as “Noether’s theorem”.It says that given a Lie group action on a Lagrangian system that leaves theaction invariant, for each element X of the Lie algebra we will have a conservedquantity

∂L

∂qδq(X)

which is independent of time along the trajectory. A basic example is whenthe Lagrangian is independent of the position variables qj , depending only onthe velocities qj , for example in the case of a free particle, when V (q) = 0. Insuch a case one has invariance under the Lie group Rd of space-translations.An infinitesimal transformation in the j-direction is given by

δqj(t) = ε

and the conserved quantity is∂L

∂qj

For the case of the free particle, this will be

∂L

∂qj= mqj

and the conservation law is conservation of momentum.Given a Lagrangian classical mechanical system, one would like to be able

to find a corresponding Hamiltonian system that will give the same equationsof motion. To do this, we proceed by defining the momenta pj as above, as theconserved quantities corresponding to space-translations, so

pj =∂L

∂qj

Then, instead of working with trajectories characterized at time t by

(q(t), q(t)) ∈ R2d

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we would like to instead use

(q(t),p(t)) ∈ R2d

where pj = ∂L∂qj

and this R2d is the Hamiltonian phase space with the conven-

tional Poisson bracket.This transformation between position-velocity and phase space is known as

the Legendre transform, and in good cases (for instance when L is quadraticin all the velocities) it is an isomorphism. In general though, this is not anisomorphism, with the Legendre transform often taking position-velocity spaceto a lower-dimensional subspace of phase space. Such cases require a much moreelaborate version of Hamiltonian formalism, known as “constrained Hamiltoniandynamics” and are not unusual: one example we will see later is that of theequations of motion of a free electromagnetic field (Maxwell’s equations).

Besides a phase space, for a Hamiltonian system one needs a Hamiltonianfunction. Choosing

h =

d∑j=1

pj qj − L(q, q)

will work, provided one can use the relation

pj =∂L

∂qj

to solve for the velocities qj and express them in terms of the momentum vari-ables. In that case, computing the differential of h one finds (for d = 1, thegeneralization to higher d is straightforward)

dh =pdq + qdp− ∂L

∂qdq − ∂L

∂qdq

=qdp− ∂L

∂qdq

So one has∂h

∂p= q,

∂h

∂q= −∂L

∂q

but these are precisely Hamilton’s equations since the Euler-Lagrange equationsimply

∂L

∂q=

d

dt

∂L

∂q= p

The Lagrangian formalism has the advantage of depending concisely just onthe choice of action functional, which does not distinguish time in the sameway that the Hamiltonian formalism does by its dependence on a choice ofHamiltonian function h. This makes the Lagrangian formalism quite useful inthe case of relativistic quantum field theories, where one would like to exploit thefull set of space-time symmetries, which can mix space and time directions. Onthe other hand, one loses the infinite dimensional group of symmetries of phase

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space for which the Poisson bracket is the Lie bracket (the so-called “canonicaltransformations”). In the Hamiltonian formalism we saw that the harmonicoscillator could be best understood using such symmetries, in particular theU(1) symmetry generated by the Hamiltonian function. The harmonic oscillatoris a more difficult problem in the Lagrangian formalism, where this symmetryis not manifest.

30.2 Path integrals

After the Legendre transform of a Lagrangian classical system to phase spaceand thus to Hamiltonian form, one can then apply the method of quantizationwe have discussed extensively earlier (known to physicists as “canonical quan-tization”). There is however a very different approach to quantization, whichcompletely bypasses the Hamiltonian formalism. This is the path integral for-malism, which is based upon a method for calculating matrix elements of thetime-evolution operator

〈qT |e−i~HT |q0〉

in the position eigenstate basis in terms of an integral over the space of pathsthat go from q0 to q1 in time T . Here |q0〉 is an eigenstate of Q with eigenvalueq0 (a delta-function at q0 in the position space representation), and |qT 〉 has Qeigenvalue qT (as in many cases, we’ll stick to d = 1 for this discussion). Thismatrix element has a physical interpretation as the amplitude for a particlestarting at q0 at t = 0 to have position qT at time T , with its norm-squaredgiving the probability density for observing the particle at position qT .

To try and derive a path-integral expression for this, one breaks up theinterval [0, T ] into N equal-sized sub-intervals and calculates

〈qT |(e−iN~HT )N |q0〉

If the Hamiltonian breaks up as H = K+V , the Trotter product formula showsthat

〈qT |e−i~HT |q0〉 = lim

N→∞〈qT |(e−

iN~KT e−

iN~V T )N |q0〉

If K(P ) can be chosen to depend only on the momentum operator P and V (Q)depends only on the operator Q then one can insert alternate copies of theidentity operator in the forms∫ ∞

−∞|q〉〈q|dq = 1,

∫ ∞−∞|p〉〈p|dp = 1

This gives a product of terms that looks like

〈qj |e−iN~K(P )T |pj〉〈pj |e−

iN~V (Q)T |qj−1〉

where the index j goes from 1 to N and the pj , qj variable will be integratedover. Such a term can be evaluated as

〈qj |pj〉〈pj |qj−1〉e−iN~K(pj)T e−

iN~V (qj−1)T

310

=1√2π~

ei~ qjpj

1√2π~

e−i~ qj−1jpje−

iN~K(pj)T e−

iN~V (qj−1)T

=1

2π~ei~pj(qj−qj−1)e−

iN~ (K(pj)+V (qj−1))T

The N factors of this kind give an overall factor of ( 12π~ )N times something

which is a discretized approximation to

ei~∫ T0

(pq−h(q(t),p(t)))dt

where the phase in the exponential is just the action. Taking into account theintegrations over qj and pj one should have something like

〈qT |e−i~HT |q0〉 = lim

N→∞(

1

2π~)N

N∏j=1

∫ ∞−∞

∫ ∞−∞

dpjdqjei~∫ T0

(pq−h(q(t),p(t)))dt

although one should not do the first and last integrals over q but fix the firstvalue of q to q0 and the last one to qT . One can try and interpret this sort ofintegration in the limit as an integral over the space of paths in phase space,thus a “phase space path integral”.

This is an extremely simple and seductive expression, apparently saying that,once the action S is specified, a quantum system is defined just by consideringintegrals ∫

Dγ ei~S[γ]

over paths γ in phase space, where Dγ is some sort of measure on this space ofpaths. Since the integration just involves factors of dpdq and the exponentialjust pdq and h, this formalism seems to share the same sort of invariance underthe infinite-dimensional group of canonical transformations (transformations ofthe phase space preserving the Poisson bracket) as the classical Hamiltonianformalism. It also appears to solve our problem with operator ordering am-biguities, since introducing products of P s and Q’s at various times will justgive a phase space path integral with the corresponding p and q factors in theintegrand, but these commute.

Unfortunately, we know from the Groenewold-van Hove theorem that thisis too good to be true. This expression cannot give a unitary representation ofthe full group of canonical transformations, at least not one that is irreducibleand restricts to what we want on transformations generated by linear functionsq and p. Another way to see the problem is that a simple argument showsthat by canonical transformations one can transform any Hamiltonian into afree-particle Hamiltonian, so all quantum systems would just be free particlesin some choice of variables. For the details of these arguments and a carefulexamination of what goes wrong, see chapter 31 of [50]. One aspect of theproblem is that, as a measure on the discrete sets of points qj , pj , points inphase space for successive values of j are not likely to be close together, sothinking of the integral as an integral over paths is not justified.

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When the Hamiltonian h is quadratic in the momentum p, the pj integralswill be Gaussian integrals that can be performed exactly. Equivalently, thekinetic energy part K of the Hamiltonian operator will have a kernel in positionspace that can be computed exactly. Using one of these, the pj integrals can beeliminated, leaving just integrals over the qj that one might hope to interpret asa path integral over paths not in phase space, but in position space. One finds,

if K = P 2

2m

〈qT |e−i~HT |q0〉 =

limN→∞

(i2π~TNm

)N2

√m

i2π~T

N∏j=1

∫ ∞−∞

dqjei~∑Nj=1(

m(qj−qj−1)2

2T/N−V (qj)

TN )

In the limit N →∞ the phase of the exponential becomes

S(γ) =

∫ T

0

dt(1

2m(q2)− V (q(t)))

One can try and properly normalize things so that this limit becomes an integral∫Dγ e

i~S[γ]

where now the paths γ(t) are paths in the position space.An especially attractive aspect of this expression is that it provides a simple

understanding of how classical behavior emerges in the classical limit as ~→ 0.The stationary phase approximation method for oscillatory integrals says that,for a function f with a single critical point at x = xc (i.e. f ′(xc) = 0) and for asmall parameter ε, one has

1√i2πε

∫ +∞

−∞dx eif/ε =

1√f ′′(c)

eif(xc)/ε(1 +O(ε))

Using the same principle for the infinite-dimensional path integral, with f = Sthe action functional on paths, and ε = ~, one finds that for ~ → 0 the pathintegral will simplify to something that just depends on the classical trajectory,since by the principle of least action, this is the critical point of S.

Such position-space path integrals do not have the problems of principleof phase space path integrals coming from the Groenewold-van Hove theorem,but they still have serious analytical problems since they involve an attempt tointegrate a wildly oscillating phase over an infinite-dimensional space. One doesnot naturally get a unitary result for the time evolution operator, and it is notclear that whatever results one gets will be independent of the details of howone takes the limit to define the infinite-dimensional integral.

Such path integrals though are closely related to integrals that are knownto make sense, ones that occur in the theory of random walks. There, a well-defined measure on paths does exist, Wiener measure. In some sense Wiener

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measure is what one gets in the case of the path integral for a free particle, buttaking the time variable t to be complex and analytically continuing

t→ it

So, one can use Wiener measure techniques to define the path integral, gettingresults that need to be analytically continued back to the physical time variable.

In summary, the path integral method has the following advantages:

• Study of the classical limit and “semi-classical” effects (quantum effectsat small ~) is straightforward.

• Calculations for free particles and for series expansions about the freeparticle limit can be done just using Gaussian integrals, and these are rela-tively easy to evaluate and make sense of, despite the infinite-dimensionalityof the space of paths.

• After analytical continuation, path integrals can be rigorously defined us-ing Wiener measure techniques, and often evaluated numerically even incases where no exact solution is known.

On the other hand, there are disadvantages:

• Some path integrals such as phase space path integrals do not at all havethe properties one might expect, so great care is required in any use ofthem.

• How to get unitary results can be quite unclear. The analytic continua-tion necessary to make path integrals well-defined can make their physicalinterpretation obscure.

• Symmetries with their origin in symmetries of phase space that aren’tjust symmetries of configuration space are difficult to see using the con-figuration space path integral, with the harmonic oscillator providing agood example. One can see such symmetries using the phase-space pathintegral, but this is not reliable.

Path integrals for anti-commuting variables can also be defined by analogywith the bosonic case, using the notion of fermionic integration discussed earlier.


For much more about Lagrangian mechanics and its relation to the Hamiltonianformalism, see [2]. More along the lines of the discussion here can be found inmost quantum mechanics and quantum field theory textbooks. For the pathintegral, Feynman’s original paper [14] or his book [15] are quite readable. Atypical textbook discussion is the one in chapter 8 of Shankar [53]. The book bySchulman [50] has quite a bit more detail, both about applications and aboutthe problems of phase-space path integrals. Yet another fairly comprehensivetreatment, including the fermionic case, is the book by Zinn-Justin [72].

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314

Chapter 31

Quantization ofInfinite-dimensional PhaseSpaces

Up until this point we have been dealing with finite-dimensional phase spacesand their quantization in terms of Weyl and Clifford algebras. We will now turnto the study of quantum systems (both bosonic and fermionic) correspondingto infinite-dimensional phase spaces. The phase spaces of interest are spacesof solutions of some partial differential equation, so these solutions are classi-cal fields. The corresponding quantum theory is thus called a “quantum fieldtheory”. In this chapter we’ll just make some general comments about the newphenomena that appear when one deals with such infinite-dimensional exam-ples, without going into any detail at all. Formulating quantum field theories ina mathematically rigorous way is a major and ongoing project in mathematicalphysics research, one far beyond the scope of this text. We will treat this subjectat a physicist’s level of rigor, while trying to give some hint of how one mightproceed with precise mathematical constructions when they exist. We will alsotry and indicate where there are issues that require much deeper or even stillunknown ideas, as opposed to those where the needed mathematical techniquesare of a conventional nature.

While finite-dimensional Lie groups and their representations are rather well-understood mathematical objects, this is not at all true for infinite-dimensionalLie groups, where mathematical results are rather fragmentary. For the caseof infinite-dimensional phase spaces, bosonic or fermionic, the symplectic or or-thogonal groups acting on these spaces will be infinite-dimensional. One wouldlike to find infinite-dimensional analogs of the role these groups and their rep-resentations play in quantum theory in the finite-dimensional case.

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31.1 Inequivalent irreducible representations

In our discussion of the Weyl and Clifford algebras in finite dimensions, an im-portant part of this story was the Stone-von Neumann theorem and its fermionicanalog, which say that these algebras each have only one interesting irreduciblerepresentation (the Schrodinger representation in the bosonic case, the spinorrepresentation in the fermionic case). Once we go to infinite dimensions, this isno longer true: there will be an infinite number of inequivalent irreducible rep-resentations, with no known complete classification of the possibilities. Beforeone can even begin to compute things like expectation values of observables,one needs to find an appropriate choice of representation, adding a new layer ofdifficulty to the problem that goes beyond that of just increasing the number ofdegrees of freedom.

To get some idea of how the Stone-von Neumann theorem can fail, onecan consider the Bargmann-Fock quantization of the harmonic oscillator with ddegrees of freedom, and note that it necessarily depends upon making a choiceof an appropriate complex structure J (see chapter 21), with the conventionalchoice denoted J0. Changing from J0 to a different J corresponds to changingthe definition of annihilation and creation operators (but in a manner thatpreserves their commutation relations). Physically, this entails a change in theHamiltonian and a change in the lowest-energy or vacuum state:

|0〉J0 → |0〉J

But |0〉J is still an element of the same state space H as |0〉J0 , and one getsthe same H by acting with annihilation and creation operators on |0〉J0 or on|0〉J . The two constructions of the same H correspond to unitarily equivalentrepresentations of the Heisenberg group H2d+1.

In the limit of d→∞, what can happen is that there can be choices of J suchthat acting with annihilation and creation operators on |0〉J0

and |0〉J gives twodifferent state spaces HJ0 and HJ , providing two inequivalent representations ofthe Heisenberg group. For quantum systems with an infinite number of degreesof freedom, one can thus have the same algebra of operators, but a differentchoice of Hamiltonian can give both a different vacuum state and a differentstate space H on which the operators act. This same phenomenon occurs bothin the bosonic and fermionic cases, as one goe to infinite dimensional Weyl orClifford algebras.

It turns out though that if one restricts the class of complex structures J toones not that different from J0, then one can recover a version of the Stone-vonNeumann theorem and have much the same behavior as in the finite-dimensionalcase. Note that for invertible linear map g on phase space, g acts on the complexstructure, taking

J0 → Jg = g · J0

One can define subgroups of the infinite-dimensional symplectic or orthogonalgroups as follows:

316

Definition (Restricted symplectic and orthogonal groups). The group of lineartransformations g of an infinite-dimensional symplectic vector space preservingthe symplectic structure and also satisfying the condition

tr(A†A) <∞

on the operatorA = [Jg, J0]

is called the restricted symplectic group and denoted Spres. The group of lineartransformations g of an infinite-dimensional inner-product space preserving theinner-product and satisfying the same condition as above on [Jg, J0] is called therestricted orthogonal group and denoted Ores.

An operator A satisfying tr(A†A) < ∞ is said to be a Hilbert-Schmidtoperator.

One then has the following replacement for the Stone-von Neumann theorem:

Theorem. Given two complex structures J1, J2 on an infinite-dimensional vec-tor space such that [J1, J2] is Hilbert-Schmidt, acting on the states

|0〉J1, |0〉J2

by annihilation and creation operators will give unitarily equivalent representa-tions of the Weyl algebra (in the bosonic case), or the Clifford algebra (in thefermionic case).

The standard reference for the proof of this statement is the original papersof Shale [51] and Shale-Stinespring [52]. A detailed discussion of the theoremcan be found in [40].

When g ∈ Spres one can construct an action of this group by automorphismson the algebra generated by annihilation and creation operators, by much thesame method as in the finite-dimensional case (see section 22.1). Elements ofthe Lie algebra of the group are represented as quadratic combinations of an-nihilation and creation operators, with the Hilbert-Schmidt condition ensuringthat these quadratic operators have well-defined commutation relations. Thisalso holds true for Ores and the fermionic annihilation and creation operators.

31.2 The anomaly and the Schwinger term

The groups Spres and Ores each have a subgroup of elements that commutewith J0 exactly, not just up to a Hilbert-Schmidt operator. One can constructa unitary representation of this group (which we’ll call U(∞)) using quadraticcombinations of annihilation and creation operators to get a representation ofthe Lie algebra, in much the same manner as in the finite-dimensional case ofsection 22.1. The group U(∞) will act trivially on the vacuum state |0〉J0

, andthe finite-dimensional groups of symmetries of quantum field theories (comingfrom, for example, the action of the rotation group on physical space) will be

317

subgroups of this group. For these, the problem to be discussed in this sectionwill not occur.

Recall though that knowing the action of the symplectic group as automor-phisms the Weyl algebra only determines its representation on the state spaceup to a phase factor. In the finite-dimensional case it turned out that this phasefactor could be chosen to be just a sign, giving a representation (the metaplecticrepresentation) that was a representation up to sign (and a true representationof a double cover of the symplectic group). In the infinite dimensional case ofSpres, it turns out that the phase factors cannot be reduced to signs, and theanalog of the metaplectic representation is a representation of Spres only up toa phase. To get a true representation, one needs to extend Spres to a larger

group Spres that is not just a cover, but has an extra dimension.In terms of Lie algebras one has

spres = spres ⊕R

with elements non-zero only in the R direction commuting with everything else.For all commutation relations, there are now possible scalar terms to keep trackof. In the finite dimensional case we saw that such terms would occur when weused normal-ordered operators (an example is the the shift by the scalar 1

2 in theHamiltonian operator for a harmonic oscillator with one degree of freedom), butwithout normal-ordering no such terms were needed. In the infinite-dimensionalcase normal-ordering is needed to avoid having a representation that acts onstates like |0〉J0

by an infinite phase change, and one can not eliminate theeffect of normal-ordering by making a well-defined finite phase change on theway the operators act.

Commuting two elements of the Lie sub-algebra u(∞) will not give a scalarfactor, but such factors can occur when one commutes the action of elements ofspres not in u(∞), in which case they are known as “Schwinger terms”. Alreadyin finite dimensions, we saw that commuting the action a2 with that of (a†)2

gave a scalar factor with respect to the normal ordered a†a operator (see section21.2), and in infinite dimensions, it is this that cannot be redefined away.

This phenomenon of new scalar terms in the commutation relations of theoperators in a quantum theory coming from a Lie algebra representation isknown as an “anomaly”, and while we have described it for the bosonic case,much the same thing happens in the fermionic case for the Lie algebra ores.This is normally considered to be something that happens due to quantiza-tion, with the “anomaly” the extra scalar terms in commutation relations notthere in the corresponding classical Poisson bracket relations. From anotherpoint of view this is a phenomenon coming not from quantization, but frominfinite-dimensionality, already visible in Poisson brackets when one makes achoice of complex structure on the phase space. It is the occurrence for infinite-dimensional phase spaces of certain inherently different ways of choosing thecomplex structure that is relevant. We will see in later chapters that in quan-tum field theories one often wants to choose a complex structure on an infinitedimensional space of solutions of a classical field equation by taking positive

318

energy complexified solutions to be eigenvectors of the complex structure witheigenvalue +i, those with negative energy to have eigenvalue −i, and it is thischoice of complex structure that introduces the anomaly phenomenon.

31.3 Higher order operators and renormaliza-tion

We have generally restricted ourselves to considering only products of the fun-damental position and momentum operators of degree less than or equal to two,since it is these that have an interpretation as the operators of a Lie algebrarepresentation. By the Groenewold-van Hove theorem, higher-order products ofposition and momentum variables have no unique quantization (the operator-ordering problem). In the finite dimensional case one can of course considerhigher-order products of operators, for instance systems with Hamiltonian op-erators of higher order than quadratic. Since the Stone-von Neumann theoremensures a unique representation, such operators will have a well-defined eigen-value problem. Unlike the quadratic case, typically no exact solution for eigen-vectors and eigenvalues will exist, but various approximation methods may beavailable. In particular, for Hamiltonians that are quadratic plus a term witha small parameter, perturbation theory methods can be used to compute apower-series approximation in the small parameter. This is an important topicin physics, covered in detail in the standard textbooks.

The standard approach to quantization of infinite-dimensional systems isto begin with “regularization”, somehow modifying the system to only havea finite-dimensional phase space. One quantizes this theory, taking advantageof the uniqueness of its representation, then tries to take a limit that recoversthe infinite-dimensional system. Such a limit will generally be quite singular,leading to an infinite result, and the process of manipulating these potentialinfinities is called “renormalization”. Techniques for taking limits of this kindin a manner that leads to a consistent and physically sensible result typicallytake up a large part of standard quantum field theory textbooks. For manytheories, no appropriate such techniques are known, and conjecturally none arepossible. For others there is good evidence that such a limit can be successfullytaken, but the details of how to do this remain unknown (with for instance a$1 million Millenium Prize offered for showing rigorously this is possible in thecase of Yang-Mills gauge theory).

In succeeding chapters we will go on to study a range of quantum fieldtheories, but due to the great complexity of the issues involved, will not addresswhat happens for non-quadratic Hamiltonians. Physically this means that wewill just be able to study theories of free particles, although with methods thatgeneralize to particles moving in non-trivial background classical fields. Fora treatment of the subject that includes interacting quantized multi-particlesystems, one of the conventional textbooks will be needed to supplement whatis here.

319


Berezin’s The Method of Second Quantization [5] develops in detail the infinite-dimensional version of the Bargmann-Fock construction, both in the bosonicand fermionic cases. Infinite-dimensional versions of the metaplectic and spinorrepresentations are given there in terms of operators defined by integral kernels.For a discussion of the infinite-dimensional Weyl and Clifford algebras, togetherwith a realization of their automorphism groups Spres and Ores (and the corre-sponding Lie algebras) in terms of annihilation and creation operators acting onthe infinite-dimensional metaplectic and spinor representations, see [40]. Thebook [43] contains an extensive discussion of the groups Spres and Ores andthe infinite-dimensional version of their metaplectic and spinor representations.It emphasizes the origin of novel infinite-dimensional phenomena here in thenature of the complex structures of interest in infinite dimensional examples.

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Chapter 32

Multi-particle Systems andNon-relativistic QuantumFields

The quantum mechanical systems we have studied so far describe a finite num-ber of degrees of freedom, which may be of a bosonic or fermionic nature. Inparticular we have seen how to describe a quantized free particle moving inthree-dimensional space. By use of the notion of tensor product, we can thendescribe any particular fixed number of such particles. We would, however, likea formalism capable of conveniently describing an arbitrary number of parti-cles. From very early on in the history of quantum mechanics, it was clearthat at least certain kinds of particles, photons, were most naturally describednot one by one, but by thinking of them as quantized excitations of a classicalsystem with an infinite number of degrees of freedom: the electromagnetic field.In our modern understanding of fundamental physics not just photons, but allelementary particles are best described in this way.

Conventional textbooks on quantum field theory often begin with relativis-tic systems, but we’ll start instead with the non-relativistic case. We’ll study asimple quantum field theory that extends the conventional single-particle quan-tum systems we have dealt with so far to deal with multi-particle systems. Thisversion of quantum field theory is what gets used in condensed matter physics,and is in many ways simpler than the relativistic case, which we’ll take up in alater chapter.

Quantum field theory is a large and complicated subject, suitable for a full-year course at an advanced level. We’ll be giving only a very basic introduc-tion, mostly just considering free fields, which correspond to systems of non-interacting particles. Most of the complexity of the subject only appears whenone tries to construct quantum field theories of interacting particles. A remark-able aspect of the theory of free quantum fields is that in many ways it is littlemore than something we have already discussed in great detail, the quantum

321

harmonic oscillator problem. However, the classical harmonic oscillator phasespace that is getting quantized in this case is an infinite dimensional one, thespace of solutions to the free particle Schrodinger equation. To describe multiplenon-interacting fermions, we just need to use fermionic oscillators.

For simplicity we’ll set ~ = 1 and start with the case of a single spatialdimension. We’ll also begin using x to denote a spatial variable instead of the qconventional when this is the coordinate variable in a finite-dimensional phasespace.

32.1 Multi-particle quantum systems as quantaof a harmonic oscillator

It turns out that quantum systems of identical particles are best understood bythinking of such particles as quanta of a harmonic oscillator system. We willbegin with the bosonic case, then later consider the fermionic case, which usesthe fermionic oscillator system.

32.1.1 Bosons and the quantum harmonic oscillator

A fundamental postulate of quantum mechanics is that given a space of statesH1 describing a bosonic single particle, a collection of N particles is describedby

(H1 ⊗ · · · ⊗ H1︸︷︷︸N−times

)S

where the superscript S means we take elements of the tensor product invariantunder the action of the group SN by permutation of the factors. We want toconsider state spaces containing an arbitrary number of particles, so we define

Definition (Bosonic Fock space, the symmetric algebra). Given a complex vec-tor space V , the symmetric Fock space is defined as

FS(V ) = C⊕ V ⊕ (V ⊗ V )S ⊕ (V ⊗ V ⊗ V )S ⊕ · · ·

This is known to mathematicians as the “symmetric algebra” S∗(V ), with

SN (V ) = (V ⊗ · · · ⊗ V︸︷︷︸N−times

)S

(recall chapter 9)

A quantum harmonic oscillator with d degrees of freedom has a state spaceconsisting of linear combinations of states with N “quanta” (i.e. states one getsby applying N creation operators to the lowest energy state), for N = 0, 1, 2, . . ..We have seen in our discussion of the quantization of the harmonic oscillator thatin the Bargmann-Fock representation, the state space is just C[z1, z2, . . . , zd],the space of polynomials in d complex variables.

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The part of the state space with N quanta has dimension(N + d− 1

N

)which grows with N . This is just the binomial coefficient giving the numberof d-variable monomials of degree N . The quanta of a harmonic oscillator areindistinguishable, which corresponds to the fact that the space of states withN quanta can be identified with the symmetric part of the tensor product of Ncopies of Cd.

More precisely, what one has is

Theorem. Given a vector space V , there is an isomorphism of algebras betweenthe symmetric algebra S∗(V ∗) (where V ∗ is the dual vector space to V ) and thealgebra C[V ] of polynomial functions on V .

We won’t try and give a detailed proof of this here, but one can exhibit theisomorphism explicitly on generators. If zj ∈ V ∗ are the coordinate functionswith respect to a basis ej of V (i.e. zj(ek) = δjk), they give a basis of V ∗ whichis also a basis of the linear polynomial functions on V . For higher degrees, onemakes the identification

zj ⊗ · · · ⊗ zj︸︷︷︸N−times

∈ SN (V ∗)↔ zNj

between N -fold symmetric tensor products and monomials in the polynomialalgebra.

Besides describing states of multiple identical quantum systems as polyno-mials or as symmetric parts of tensor products, a third description is useful.This is the so-called “occupation number representation”, where states are la-beled by d non-negative integers nj = 0, 1, 2, · · · , with an identification with thepolynomial description as follows:

|n1, n2, . . . , nj , . . . , nd〉 ↔ zn11 zn2

2 · · · znjj · · · z

ndd

So, starting with a single particle state space H1, we have three equivalentdescriptions of the Fock space describing multi-particle bosonic states

• As a symmetric algebra S∗(H∗1) defined in terms of tensor products of theH∗1, the dual vector space to H1.

• As polynomial functions on H1.

• As linear combinations of states labeled by occupation numbers nj =0, 1, 2, . . ..

For each of these descriptions, one can define d annihilation or creationoperators aj , a

†j , satisfying the canonical commutation relations, and these will

generate an algebra of operators on the Fock space.

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32.1.2 Fermions and the fermionic oscillator

For the case of fermionic particles, there’s an analogous Fock space constructionusing tensor products:

Definition (Fermionic Fock space, exterior algebra). Given a complex vectorspace V , the Fermionic Fock space is

FA(V ) = C⊕ V ⊕ (V ⊗ V )A ⊕ (V ⊗ V ⊗ V )A ⊕ · · ·

where the superscript A means the subspace of the tensor product that justchanges sign under interchange of two factors. This is known to mathemati-cians as the “exterior algebra” Λ∗(V ), with

ΛN (V ) = (V ⊗ · · · ⊗ V︸︷︷︸N−times

)A

As in the bosonic case, one can interpret Λ∗(V ∗) as polynomials on V , butnow polynomials in anti-commuting variables.

For the case of fermionic particles with single particle state space H1, onecan again define the multi-particle state space in three equivalent ways:

• Using tensor products and anti-symmetry, as the exterior algebra Λ∗(H∗1).

• As polynomial functions in anti-commuting coordinates on H1. One canalso think of such polynomials as anti-symmetric multilinear functions onthe product of copies of H1.

• In terms of occupation numbers nj , where now the only possibilities arenj = 0, 1.

For each of these descriptions, one can define d annihilation or creation oper-ators aj , a

†j satisfying the canonical anti-commutation relations, and these will

generate an algebra of operators on the Fock space.

32.2 Solutions to the free particle Schrodingerequation

To describe multi-particle quantum systems we will take as our single-particlestate space H1 the space of solutions of the free-particle Schrodinger equation,as already studied in chapter 10. As we saw in that chapter, one can use a “finitebox” normalization, which gives a discrete, countable basis for this space andthen try and take the limit of infinite box size. To fully exploit the symmetries ofphase space though, we will need the “continuum normalization”, which requiresconsidering not just functions but distributions.

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32.2.1 Box normalization

Recall that for a free particle in one dimension the state space H consists ofcomplex-valued functions on R, with observables the self-adjoint operators formomentum

P = −i ddx

and energy (the Hamiltonian)

H =P 2

2m= − 1

2m

d2

dx2

Eigenfunctions for both P and H are the functions of the form

ψp(x) ∝ eipx

for p ∈ R, with eigenvalues p for P and p2

2m for H.Note that these eigenfunctions are not normalizable, and thus not in the

conventional choice of state space as L2(R). One way to deal with this issue isto do what physicists sometimes refer to as “putting the system in a box”, byimposing periodic boundary conditions

ψ(x+ L) = ψ(x)

for some number L, effectively restricting the relevant values of x to be consid-ered to those on an interval of length L. For our eigenfunctions, this conditionis

eip(x+L) = eipx

so we must haveeipL = 1

which implies that

p =2π

Ll

for l an integer. Then p will take on a countable number of discrete valuescorresponding to the l ∈ Z, and

|l〉 = ψl(x) =1√Leipx =

1√Lei

2πlL x

will be orthornormal eigenfunctions satisfying

〈l′|l〉 = δll′

This “box” normalization is one form of what physicists call an “infrared cutoff”,a way of removing degrees of freedom that correspond to arbitrarily large sizes,in order to make a problem well-defined. To get a well-defined problem onestarts with a fixed value of L, then one studies the limit L→∞.

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The number of degrees of freedom is now countable, but still infinite. In orderto get a completely well-defined problem, one typically needs to first make thenumber of degrees of freedom finite. This can be done with an additional cutoff,an “ultraviolet cutoff”, which means restricting attention to |p| ≤ Λ for somefinite Λ, or equivalently |l| < ΛL

2π . This makes the state space finite dimensionaland one then studies the Λ→∞ limit.

For finite L and Λ our single-particle state space H1 is finite dimensional,with orthonormal basis elements ψl(x). An arbitrary solution to the Schrodingerequation is then given by

ψ(x, t) =

+ ΛL2π∑

l=−ΛL2π

αlei 2πlL xe−i

4π2l2

2mL2 t

for arbitrary complex coefficients αl and can be completely characterized by itsinitial value at t = 0

ψ(x, 0) =

+ ΛL2π∑

l=−ΛL2π

α(l)ei2πlL x

Vectors in H1 have coordinates αl ∈ C with respect to our chosen basis, andthese coordinates are in the dual space H∗1.

Multi-particle states are now described by the Fock spaces FS(H∗1) or FA(H∗1),depending on whether the particles are bosons or fermions. In the occupationnumber representation of the Fock space, orthonormal basis elements are

| · · · , npj−1 , npj , npj+1 , · · · 〉

where the subscript j indexes the possible values of the momentum p (which arediscretized in units of 2π

L , and in the interval [−Λ,Λ]). The occupation numbernpj is the number of particles in the state with momentum pj . In the bosoniccase it takes values 0, 1, 2, · · · ,∞, in the fermionic case it takes values 0 or 1.The state with all occupation numbers equal to zero is denoted

| · · · , 0, 0, 0, · · · 〉 = |0〉

and called the “vacuum” state.For each pj we can define annihilation and creation operators apj and a†pj .

These satisfy the commutation relations

[apj , a†pk

] = δjk

and act on states in the occupation number representation as

apj | · · · , npj−1, npj , npj+1

, · · · 〉 =√npj | · · · , npj−1

, npj − 1, npj+1, · · · 〉

a†pj | · · · , npj−1, npj , npj+1

, · · · 〉 =√npj + 1| · · · , npj−1

, npj + 1, npj+1, · · · 〉

Observables one can build out of these operators include

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• The number operator

N =∑k

a†pkapk

which will have as eigenvalues the total number of particles

N | · · · , npj−1 , npj , npj+1 , · · · 〉 = (∑k

npk)| · · · , npj−1 , npj , npj+1 , · · · 〉

• The momentum operator

P =∑k

pka†pkapk

with eigenvalues the total momentum of the multiparticle system.

P | · · · , npj−1 , npj , npj+1 , · · · 〉 = (∑k

nkpk)| · · · , npj−1 , npj , npj+1〉

• The Hamiltonian

H =∑k

p2k

2ma†pkapk

which has eigenvalues the total energy

H| · · · , npj−1 , npj , npj+1 , · · · 〉 = (∑k

nkp2k

2m)| · · · , npj−1 , npj , npj+1 , · · · 〉

With ultraviolet and infrared cutoffs in place, the possible values of pj arefinite in number, H1 is finite dimensional and this is nothing but the standardquantized harmonic oscillator (with a Hamiltonian that has different frequencies

ω(pj) =p2j

2m

for different values of j). In the limit as one or both cutoffs are removed,H1 becomes infinite dimensional, the Stone-von Neumann theorem no longerapplies, and we are in the situation discussed in chapter 31. State spaces withdifferent choices of vacuum state |0〉 can be unitarily inequivalent, with not justthe dynamics of states in the state space dependent on the Hamiltonian, but thestate space itself depending on the Hamiltonian (through the characterizationof |0〉 as lowest energy state). Even for the free particle, we have here definedthe Hamiltonian as the normal-ordered version, which for finite dimensional H1

differs from the non-normal-ordered one just by a constant, but as cut-offs areremoved this constant becomes infinite, requiring careful treatment of the limit.

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32.2.2 Continuum normalization

A significant problem introduced by using cutoffs such as the box normalizationis that these ruin some of the space-time symmetries of the system. The one-particle space with an infrared cutoff is a space of functions on a discrete set ofpoints, and this set of points will not have the same symmetries as the usualcontinuous momentum space (for instance in three dimensions it will not carryan action of the rotation group SO(3)). In our study of quantum field theorywe would like to exploit the action of space-time symmetry groups on the statespace of the theory, so need a formalism that preserves such symmetries.

In our earlier discussion of the free particle, we saw that physicists oftenwork with a “continuum normalization” such that

|p〉 = ψp(x) =1√2πeipx, 〈p′|p〉 = δ(p− p′)

where formulas such as the second one need to be interpreted in terms of dis-tributions. In quantum field theory we want to be able to think of each valueof p as corresponding to a classical degree of freedom that gets quantized, andthis “continuum” normalization will then correspond to an uncountable numberof degrees of freedom, requiring great care when working in such a formalism.This will however allow us to readily see the action of space-time symmetrieson the states of the quantum field theory and to exploit the duality betweenposition and momentum space embodied in the Fourier transform.

In the continuum normalization, an arbitrary solution to the free-particleSchrodinger equation is given by

ψ(x, t) =1√2π

∫ ∞−∞

α(p)eipxe−ip2

2m tdp

for some function complex-valued function α(p) on momentum space. Suchsolutions are in one-to-one correspondence with initial data

ψ(x, 0) =1√2π

∫ ∞−∞

α(p)eipxdp

This is exactly the Fourier inversion formula, expressing a function ψ(x, 0) in

terms of its Fourier transform ψ(x, 0)(p) = α(p). Note that we want to con-sider not just square integrable functions α(p), but non-integrable functionslike α(p) = 1 (which corresponds to ψ(x, 0) = δ(x)), and distributions such asα(p) = δ(p), which corresponds to ψ(x, 0) = 1.

We will generally work with this continuum normalization, taking as oursingle-particle spaceH1 the space of complex valued functions ψ(x, 0) on R. Onecan think of the |p〉 as an orthornomal basis of H1, with α(p) the coordinatefunction for the |p〉 basis vector. α(p) is then an element of H∗1, the linearfunction on H1 given by taking the coefficient of the |p〉 basis vector.

Quantization should take α(p) ∈ H∗1 to corresponding annihilation and cre-ation operators a(p), a†(p). Such operators though need to be thought of as

328

operator-valued distributions: what is really a well-defined operator is not a(p),but ∫ +∞

−∞f(p)a(p)dp

for sufficiently well-behaved functions f(p). From the point of view of quanti-zation of H∗1, it is vectors in H∗1 that one can write in the form

α(f) =

∫ +∞

−∞f(p)α(p)dp

that have well-defined quantizations as operators.Typically though we will follow the the standard physicist’s practice of writ-

ing formulas which are straight-forward generalizations of the finite-dimensionalcase, with sums becoming integrals. Some examples are

N =

∫ +∞

−∞a(p)†a(p)dp

for the number operator,

P =

∫ +∞

−∞pa(p)†a(p)dp

for the momentum operator, and

H =

∫ +∞

−∞

p2

2ma(p)†a(p)dp

for the Hamiltonian operator.To properly manipulate these formulas one must be aware that they should

be interpreted as formulas for distributions, and that in particular products ofdistributions need to be treated with care. Expressions like a(p)†a(p) will onlygive sensible operators when integrated against well-behaved functions. Whatwe really have is for instance an operator N(f) which we will write formally as

N(f) =

∫ +∞

−∞a(p)†a(p)dp

but which is only defined for some particularly well-behaved f (and in particularis not defined for the non-integrable choice of f = 1).

32.3 Quantum field operators

The formalism developed so far works well to describe states of multiple freeparticles, but does so purely in terms of states with well-defined momenta, withno information at all about their position. To get operators that know aboutposition, one can Fourier transform the annihilation and creation operators formomentum eigenstates as follows (we’ll begin with the box normalization):

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Definition (Quantum field operator). The quantum field operators for the freeparticle system are

ψ(x) =∑p

ψp(x)ap =∑p

1√Leipxap

and its adjoint

ψ†(x) =∑p

ψ∗p(x)a†k =∑k

1√Le−ikxa†k

(where k takes the discrete values kj = 2πjL ).

Note that these are not self-adjoint operators, and thus not themselves ob-servables. To get some idea of their behavior, one can calculate what they doto the vacuum state. One has

ψ(x)|0〉 = 0

ψ†(x)|0〉 =1√L

∑p

e−ipx| · · · , 0, np = 1, 0, · · · 〉

While this sum makes sense as long as it is finite, when cutoffs are removed it isclear that ψ†(x) will have a rather singular limit as an infinite sum of operators.It can be in some vague sense thought of as the (ill-defined) operator that createsa particle localized precisely at x.

The field operators allow one to recover conventional wave-functions, forsingle and multiple-particle states. One sees by orthonormality of the occupationnumber basis states that

〈· · · , 0, np = 1, 0, · · · |ψ†(x)|0〉 =1√Le−ipx = ψp(x)

the complex conjugate wave function of the single-particle state of momentump. An arbitrary one particle state |Ψ1〉 with wave function ψ(x) is a linearcombination of such states, and taking complex conjugates one finds

〈0|ψ(x)|Ψ1〉 = ψ(x)

Similarly, for a two-particle state of identical particles with momenta pj1 andpj2 one finds

〈0|ψ(x1)ψ(x2)| · · · , 0, npj1 = 1, 0, · · · , 0, npj2 = 1, 0, · · · 〉 = ψpj1 ,pj2 (x1, x2)

whereψpj1 ,pj2 (x1, x2)

is the wavefunction (symmetric under interchange of x1 and x2 for bosons) forthis two particle state. For a general two-particle state |Ψ2〉 with wavefunctionψ(x1, x2) one has

〈0|ψ(x1)ψ(x2)|Ψ2〉 = ψ(x1, x2)

330

and one can easily generalize this to see how field operators are related to wave-functions for an arbitrary number of particles.

Cutoffs ruin translational invariance and calculations with them quickly be-come difficult. We’ll now adopt the physicist’s convention of working directlyin the continuous case with no cutoff, at the price of having formulas that onlymake sense as distributions. One needs to be aware that the correct interpreta-tion of such formulas may require going back to the cutoff version.

In the continuum normalization we take as normalized eigenfunctions for thefree particle

|p〉 = ψp(x) =1√2πeipx

with

〈p′|p〉 =1

2π

∫ ∞−∞

ei(p−p′)xdx = δ(p− p′)

The annihilation and creation operators satisfy

[a(p), a†(p′)] = δ(p− p′)

The field operators are then

ψ(x) =1√2π

∫ ∞−∞

eipxa(p)dx

ψ†(x) =1√2π

∫ ∞−∞

e−ipxa†(p)dx

and one can compute the commutators

[ψ(x), ψ(x′)] = [ψ†(x), ψ†(x′)] = 0

[ψ(x), ψ†(x′)] =1

2π

∫ ∞−∞

∫ ∞−∞

eipxe−ip′x′ [a(p), a†(p′)]dpdp′

=1

2π

∫ ∞−∞

∫ ∞−∞

eipxe−ip′x′δ(p− p′)dpdp′

=1

2π

∫ ∞−∞

eip(x−x′)dp

=δ(x− x′)

One should really think of such continuum-normalized field operators asoperator-valued distributions, with the distributions defined on some appropri-ately chosen function space, for instance the Schwartz space of functions suchthat the function and its derivatives fall off faster than any power at ±∞. Givensuch functions f, g, one gets operators

ψ(f) =

∫ ∞−∞

f(x)ψ(x)dx, ψ†(g) =

∫ ∞−∞

g(x)ψ†(x)dx

331

and the commutator relation above means

[ψ(f), ψ†(g)] =

∫ ∞−∞

∫ ∞−∞

f(x)g(x′)δ(x− x′)dxdx′ =

∫ ∞−∞

f(x)g(x)dx

A major source of difficulties when manipulating quantum fields is that powersof distributions are not necessarily defined, so one has trouble making sense ofrather innocuous looking expressions like

(ψ(x))4

There are observables that one can define simply using field operators. Theseinclude:

• The number operator N . One can define a number density operator

n(x) = ψ†(x)ψ(x)

and integrate it to get an operator with eigenvalues the total number ofparticles in a state

N =

∫ ∞−∞

n(x)dx

=

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

1√2πe−ip

′xa†(p′)1√2πeipxa(p)dpdp′dx

=

∫ ∞−∞

∫ ∞−∞

δ(p− p′)a†(p′)a(p)dpdp′

=

∫ ∞−∞

a†(p)a(p)dp

• The total momentum operator P . This can be defined in terms of fieldoperators as

P =

∫ ∞−∞

ψ†(x)(−i ddx

)ψ(x)dx

=

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

1√2πe−ip

′xa†(p′)(−i)(ip) 1√2πeipxa(p)dpdp′dx

=

∫ ∞−∞

∫ ∞−∞

δ(p− p′)pa†(p′)a(p)dpdp′

=

∫ ∞−∞

pa†(p)a(p)dp

• The Hamiltonian H. This can be defined much like the momentum, justchanging

−i ddx→ − 1

2m

d2

dx2

332

to find

H =

∫ ∞−∞

ψ†(x)(− 1

2m

d2

dx2)ψ(x)dx =

∫ ∞−∞

p2

2ma†(p)a(p)dp

All of these formulas really need to be interpreted as operator valued distribu-tions: what really makes sense is not N , but N(f) for some class of functionsf , which can formally be written as

N(f) =

∫ ∞−∞

f(x)ψ†(x)ψ(x)dx

We will see that one can more generally use quadratic expressions in fieldoperators to define an observable O corresponding to a one-particle quantummechanical observable O by

O =

∫ ∞−∞

ψ†(x)Oψ(x)dx

In particular, to describe an arbitrary number of particles moving in an externalpotential V (x), one takes the Hamiltonian to be

H =

∫ ∞−∞

ψ†(x)(− 1

2m

d2

dx2+ V (x))ψ(x)dx

If one can solve the one-particle Schrodinger equation for a complete set oforthonormal wave functions ψn(x), one can describe this quantum system usingthe same techniques as for the free particle. A creation-annihilation operatorpair an, a

†n is associated to each eigenfunction, and quantum fields are defined

by

ψ(x) =∑n

ψn(x)an, ψ†(x) =∑n

ψn(x)a†n

For Hamiltonians just quadratic in the quantum fields, quantum field the-ories are quite tractable objects. They are in some sense just free quantumoscillator systems, with all of their symmetry structure intact, but taking thenumber of degrees of freedom to infinity. Higher order terms though makequantum field theory a difficult and complicated subject, one that requires ayear-long graduate level course to master basic computational techniques, andone that to this day resists mathematician’s attempts to prove that many ex-amples of such theories have even the basic expected properties. In the theoryof charged particles interacting with an electromagnetic field, when the electro-magnetic field is treated classically one still has a Hamiltonian quadratic in thefield operators for the particles. But if the electromagnetic field is treated as aquantum system, it acquires its own field operators, and the Hamiltonian is nolonger quadratic in the fields, a vastly more complicated situation described asan“interacting quantum field theory”.

Even if one restricts attention to the quantum fields describing one kind ofparticles, there may be interactions between particles that add terms to the

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Hamiltonian, and these will be higher order than quadratic. For instance, ifthere is an interaction between such particles described by an interaction energyv(y − x), this can be described by adding the following quartic term to theHamiltonian

1

2

∫ ∞−∞

∫ ∞−∞

ψ†(x)ψ†(y)v(y − x)ψ(y)ψ(x)dxdy

The study of “many-body” quantum systems with interactions of this kind is amajor topic in condensed matter physics.

One can easily extend the above to three spatial dimensions, getting field op-erators ψ(x) and ψ†(x), defined by integrals over three-dimensional momentumspace. For instance, in the continuum normalization

ψ(x) = (1

2π)

32

∫R3

eip·xa(p)d3p

and the Hamiltonian for the free field is

H =

∫R3

ψ†(x)(− 1

2m∇2)ψ(x)d3x

More remarkably, one can also very easily write down theories of quantumsystems with an arbitrary number of fermionic particles, just by changing com-mutators to anti-commutators for the creation-annihilation operators and usingfermionic instead of bosonic oscillators. One gets fermionic fields that satisfyanti-commutation relations

[ψ(x), ψ†(x′)]+ = δ(x− x′)

and states that in the occupation number representation have nk = 0, 1.


This material is discussed in essentially any quantum field theory textbook.Many do not explicitly discuss the non-relativistic case, two that do are [23]and [31]. Two books aimed at mathematicians that cover the subject muchmore carefully than those for physicists are [20] and [11].

A good source for learning about quantum field theory from the point of viewof non-relativistic many-body theory is Feynman’s lecture notes on statisticalmechanics [18].

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Chapter 33

Field Quantization andDynamics forNon-relativistic QuantumFields

In finite dimensions, we saw that we could think of phase space M as the spaceparametrizing solutions to the equations of motion of a classical system, thatlinear functions on this space carried the structure of a Lie algebra (the Heisen-berg Lie algebra), and that quantization was given by finding an irreducibleunitary representation of this Lie algebra.

Instead of motivating the definition of quantum fields by starting with an-nihilation and creation operators for a free particle of fixed momentum, onecan more simply just define them as what one gets by taking the space H1 ofsolutions of the free single particle Schrodinger equation as a classical phasespace, and quantizing to get a unitary representation (of a Heisenberg algebrathat is now infinite-dimensional). This procedure is sometimes called “secondquantization”, with “first quantization” what was done when one started withthe classical phase space for a single particle and quantized to get the space H1

of wavefunctions.In this chapter we’ll consider the properties of quantum fields from this point

of view, including seeing how quantization of classical Hamiltonian dynamicsgives the dynamics of quantum fields.

33.1 Quantization of classical fields

The Schrodinger equation is first-order in time, so solutions are determined bythe initial values ψ(x) = ψ(x, 0) of the wave-function at t = 0 and elements ofthe spaceH1 of solutions can be identified with their initial-value data, the wave-

335

function at t = 0. Note that this is unlike typical finite-dimensional classicalmechanical systems, where the equation of motion is second-order in time, withsolutions determined by two pieces of initial-value data, the coordinates andmomenta (since one needs initial velocities as well as positions). Taking M = H1

as a classical phase space, it has the property that there is no natural splitting ofcoordinates into position-like variables and momentum-like variables, and thusno natural way of setting up an infinite-dimensional Schrodinger representationwhere states would be functionals of position-like variables.

On the other hand, since wave-functions are complex valued, H1 is alreadya complex vector space, and we can quantize by the Bargmann-Fock methodusing this complex structure. This is quite unlike our previous examples ofquantization, where we started with a real phase space and needed to choose acomplex structure (to get annihilation and creation operators).

Using Fourier transforms we can think of H1 either as a space of functionsψ of position x, or as a space of functions ψ of momentum p. This correspondsto two possible choices of orthonormal bases of the function space H1: the |p〉(plane waves of momentum p) or the |x〉 (delta-functions at position x). Inthe finite dimensional case it is the coordinate functions qj , pj on phase space,which lie in the dual phase space M = M∗, that get mapped to operatorsQj , Pj under quantization. Here what corresponds to the qj , pj is either theα(p) (coordinates with respect to the |p〉 basis) which quantize to annihilationoperators a(p), or the ψ(x) (field value at x, coordinates with respect to the |x〉basis) which quantize to field operators ψ(x).

As described in chapter 32 though, what is really well-defined is not thequantization of ψ(x), but of ψ(f) for some class of functions f . ψ(x) is thelinear function on the space of solutions H1 given by

ψ(x) : ψ ∈ H1 → ψ(x, 0)

but to get a well-defined operator, one wants the quantization not of this, butof elements of H∗1 of the form

ψ(f) : ψ ∈ H1 →∫ ∞−∞

f(x)ψ(x, 0)dx

To get all elements of H∗1 we will also need the ψ(x) and

ψ(g) : ψ ∈ H1 →∫ ∞−∞

g(x)ψ(x, 0)dx

Despite the potential for confusion, we will write ψ(x) for the distribution givenby evaluation at x, which corresponds to taking f to be the delta-functionδ(x− x′). This convenient notational choice means that one needs to be awarethat ψ(x) may be a complex number, or may be the “evaluation at x” linearfunction on H1.

So quantum fields should be “operator-valued distributions” and the propermathematical treatment of this situation becomes quite challenging, with one

336

class of problems coming from the theory of distributions. What class of func-tions should appear in the space H1? What class of linear functionals on thisspace should be used? What properties should the operators ψ(f) satisfy?These issues are far beyond what we can discuss here, and they are not purelymathematical, with the fact that the product of two distributions does not havean unambiguous sense one indication of the difficulties of quantum field theory.

To understand the Poisson bracket structure on functions on H1, one shouldrecall that in the Bargmann-Fock quantization we found that, choosing a com-plex structure and complex coordinates zj on phase space, the non-zero Poissonbracket relations were

zj , izl = δjl

If we use the |p〉 basis for H1, our complex coordinates will be α(p), α(p)with Poisson bracket

α(p), α(p′) = α(p), α(p′) = 0, α(p), iα(p′) = δ(p− p′)

and under quantization we have

α(p)→ a(p), α(p)→ a†(p), 1→ 1

Multiplying these operators by −i gives a Heisenberg Lie algebra representationΓ′ (unitary on the real and imaginary parts of α(p)) with

Γ′(α(p)) = −ia(p), Γ′(α(p)) = −ia†(p)

and the commutator relations

[a(p), a(p′)] = [a†(p), a†(p′)] = 0, [a(p), a†(p′)] = δ(p− p′)1

Using instead the |x〉 basis forH1, our complex coordinates will be ψ(x), ψ(x)with Poisson brackets

ψ(x), ψ(x′) = ψ(x), ψ(x′) = 0, ψ(x), iψ(x′) = δ(x− x′)

and quantization takes

ψ(x)→ −iψ(x), ψ(x)→ −iψ†(x), 1→ −i1

This gives a Heisenberg Lie algebra representation Γ′ (unitary on the real andimaginary parts of ψ(x)) with commutator relations

[ψ(x), ψ(x′)] = [ψ†(x), ψ†(x′)] = 0 [ψ(x), ψ†(x′)] = δ(x− x′)1

To get well-defined operators, these formulas need to be interpreted in termsof distributions. What we should really consider is, for functions f, g in someproperly chosen class, elements

ψ(f) + ψ(g) =

∫ ∞−∞

(f(x)ψ(x) + g(x)ψ(x))dx

337

of H∗1 with Poisson bracket relations

ψ(f1) + ψ(g1), ψ(f2) + ψ(g2) = −i∫ ∞−∞

(f1(x)g2(x)− f2(x)g1(x))dx

Here the right-hand size of the equation is the symplectic form Ω for thedual phase space M = H∗1, and this should be thought of as an infinite-dimensional version of formula 20.2. This is the Lie bracket relation for aninfinite-dimensional Heisenberg Lie algebra, with quantization giving a repre-sentation of the Lie algebra, with commutation relations for field operators

[ψ(f1) + ψ†(g1), ψ(f2) + ψ†(g2)] =

∫ ∞−∞

(f1(x)g2(x)− f2(x)g1(x))dx · 1

Pretty much exactly the same formalism works to describe fermions, with thesameH1 and the same choice of bases. The only difference is that the coordinatefunctions are now taken to be anti-commuting, satisfying the fermionic Poissonbracket relations of a super Lie algebra rather than a Lie algebra. After quantiza-tion, the fields ψ(x), ψ†(x) or the annihilation and creation operators a(p), a†(p)satisfy anticommutation relations and generate an infinite-dimensional Cliffordalgebra, rather than the Weyl algebra of the bosonic case.

33.2 Dynamics of the free quantum field

In classical Hamiltonian mechanics, the Hamiltonian function h determines howan observable f evolves in time by the differential equation

d

dtf = f, h

Quantization takes f to an operator f , and h to a self-adjoint operator H.Multiplying this by −i gives a skew-adjoint operator that exponentiates (we’llassume here H time-independent) to the unitary operator

U(t) = e−iHt

that determines how states (in the Schrodinger picture) evolve under time trans-lation. In the Heisenberg picture states stay the same and operators evolve, withtheir time evolution given by

f(t) = eiHtf(0)e−iHt

Such operators satisfyd

dtf = [f ,−iH]

which is the quantization of the classical dynamical equation.To describe the time evolution of a quantum field theory system, it is gener-

ally easier to work with the Heisenberg picture (in which the time dependence

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is in the quantum field operators) than the Schrodinger picture (in which thetime dependence is in the states). This is especially true in relativistic systemswhere one wants to as much as possible treat space and time on the same foot-ing. It is however also true in non-relativistic cases due to the complexity ofthe description of the states (inherent since one is trying to describe arbitrarynumbers of particles) versus the description of the operators, which are builtsimply out of the quantum fields.

The classical phase space to be quantized is the space H1 of solutions ofthe free particle Schrodinger equation, parametrized by the initial data of acomplex-valued wavefunction ψ(x, 0) ≡ ψ(x), with Poisson bracket

ψ(x), iψ(x′) = δ(x− x′)

Time translation on this space is given by the Schrodinger equation, which saysthat wavefunctions will evolve with time dependence given by

∂

∂tψ(x, t) =

i

2m

∂2

∂x2ψ(x, t)

If we take our hamiltonian function on H1 to be

h =

∫ +∞

−∞ψ(x)

−1

2m

∂2

∂x2ψ(x)dx

then we will get the single-particle Schrodinger equation from the Hamiltoniandynamics, since

∂

∂tψ(x, t) =ψ(x, t), h

=ψ(x, t),

∫ +∞

−∞ψ(x′, t)

−1

2m

∂2

∂x′2ψ(x′, t)dx′

=−1

2m

∫ +∞

−∞(ψ(x, t), ψ(x′, t) ∂

2

∂x′2ψ(x′, t)+

ψ(x′, t)ψ(x, t),∂2

∂x′2ψ(x′, t))dx′)

=−1

2m

∫ +∞

−∞(−iδ(x− x′) ∂2

∂x′2ψ(x′, t) + ψ(x′, t)

∂2

∂x′2ψ(x, t), ψ(x′, t))dx′)

=i

2m

∂2

∂x2ψ(x, t)

Here we have used the derivation property of the Poisson bracket and the lin-

earity of the operator ∂2

∂x′2.

Note that there are other forms of the same Hamiltonian function, relatedto the one we chose by integration by parts. One has

ψ(x)d2

dx2ψ(x) =

d

dx(ψ(x)

d

dxψ(x))− | d

dxψ(x)|2

=d

dx(ψ(x)

d

dxψ(x)− (

d

dxψ(x))ψ(x)) + (

d2

dx2ψ(x))ψ(x)

339

so neglecting integrals of derivatives (assuming boundary terms go to zero atinfinity), one could have used

h =1

2m

∫ +∞

−∞| ddxψ(x)|2dx or h =

−1

2m

∫ +∞

−∞(d2

dx2ψ(x))ψ(x)dx

Instead of working with position space fields ψ(x, t) we could work withtheir momentum space components. Recall that we can write solutions to theSchrodinger equation as

ψ(x, t) =1√2π

∫ ∞−∞

α(p, t)eipxdp

where

α(p, t) = α(p)e−ip2

2m t

Using these as our coordinates on the H1, dynamics is given by

∂

∂tα(p, t) = α(p, t), h = −i p

2

2mα(p, t)

and one can easily see that one can choose

h =

∫ ∞−∞

p2

2m|α(p)|2dp

as Hamiltonian function in momentum space coordinates. This is the sameexpression one would get by substituting the expression for ψ in terms of α andcalculating h from its formula as a quadratic polynomial in the fields.

In momentum space, quantization is simply given by

α(p)→ a(p), h→ H =

∫ ∞−∞

p2

2ma†(p)a(p)dp

where we have normal-ordered H so that the vacuum energy is zero.In position space the expression for the Hamiltonian operator (again normal-

ordered) will be:

H =

∫ +∞

−∞ψ†(x)

−1

2m

∂2

∂x2ψ(x)dx

Using this quantized form, essentially the same calculation as before (now withoperators and commutators instead of functions and Poisson brackets) showsthat the quantum field dynamical equation

d

dtψ(x, t) = −i[ψ(x, t), H]

becomes∂

∂tψ(x, t) =

i

2m

∂2

∂x2ψ(x, t)

340

The field operator ψ(x, t) satisfies the Schrodinger equation which now ap-pears as a differential equation for operators rather than for wavefunctions.One can explicitly solve such a differential equation just as for wavefunctions,by Fourier transforming and turning differentiation into multiplication. If theoperator ψ(x, t) is related to the operator a(p, t) by

ψ(x, t) =1√2π

∫ ∞−∞

eipxa(p, t)dp

then the Schrodinger equation for the a(p, t) will be

∂

∂ta(p, t) =

−ip2

2ma(p, t)

with solution

a(p, t) = e−ip2

2m ta(p, 0)

The solution for the field will then be

ψ(x, t) =1√2π

∫ ∞−∞

eipxe−ip2

2m ta(p)dp

where the operators a(p) ≡ a(p, 0) are the initial values.We will not enter into the important topic of how to compute observables

in quantum field theory that can be connected to experimentally importantquantities such as scattering cross-sections. A crucial role in such calculationsis played by the following observables:

Definition (Green’s function or propagator). The Green’s function or propa-gator for a quantum field theory is the amplitude, for t > t′

G(x, t, x′, t′) = 〈0|ψ(x, t)ψ†(x′, t′)|0〉

The physical interpretation of these functions is that they describe the am-plitude for a process in which a one-particle state localized at x is created at timet′, propagates for a time t− t′, and then its wave-function is compared to thatof a one-particle state localized at x. Using the solution for the time-dependentfield operator given earlier we find

G(x, t, x′, t′) =1

2π

∫ +∞

−∞

∫ +∞

−∞〈0|eipxe−i

p2

2m ta(p)e−ip′xei

p′22m t

′a†(p′)|0〉dpdp′

=

∫ +∞

−∞

∫ +∞

−∞eipxe−i

p2

2m te−ip′x′ei

p′22m t

′δ(p− p′)dpdp′

=

∫ +∞

−∞eip(x−x

′)e−ip2

2m (t−t′)dp

One can evaluate this integral, finding

G(x′, t′, x, t) = (−im

2π(t′ − t))

32 e

−im2π(t′−t) (x′−x)2

341

and thatlimt→t′

G(x′, t′, x, t) = δ(x′ − x)

While we have worked purely in the Hamiltonian formalism, one could in-stead start with a Lagrangian for this system. A Lagrangian that will give theSchrodinger equation as an Euler-Lagrange equation is

L = iψ∂

∂tψ − h = iψ

∂

∂tψ + ψ

1

2m

∂2

∂x2ψ

or, using integration by parts to get an alternate form of h mentioned earlier

L = iψ∂

∂tψ − 1

2m| ∂∂xψ|2

If one tries to define a canonical momentum for ψ as ∂L∂ψ

one just gets iψ.

This justifies the Poisson bracket relation

ψ(x), iψ(x′) = δ(x− x′)

but, as expected for a case where the equation of motion is first-order in time,such a canonical momentum is not independent of ψ and the space of the wave-functions ψ is already a phase space. One could try and quantize this systemby path integral methods, for instance computing the propagator by doing theintegral ∫

Dγ ei~S[γ]

over paths γ from (x, t) to (x′, t′). However one needs to keep in mind thewarnings given earlier about path integrals over phase space, since that is whatone has here.


As with the last chapter, the material here is discussed in essentially any quan-tum field theory textbook, with two that explicitly discuss the non-relativisticcase [23] and [31]. For a serious mathematical treatment of quantum fields asdistribution-valued operators, a standard reference is [57].

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Chapter 34

Symmetries andNon-relativistic QuantumFields

In our study of the harmonic oscillator (chapter 21) we found that the sym-metries of the system could be studied using quadratic functions on the phasespace. Classically these gave a Lie algebra under the Poisson bracket, and quan-tization provided a unitary representation Γ′ of the Lie algebra, with quadraticfunctions becoming quadratic operators. In the case of fields, the same patternholds, with the phase space now an infinite dimensional space, the single particleHilbert space H1. Certain specific quadratic functions of the fields will providea Lie algebra under the Poisson bracket, with quantization then providing aunitary representation of the Lie algebra in terms of quadratic field operators.

In chapter 33 we saw how this works for time translation symmetry, whichdetermines the dynamics of the theory. For the case of a free particle, thefield theory Hamiltonian is a quadratic function of the fields, providing a basicexample of how such functions generate a unitary representation on the statesof the quantum theory by use of a quadratic combination of the quantum fieldoperators. In this chapter we will see how other group actions on the spaceH1 also lead to quadratic operators and unitary transformations on the fullquantum field theory. We would like to find a formula to these, somethingthat will be simplest to do in the case that the group acts on phase space asunitary transformations, preserving the complex structure used in Bargmann-Fock quantization.

34.1 Internal symmetries

Since the phase space H1 is a space of complex functions, there is a group thatacts unitarily on this space is the group U(1) of phase transformations of the

343

complex values of the function. Such a group action that acts trivially on thespatial coordinates but non-trivially on the values of ψ(x) is called an “internalsymmetry”. If the fields ψ have multiple components, taking values in Cm,there will be a unitary action of the larger group U(m).

34.1.1 U(1) symmetry

In chapter 2 we saw that the fact that irreducible representations of U(1) arelabeled by integers is what is responsible for the term “quantization”: sincequantum states are representations of this group, they break up into statescharacterized by integers, with these integers counting the number of “quanta”.In the non-relativistic quantum field theory, the integer will just be the totalparticle number. Such a theory can be thought of as an harmonic oscillatorwith an infinite number of degrees of freedom, and the total particle number isjust the total occupation number, summed over all degrees of freedom.

The U(1) action on the fields ψ(x) which provide coordinates on H1 is givenby

ψ(x)→ eiθψ(x), ψ(x)→ e−iθψ(x)

(recall that the fields are in the dual space to H1, so the action is the inverse tothe action of U(1) on H1 itself by multiplication by e−iθ).

To understand the infinitesimal generator of this symmetry, first recall thesimple case of a harmonic oscillator in one variable, identifying the phase spaceR2 with C so the coordinates are z, z, with a U(1) action

z → eiθz, z → e−iθz

The Poisson bracket isz, z = −i

which implieszz, z = iz, zz, z = −iz

Quantizing takes z → a, z → a† and we saw in chapter 21 that we have achoice to make for the unitary operator that will be the quantization of zz:

•zz → − i

2(a†a+ aa†)

This will have eigenvalues −i(n+ 12 ), n = 0, 1, 2 . . . .

•zz → −ia†a

This is the normal-ordered form, with eigenvalues −in.

With either choice, we get a number operator

N =1

2(a†a+ aa†), or N =

1

2: (a†a+ aa†) := a†a

344

In both cases we have[N, a] = −a, [N, a†] = a†

soe−iθNaeiθN = eiθa, e−iθNa†eiθN = e−iθa†

Either choice of N will give the same action on operators. Hoever, on statesonly the normal-ordered one will have the desirable feature that

N |0〉 = 0, e−iNθ|0〉 = |0〉

Since we now want to treat fields, adding together an infinite number of suchoscillator degrees of freedom, we will need the normal-ordered version in orderto not get ∞ · 1

2 as the number eigenvalue for the vacuum state.In momentum space, we simply do the above for each value of p and sum,

getting

N =

∫ +∞

−∞a†(p)a(p)dp

where one needs to keep in mind that this is really an operator valued distribu-tion, which must be integrated against some weighting function on momentumspace to get a well-defined operator. What really makes sense is

N(f) =

∫ +∞

−∞a†(p)a(p)f(p)dp

for a suitable class of functions f .Instead of working with a(p), the quantization of the Fourier transform of

ψ(x), one could work with ψ(x) itself, and write

N =

∫ +∞

−∞ψ†(x)ψ(x)dx

with the Fourier transform relating the two formulas for N . ψ†(x)ψ(x) is also anoperator valued distribution, with the interpretation of measuring the numberdensity at x.

On field operators, N satisfies

[N , ψ] = −ψ, [N , ψ†] = ψ†

so ψ acts on states by reducing the eigenvalue of N by one, while ψ† acts onstates by increasing the eigenvalue of N by one. Exponentiating, one has

e−iθN ψeiθN = eiθψ, e−iθN ψ†eiθN = e−iθψ†

which are the quantized versions of the U(1) action on the phase space coordi-nates

ψ(x)→ eiθψ(x), ψ(x)→ e−iθψ(x)

345

that we began our discussion with.

An important property of N that can be straightforwardly checked is that

[N , H] = [N ,

∫ +∞

−∞ψ†(x)

−1

2m

∂2

∂x2ψ(x)dx] = 0

This implies that particle number is a conserved quantity: if we start out witha state with a definite particle number, this will remain constant. Note that theorigin of this conservation law comes from the fact that N is the quantized gen-erator of the U(1) symmetry of phase transformations on complex-valued fieldsψ. If we start with any hamiltonian function h on H1 that is invariant under theU(1) (i.e. built out of terms with an equal number of ψs and ψs), then for such

a theory N will commute with H and particle number will be conserved. Notethough that one needs to take some care with arguments like this, which assumethat symmetries of the classical phase space give rise to unitary representationsin the quantum theory. The need to normal-order operator products, workingwith operators that differ from the most straightforward quantization by an in-finite constant, can cause a failure of symmetries to be realized as expected inthe quantum theory, a phenomenon known as an “anomaly” in the symmetry.

In quantum field theories, due to the infinite number of degrees of freedom,the Stone-von Neumann theorem does not apply, and one can have unitarilyinequivalent representations of the algebra generated by the field operators,leading to new kinds of behavior not seen in finite dimensional quantum systems.In particular, one can have a space of states where the lowest energy state |0〉does not have the property

N |0〉 = 0, e−iθN |0〉 = |0〉

but instead gets taken by e−iθN to some other state, with

N |0〉 6= 0, e−iθN |0〉 ≡ |θ〉 6= |0〉 (for θ 6= 0)

In this case, the vacuum state is not an eigenstate of N so does not have awell-defined particle number. If [N , H] = 0, the states |θ〉 will all have the sameenergy as |0〉 and there will be a multiplicity of different vacuum states, labeledby θ. In such a case the U(1) symmetry is said to be “spontaneously broken”.This phenomenon occurs when non-relativistic quantum field theory is used todescribe a superconductor. There the lowest energy state will be a state withouta definite particle number, with electrons pairing up in a way that allows themto lower their energy, “condensing” in the lowest energy state.

34.1.2 U(m) symmetry

By taking fields with values in Cm, or, equivalently, m different species ofcomplex-valued field ψj , j = 1, 2, . . . ,m, one can easily construct quantum field

346

theories with larger internal symmetry groups than U(1). Taking as Hamilto-nian function

h =

∫ +∞

−∞

m∑j=1

ψj(x)−1

2m

∂2

∂x2ψj(x)dx

one can see that this will be invariant not just under U(1) phase transformations,but also under transformations

ψ1

ψ2

...ψm

→ U

ψ1

ψ2

...ψm

where U is an m by m unitary matrix. The Poisson brackets will be

ψj(x), ψk(x′) = −iδ(x− x′)δjk

and are also invariant under such transformations by U ∈ U(m).As in the U(1) case, one can begin by considering the case of one particular

value of p or of x, for which the phase space is Cm, with coordinates zb, zb. As wesaw in section 22.1, the m2 quadratic combinations zjzk for j = 1, . . . ,m, k =1, . . . ,m will generalize the role played by zz in the m = 1 case, with theirPoisson bracket relations exactly the Lie bracket relations of the Lie algebrau(m) (or, considering all complex linear combinations, gl(m,C)).

After quantization, these quadratic combinations become quadratic combi-nations in annihilation and creation operators aj , a

†j satisfying

[aj , a†k] = δjk

Recall (theorem 22.2) that for m by m matrices X and Y one will have

[

m∑j,k=1

a†jXjkak,

m∑j,k=1

a†jYjkak] =

m∑j,k=1

a†j [X,Y ]jkak

So, for each X in the Lie algebra gl(m,C), quantization will give us a represen-tation of gl(m,C) where X acts as the operator

m∑j,k=1

a†jXjkak

When the matrices X are chosen to be skew-adjoint (Xjk = −Xkj) this con-struction will give us a unitary representation of u(m).

As in the U(1) case, one gets an operator in the quantum field theory just bysumming over either the a(p) in momentum space, or the fields in configurationspace, finding for each X ∈ u(m) an operator

X =

∫ +∞

−∞

m∑j,k=1

ψ†j (x)Xjkψk(x)dx

347

that provides a representation of u(m) and U(m) on the quantum field theorystate space. This representation takes

eX ∈ U(m)→ U(eX) = eX = e∫ +∞−∞

∑mb,c=1 ψ

†j (x)Xjkψk(x)dx

When, as for the free-particle h we chose, the Hamiltonian is invariant underU(m) transformations of the fields ψj , then we will have

[X, H] = 0

In this case, if |0〉 is invariant under the U(m) symmetry, then energy eigenstatesof the quantum field theory will break up into irreducible representations ofU(m) and can be labeled accordingly. As in the U(1) case, the U(m) symmetrymay be spontaneously broken, with

X|0〉 6= 0

for some directions X in u(m). When this happens, just as in the U(1) casestates did not have well-defined particle number, now they will not carry well-defined irreducible U(m) representation labels.

34.2 Spatial symmetries

We saw in chapter 17 that the action of the group E(3) on physical spaceR3 induces a unitary action on the space H1 of solutions to the free-particleSchrodinger equation. Quantization of this phase space with this group actionproduces a quantum field theory state space carrying a unitary representationof the group E(3). There are three different actions of the group E(3) that oneneeds to keep straight here. Given an element (a, R) ∈ E(3) one has:

1. an action on R3, preserving the inner product on R3

x→ Rx + a

2. A unitary action on H1 given by

ψ(x)→ u(a, R)ψ(x) = ψ(R−1(x− a))

on wavefunctions, or, on Fourier transforms by

ψ(p)→ u(a, R)ψ(p) = e−ia·R−1pψ(R−1p)

Recall that this is not an irreducible representation of E(3), but one canget an irreducible representation by taking distributional wave-functionsψE with support on the sphere |p|2 = 2mE.

348

For the case of two-component wave functions ψ =

(ψ1

ψ2

)satisfying the

Pauli equation (see chapter 29), one has to use the double cover of E(3),with elements (a,Ω),Ω ∈ SU(2) and on these the action is

ψ(x)→ u(a,Ω)ψ(x) = Ωψ(R−1(x− a))

and

ψ(p)→ u(a,Ω)ψ(p) = e−ia·R−1pΩψ(R−1p)

where R = Φ(Ω) is the SO(3) group element corresponding to Ω.

The infinitesimal version of this unitary action is given by the operators−iP and −iL (in the two-component case, instead of −iL, one needs−iJ = −i(L + S)).

3. The quantum field theory one gets by treating H1 as a classical phasespace, and quantizing using an appropriate infinite-dimensional version ofthe Bargmann-Fock representation comes with another unitary represen-tation U(a,R), on the quantum field theory state space H. This is be-cause the representation u(a,R) preserves the symplectic structure on H1,and the Bargmann-Fock construction gives a representation of the groupof such symplectic transformations, of which E(3) is a finite-dimensionalsubgroup.

It is the last of these that we want to understand here, and as usual forquantum field theory, we don’t want to try and explicitly construct the statespace H and see the E(3) action on that construction, but instead want to usethe analog of the Heisenberg picture in the time-translation case, taking thegroup to act on operators. For each (a, R) ∈ E(3) we want to find operatorsU(a,R) that will be built out of the field operators, and act on the field operatorsas

ψ(x)→ U(a,R)ψ(x)U(a,R)−1 = ψ(Rx + a) (34.1)

Note that here the way the group acts on the argument of the operator-valueddistribution is opposite to the way that it acts on the argument of a solutionin H1. This is because ψ(x) is an operator associated not to an element of H1,but to a distribution on this space, in particular the distribution ψ(x), heremeaning “evaluation of the solution ψ at x. The group will act oppositely onsuch linear functions on H1 to its action on elements of H1. For a more generaldistribution of the form

ψ(f) =

∫R3

f(x)ψ(x)d3x

E(3) will act on f by

f → (a, R) · f(x) = f(R−1(x− a))

349

and on ψ(f) by

ψ(f)→ (a, R) · ψ(f) =

∫R3

f(R−1(x− a))ψ(x)d3x

The distribution ψ(x) corresponds to taking f as the delta-function, and it willtransform as

ψ(x)→ (a, R) · ψ(x) = ψ(Rx + a))

For spatial translations, we want to construct momentum operators−iP thatgive a Lie algebra representation of the translation group, and the operators

U(a,1) = e−ia·P

after exponentiation. Note that these are not the momentum operators P thatact on H1, but are operators in the quantum field theory that will be built outof quadratic combinations of the field operators. By equation 34.1 we want

e−ia·Pψ(x)eia·P = ψ(x + a)

Such an operator P can be constructed in terms of quadratic operators in thefields in the same way as the Hamiltonian H was in section 33.2, although thereis an opposite choice of sign for time versus space translations (−iH = ∂

∂t , and

−iP = − ∂∂x ) for reasons that appear when we combine space and time later in

special relativity. The calculation proceeds by just replacing the single-particleHamiltonian operator by the single-particle momentum operator P = −i∇. Soone has

P =

∫R3

ψ†(x)(−i∇)ψ(x)d3x

In the last chapter we saw that, in terms of annihilation and creation operators,this operator is just

P =

∫R3

p a†(p)a(p)d3p

which is just the integral over momentum space of the momentum times thenumber-density operator in momentum space.

For spatial rotations, we found in chapter 17 that these had generators theangular momentum operators

L = X×P = X× (−i∇)

acting on H1. Just as for energy and momentum, we can construct angularmomentum operators in the quantum field theory as quadratic field operatorsby

L =

∫R3

ψ†(x)(x× (−i∇))ψ(x)d3x

These will generate the action of rotations on the field operators. For instance,if R(θ) is a rotation about the x3 axis by angle θ, we will have

ψ(R(θ)x) = e−iθL3 ψ(x)eiθL3

350

Note that these constructions are infinite-dimensional examples of theorem22.2 which showed how to take an action of the unitary group on phase space(preserving Ω) and produce a representation of this group on the state spaceof the quantum theory. In our study of quantum field theory, we will be con-tinually exploiting this construction, for groups acting unitarily on the infinite-dimensional phase space H1 of solutions of some linear field equations.

34.3 Fermions

Everything that was done in this chapter carries over straightforwardly to thecase of a fermionic non-relativistic quantum field theory of free particles. Fieldoperators will in this case generate an infinite-dimensional Clifford algebra andthe quantum state space will be an infinite-dimensional version of the spinorrepresentation. All the symmetries considered in this chpter also appear in thefermionic case, and have Lie algebra representations constructed using quadraticcombinations of the field operators in just the same way as in the bosonic case.In section 27.3 we saw in finite dimensions how unitary group actions on thefermionic phase space gave a unitary representation on the fermionic oscillatorspace, by the same method of annihilation and creation operators as in thebosonic case. The construction of the Lie algebra representation operators inthe fermionic case is an infinite-dimensional example of that method.


The material of this chapter is often developed in conventional quantum fieldtheory texts in the context of relativistic rather than non-relativistic quantumfield theory. Symmetry generators are also more often derived via Lagrangianmethods (Noether’s theorem) rather than the Hamiltonian methods used here.For an example of a detailed discussion relatively close to this one, see [23].

351

352

Chapter 35

Minkowski Space and theLorentz Group

For the case of non-relativistic quantum mechanics, we saw that systems withan arbitrary number of particles, bosons or fermions, could be described bytaking as the Hamiltonian phase space the state space H1 of the single-particlequantum theory (e.g. the space of complex-valued wave-functions on R3 in thebosonic case). This phase space is infinite-dimensional, but it is linear and itcan be quantized using the same techniques that work for the finite-dimensionalharmonic oscillator. This is an example of a quantum field theory since it is aspace of functions (fields, to physicists) that is being quantized.

We would like to find some similar way to proceed for the case of rela-tivistic systems, finding relativistic quantum field theories capable of describ-ing arbitrary numbers of particles, with the energy-momentum relationshipE2 = |p|2c2 + m2c4 characteristic of special relativity, not the non-relativistic

limit |p| mc where E = |p|22m . In general, a phase space can be thought of as

the space of initial conditions for an equation of motion, or equivalently, as thespace of solutions of the equation of motion. In the non-relativistic field theory,the equation of motion is the first-order in time Schrodinger equation, and thephase space is the space of fields (wave-functions) at a specified initial time,say t = 0. This space carries a representation of the time-translation group R,the space-translation group R3 and the rotation group SO(3). To construct arelativistic quantum field theory, we want to find an analog of this space. It willbe some sort of linear space of functions satisfying an equation of motion, andwe will then quantize by applying harmonic oscillator methods.

Just as in the non-relativistic case, the space of solutions to the equationof motion provides a representation of the group of space-time symmetries ofthe theory. This group will now be the Poincare group, a ten-dimensionalgroup which includes a four-dimensional subgroup of translations in space-time,and a six-dimensional subgroup (the Lorentz group), which combines spatialrotations and “boosts” (transformations mixing spatial and time coordinates).

353

The representation of the Poincare group on the solutions to the relativisticwave equation will in general be reducible. Irreducible such representations willbe the objects corresponding to elementary particles. Our first goal will be tounderstand the Lorentz group, in later sections we will find representations ofthis group, then move on to the Poincare group and its representations.

35.1 Minkowski space

Special relativity is based on the principle that one should consider considerspace and time together, and take them to be a four-dimensional space R4 withan indefinite inner product:

Definition (Minkowski space). Minkowski space M4 is the vector space R4

with an indefinite inner product given by

(x, y) ≡ x · y = −x0y0 + x1y1 + x2y2 + x3y3

where (x0, x1, x2, x3) are the coordinates of x ∈ R4, (y0, y1, y2, y3) the coordi-nates of y ∈ R4.

Digression. We have chosen to use the −+ ++ instead of the more common+−−− sign convention for the following reasons:

• Analytically continuing the time variable x0 to ix0 gives a positive definiteinner product.

• Restricting to spatial components, there is no change from our previousformulas for the symmetries of Euclidean space E(3).

• Only for this choice will we have a purely real spinor representation (sinceCliff(3, 1) = M(22,R) 6= Cliff(1, 3)).

• Weinberg’s quantum field theory textbook [66] uses this convention (al-though, unlike him, we’ll put the 0 component first).

This inner product will also sometimes be written using the matrix

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

as

x · y =

3∑µ,ν=0

ηµνxµxν

354

Digression (Upper and lower indices). In many physics texts it is conventionalin discussions of special relativity to write formulas using both upper and lowerindices, related by

xµ =

3∑ν=0

ηµνxν = ηµνx

ν

with the last form of this using the Einstein summation convention.

One motivation for introducing both upper and lower indices is that specialrelativity is a limiting case of general relativity, which is a fully geometricaltheory based on taking space-time to be a manifold M with a metric g thatvaries from point to point. In such a theory it is important to distinguish betweenelements of the tangent space Tx(M) at a point x ∈M and elements of its dual,the co-tangent space T ∗x (M), while using the fact that the metric g provides aninner product on Tx(M) and thus an isomorphism Tx(M) ' T ∗x (M). In thespecial relativity case, this distinction between Tx(M) and T ∗x (M) just comesdown to an issue of signs, but the upper and lower index notation is useful forkeeping track of those.

A second motivation is that position and momenta naturally live in dualvector spaces, so one would like to distinguish between the vector space M4 ofpositions and the dual vector space of momenta. In the case though of a vectorspace like M4 which comes with a fixed inner product ηµν , this inner productgives a fixed identification of M4 and its dual, an identification that is also anidentification as representations of the Lorentz group. For simplicity, we willnot here try and distinguish by notation whether a vector is in M4 or its dual,so will just use lower indices, not both upper and lower indices.

The coordinates x = x1, x2, x3 are interpreted as spatial coordinates, and thecoordinate x0 is a time coordinate, related to the conventional time coordinatet with respect to chosen units of time and distance by x0 = ct where c is thespeed of light. Mostly we will assume units of time and distance have beenchosen so that c = 1.

Vectors v ∈M4 such that |v|2 = v · v > 0 are called “spacelike”, those with|v|2 < 0 “time-like” and those with |v|2 = 0 are said to lie on the “light-cone”.Suppressing one space dimension, the picture to keep in mind of Minkowskispace looks like this:

355

We can take Fourier transforms with respect to the four space-time variables,which will take functions of x0, x1, x2, x3 to functions of the Fourier transformvariables p0, p1, p2, p3. The definition we will use for this Fourier transform willbe

f(p) =1

(2π)2

∫M4

e−ip·xf(x)d4x

=1

(2π)2

∫M4

e−i(−p0x0+p1x1+p2x2+p3x3)f(x)dx0d3x

and the Fourier inversion formula is

f(x) =1

(2π)2

∫M4

eip·xf(p)d4p

Note that our definition puts one factor of 1√2π

with each Fourier (or inverse

Fourier) transform with respect to a single variable. A common alternate con-vention among physicists is to put all factors of 2π with the p integrals (and

thus in the inverse Fourier transform), none in the definition of f(p), the Fouriertransform itself.

The reason why one conventionally defines the Hamiltonian operator as i ∂∂tbut the momentum operator with components −i ∂

∂xjis due to the sign change

between the time and space variables that occurs in this Fourier transform inthe exponent of the exponential.

356

35.2 The Lorentz group and its Lie algebra

Recall that in 3 dimensions the group of linear transformations of R3 pre-serving the standard inner product was the group O(3) of 3 by 3 orthogonalmatrices. This group has two disconnected components: SO(3), the subgroupof orientation preserving (determinant +1) transformations, and a componentof orientation reversing (determinant −1) transformations. In Minkowksi space,one has

Definition (Lorentz group). The Lorentz group O(3, 1) is the group of lineartransformations preserving the Minkowski space inner product on R4.

In terms of matrices, the condition for a 4 by 4 matrix Λ to be in O(3, 1)will be

ΛT

−1 0 0 00 1 0 00 0 1 00 0 0 1

Λ =

−1 0 0 00 1 0 00 0 1 00 0 0 1

The Lorentz group has four components, with the component of the iden-

tity a subgroup called SO(3, 1) (which some call SO+(3, 1)). The other threecomponents arise by multiplication of elements in SO(3, 1) by P, T, PT where

P =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

is called the “parity” transformation, reversing the orientation of the spatialvariables, and

T =

−1 0 0 00 1 0 00 0 1 00 0 0 1

reverses the time orientation.

The Lorentz group has a subgroup SO(3) of transformations that just acton the spatial components, given by matrices of the form

Λ =

1 0 0 000 R0

where R is in SO(3). For each pair j, k of spatial directions one has the usualSO(2) subgroup of rotations in the j−k plane, but now in addition for each pair0, j of the time direction with a spatial direction, one has SO(1, 1) subgroups

357

of matrices of transformations called “boosts” in the j direction. For example,for j = 1, one has the subgroup of SO(3, 1) of matrices of the form

Λ =

coshφ sinhφ 0 0sinhφ coshφ 0 0

0 0 1 00 0 0 1

for φ ∈ R.

The Lorentz group is six-dimensional. For a basis of its Lie algebra one cantake six matrices Mµν for µ, ν ∈ 0, 1, 2, 3 and j < k. For the spatial indices,these are

M12 =

0 0 0 00 0 −1 00 1 0 00 0 0 0

, M13 =

0 0 0 00 0 0 10 0 0 00 −1 0 0

, M23 =

0 0 0 00 0 0 00 0 0 −10 0 1 0

which correspond to the basis elements of the Lie algebra of SO(3) that we sawin an earlier chapter. One can rename these using the same names as earlier

l1 = M23, l2 = M13, l3 = M12

and recall that these satisfy the so(3) commutation relations

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

and correspond to infinitesimal rotations about the three spatial axes.Taking the first index 0, one gets three elements corresponding to infinitesi-

mal boosts in the three spatial directions

M01 =

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

, M02 =

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

, M03 =

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

These can be renamed as

k1 = M01, k2 = M02, k3 = M03

One can easily calculate the commutation relations between the kj and lj , whichshow that the kj transform as a vector under infinitesimal rotations. For in-stance, for infinitesimal rotations about the x1 axis, one finds

[l1, k1] = 0, [l1, k2] = k3, [l1, k3] = −k2

Commuting infinitesimal boosts, one gets infinitesimal spatial rotations

[k1, k2] = −l3, [k3, k1] = −l2, [k2, k3] = −l1

358

Digression. A more conventional notation in physics is to use Jj = ilj forinfinitesimal rotations, and Kj = ikj for infinitesimal boosts. The intention ofthe different notation used here is to start with basis elements of the real Liealgebra so(3, 1), (the lj and kj) which are purely real objects, before complexifyingand considering representations of the Lie algebra.

Taking the following complex linear combinations of the lj and kj

Aj =1

2(lj + ikj), Bj =

1

2(lj − ikj)

one finds[A1, A2] = A3, [A3, A1] = A2, [A2, A3] = A1

and[B1, B2] = B3, [B3, B1] = B2, [B2, B3] = B1

This construction of the Aj , Bj requires that we complexify (allow complexlinear combinations of basis elements) the Lie algebra so(3, 1) of SO(3, 1) andwork with the complex Lie algebra so(3, 1) ⊗ C. It shows that this Lie alge-bra splits into a product of two sub-Lie algebras, which are each copies of the(complexified) Lie algebra of SO(3), so(3)⊗C. Since

so(3)⊗C = su(2)⊗C = sl(2,C)

we haveso(3, 1)⊗C = sl(2,C)× sl(2,C)

In the next section we’ll see the origin of this phenomenon at the group level.

35.3 Spin and the Lorentz group

Just as the groups SO(n) have double covers Spin(n), the group SO(3, 1) has adouble cover, which we will show can be identified with the group SL(2,C) of2 by 2 complex matrices with unit determinant. This group will have the sameLie algebra as the SO(3, 1), and we will sometimes refer to either group as the“Lorentz group”.

Recall that for SO(3) the spin double cover Spin(3) can be identified witheither Sp(1) (the unit quaternions) or SU(2), and then the action of Spin(3)as SO(3) rotations of R3 was given by conjugation of imaginary quaternions orcertain 2 by 2 complex matrices respectively. In the SU(2) case this was doneexplicitly by identifying

(x1, x2, x3)↔(

x3 x1 − ix2

x1 + ix2 −x3

)and then showing that conjugating this matrix by an element of SU(2) was alinear map leaving invariant

det

(x3 x1 − ix2

x1 + ix2 −x3

)= −(x2

1 + x22 + x2

3)

359

and thus a rotation in SO(3).The same sort of thing works for the Lorentz group case. Now we identify

R4 with the space of 2 by 2 complex self-adjoint matrices by

(x0, x1, x2, x3)↔(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)and observe that

det

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)= x2

0 − x21 − x2

2 − x23

This provides a very useful way to think of Minkowski space: as complex self-adjoint 2 by 2 matrices, with norm-squared minus the determinant of the matrix.

The linear transformation(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)→ Λ

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Λ†

for Λ ∈ SL(2,C) preserves the determinant and thus the inner-product, since

det(Λ

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Λ†) =(det Λ) det

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)(det Λ†)

=x20 − x2

1 − x22 − x2

3

It also takes self-adjoint matrices to self-adjoints, and thus R4 to R4, since

(Λ

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Λ†)† =(Λ†)†

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)†Λ†

=Λ

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Λ†

Note that both Λ and −Λ give the same linear transformation when they act byconjugation like this. One can show that all elements of SO(3, 1) arise as suchconjugation maps, by finding appropriate Λ that give rotations or boosts in theµ− ν planes, since these generate the group.

Recall that the double covering map

Φ : SU(2)→ SO(3)

was given for Ω ∈ SU(2) by taking Φ(Ω) to be the linear transformation inSO(3) (

x3 x1 − ix2

x1 + ix2 −x3

)→ Ω

(x3 x1 − ix2

x1 + ix2 −x3

)Ω−1

We have found an extension of this map to a double covering map from SL(2,C)to SO(3, 1). This restricts to Φ on the subgroup SU(2) of SL(2,C) matricessatisfying Λ† = Λ−1.

360

Digression (The complex group Spin(4,C) and its real forms). Recall thatwe found that Spin(4) = Sp(1) × Sp(1), with the corresponding SO(4) trans-formation given by identifying R4 with the quaternions H and taking not justconjugations by unit quaternions, but both left and right multiplication by dis-tinct unit quaternions. Rewriting this in terms of complex matrices instead ofquaternions, we have Spin(4) = SU(2) × SU(2), and a pair Ω1,Ω2 of SU(2)matrices acts as an SO(4) rotation by(

x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)→ Ω1

(x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)Ω2

preserving the determinant x20 + x2

1 + x22 + x2

3.For another example, consider the identification of R4 with 2 by 2 real ma-

trices given by

(x0, x1, x2, x3)↔(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)Given a pair of matrices Ω1,Ω2 in SL(2,R), the linear transformation(

x0 + x3 x2 + x1

x2 − x1 x0 − x3

)→ Ω1

(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)Ω2

preserves the reality condition on the matrix, and preserves

det

(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)= x2

0 + x21 − x2

2 − x23

so gives an element of SO(2, 2) and we see that Spin(2, 2) = SL(2,R) ×SL(2,R).

The three different examples

Spin(4) = SU(2)× SU(2), Spin(3, 1) = SL(2,C)

andSpin(2, 2) = SL(2,R)× SL(2,R)

that we have seen are all so-called “real forms” of a fact about complex groupsthat one can get by complexifying any of the examples, i.e. considering elements(x0, x1, x2, x3) ∈ C4, not just in R4. For instance, in the Spin(4) case, takingthe x0, x1, x2, x3 in the matrix(

x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)to have arbitrary complex values z0, z1, z2, z3 one gets arbitrary 2 by 2 complexmatrices, and the transformation(

z0 − iz3 −z2 − iz1

z2 − iz1 z0 + iz3

)→ Ω1

(z0 − iz3 −z2 − iz1

z2 − iz1 z0 + iz3

)Ω2

361

preserves this space as well as the determinant (z20 +z2

1 +z22 +z2

3) for Ω1 and Ω2

not just in SU(2), but in the larger group SL(2,C). So we find that the groupSO(4,C) of complex orthogonal transformations of C4 has spin double cover

Spin(4,C) = SL(2,C)× SL(2,C)

Since spin(4,C) = so(3, 1)⊗C, this relation between complex Lie groups corre-sponds to the Lie algebra relation

so(3, 1)⊗C = sl(2,C)× sl(2,C)

we found explicitly earlier when we showed that by taking complex coefficientsof generators lj and kj of so(3, 1) we could find generators Aj and Bj of twodifferent sl(2,C) sub-algebras.


Those not familiar with special relativity should consult a textbook on thesubject for the physics background necessary to appreciate the significance ofMinkowski space and its Lorentz group of invariances. An example of a suitablesuch book aimed at mathematics students is Woodhouse’s Special Relativity [70].

Most quantum field theory textbooks have some sort of discussion of theLorentz group and its Lie algebra, although the issue of its complexification isoften not treated. A typical example is Peskin-Schroeder [42], see the beginningof Chapter 3. Another example is Quantum Field Theory in a Nutshell byTony Zee, see Chapter II.3 [71] (and test your understanding by interpretingproperly some of the statements included there such as “The mathematicallysophisticated say the algebra SO(3, 1) is isomorphic to SU(2)⊗ SU(2)”).

362

Chapter 36

Representations of theLorentz Group

Having seen the importance in quantum mechanics of understanding the repre-sentations of the rotation group SO(3) and its double cover Spin(3) = SU(2)one would like to also understand the representations of the Lorentz group. We’llconsider this question for the double cover SL(2,C). As in the three-dimensionalcase, only some of these will also be representations of SO(3, 1). One differencefrom the SO(3) case is that these will be non-unitary representations, so do notby themselves provide physically sensible state spaces. All finite-dimensionalirreducible representations of the Lorentz group are non-unitary (except for thetrivial representation). The Lorentz group does have unitary irreducible repre-sentations, but these are infinite-dimensional and a topic we will not cover.

36.1 Representations of the Lorentz group

In the SU(2) case we found irreducible unitary representations (πn, Vn) of di-

mension n+1 for n = 0, 1, 2, . . .. These could also be labeled by s = n2 . called the

“spin” of the representation, and we will do that from now on. These represen-tations can be realized explicitly as homogeneous polynomials of degree n = 2sin two complex variables z1, z2. For the case of Spin(4) = SU(2)× SU(2), theirreducible representation will just be tensor products

V s1 ⊗ V s2

of SU(2) irreducibles, with the first SU(2) acting on the first factor, the secondon the second factor. The case s1 = s2 = 0 is the trivial representation, s1 =12 , s2 = 0 is one of the half-spinor representations of Spin(4) on C2, s1 = 0, s2 =12 is the other, and s1 = s2 = 1

2 is the representation on four-dimensional(complexified) vectors.

Turning now to Spin(3, 1) = SL(2,C), one can take the SU(2) matricesacting on homogeneous polynomials to instead be SL(2,C) matrices and still

363

have irreducible representations of dimension 2s + 1 for s = 0, 12 , 1, . . .. These

will now be representations (πs, Vs) of SL(2,C). There are several things that

are different though about these representations:

• They are not unitary (except in the case of the trivial representation).

For example, for the defining representation V12 on C2 and the Hermitian

inner product < ·, · >

<

(ψ1

ψ2

),

(ψ′1ψ′2

)>=

(ψ1 ψ2

)·(ψ′1ψ′2

)= ψ1ψ

′1 + ψ2ψ

′2

is invariant under SU(2) transformations Ω since

< Ω

(ψ1

ψ2

),Ω

(ψ′1ψ′2

)>=

(ψ1 ψ2

)Ω† · Ω

(ψ′1ψ′2

)and Ω†Ω = 1 by unitarity. This is no longer true for Ω ∈ SL(2,C).

• The condition that matrices Ω ∈ SL(2,C) do satisfy is that they havedeterminant 1. It turns out that Ω having determinant one is equivalentto the condition that Ω preserves the anti-symmetric bilinear form on C2,i.e.

ΩT(

0 1−1 0

)Ω =

(0 1−1 0

)To see this, take

Ω =

(α βγ δ

)and calculate

ΩT(

0 1−1 0

)Ω =

(0 αδ − βγ

βγ − αδ 0

)= (det Ω)

(0 1−1 0

)As a result, on representations V

12 of SL(2,C) we do have a non-degenerate

bilinear form

(

(ψ1

ψ2

),

(ψ′1ψ′2

))→

(ψ1 ψ2

)( 0 1−1 0

)(ψ′1ψ′2

)= ψ1ψ

′2 − ψ2ψ

′1

that is invariant under the SL(2,C) action on V12 and can be used to

identify the representation and its dual.

Such a non-degenerate bilinear form is called a “symplectic form”, and wehave already made extensive use of these as a fundamental structure thatoccurs on a Hamiltonian phase space. For the simplest case of the phasespace R2 for one degree of freedom, such a form is given with respect toa basis as the same matrix

ε =

(0 1−1 0

)364

that we find here giving the symplectic form on a representation space C2

of SL(2,C). In the phase space case, everything was real, and the invari-ance group of ε was the real symplectic group Sp(2,R) = SL(2,R). Whatoccurs here is just the complexification of this story, with the symplecticform now on C2, and the invariance group now SL(2,C).

• In the case of SU(2) representations, the complex conjugate representa-tion one gets by taking as representation matrices π(g) instead of π(g) isequivalent to the original representation (the same representation, with adifferent basis choice, so matrices changed by a conjugation). To see thisfor the spin-1

2 representation, note that SU(2) matrices are of the form

Ω =

(α β

−β α

)and one has (

0 1−1 0

)(α β

−β α

)(0 1−1 0

)−1

=

(α β−β α

)so the matrix (

0 1−1 0

)is the change of basis matrix relating the representation and its complexconjugate.

This is no longer true for SL(2,C). One cannot complex conjugate arbi-trary 2 by 2 complex matrices of unit determinant by a change of basis,and representations πs will not be equivalent to their complex conjugatesπs.

The classification of irreducible finite dimensional SU(2) representation wasdone earlier in this course by considering its Lie algebra su(2), complexified togive us raising and lowering operators, and this complexification is sl(2,C). Ifyou take a look at that argument, you see that it mostly also applies to irre-ducible finite-dimensional sl(2,C) representations. There is a difference though:now flipping positive to negative weights (which corresponds to change of signof the Lie algebra representation matrices, or conjugation of the Lie group rep-resentation matrices) no longer takes one to an equivalent representation. Itturns out that to get all irreducibles, one must take both the representationswe already know about and their complex conjugates. Using the fact that thetensor product of one of each type of irreducible is still an irreducible, one canshow (we won’t do this here) that the complete list of irreducible representationsof sl(2,C) is given by

Theorem (Classification of finite dimensional sl(2,C) representations). Theirreducible representations of sl(2,C) are labeled by (s1, s2) for sj = 0, 1

2 , 1, . . ..

365

These representations are built out of the representations (πs, Vs) with the ir-

reducible (s1, s2) given by

(πs1 ⊗ πs2 , V s1 ⊗ V s2)

and having dimension (2s1 + 1)(2s2 + 1).

All these representations are also representations of the group SL(2,C) andone has the same classification theorem for the group, although we will not tryand prove this. We will also not try and study these representations in general,but will restrict attention to the four cases of most physical interest.

• (0, 0): The trivial representation on C, also called the “spin 0” or scalarrepresentation.

• ( 12 , 0): These are called left-handed (for reasons we will see later on) “Weyl

spinors”. We will often denote the representation space C2 in this case asSL, and write an element of it as ψL.

• (0, 12 ): These are called right-handed Weyl spinors. We will often denote

the representation space C2 in this case as SR, and write an element of itas ψR.

• ( 12 ,

12 ): This is called the “vector” representation since it is the complexifi-

cation of the action of SL(2,C) as SO(3, 1) transformations of space-timevectors that we saw earlier. Recall that for Ω ∈ SL(2,C) this action was(

x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)→ Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†

Since Ω† is the conjugate transpose this is the action of SL(2,C) on therepresentation SL ⊗ SR. This representation is on a vector space C4 =M(2,C), but preserves the subspace of self-adjoint matrices that we haveidentified with the Minkowski space R4.

The reducible 4 complex dimensional representation ( 12 , 0)⊕ (0, 1

2 ) is known asthe representation on “Dirac spinors”. As explained earlier, of these representa-tions, only the trivial one is unitary. Only the trivial and vector representationsare representations of SO(3, 1) as well as SL(2,C).

One can manipulate spinors like tensors, distinguishing between a spinorspace and its dual by upper and lower indices, and using the SL(2,C) invariantbilinear form ε to raise and lower indices. With complex conjugates and duals,there are four kinds of irreducible SL(2,C) representations on C2 to keep trackof

• SL: This is the standard defining representation of SL(2,C) on C2, withΩ ∈ SL(2,C) acting on ψL ∈ SL by

ψL → ΩψL

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A standard index notation for such things is called the “van der Waer-den notation”. It uses a lower index α taking values 1, 2 to label thecomponents

ψL =

(ψ1

ψ2

)= ψα

and in this notation Ω acts by

ψα → Ωβαψβ

For instance, the element

Ω = e−iθ2σ3

that acts on vectors by a rotation by an angle θ around the z-axis acts onSL by (

ψ1

ψ2

)→ e−i

θ2σ3

(ψ1

ψ2

)• S∗L: This is the dual of the defining representation, with Ω ∈ SL(2,C)

acting on ψ∗L ∈ S∗L byψ∗L → (Ω−1)Tψ∗L

This is a general property of representations: given any finite-dimensionalrepresentation (π(g), V ), the pairing between V and its dual V ∗ is pre-served by acting on V ∗ by matrices (π(g)−1)T , and these provide a repre-sentation ((π(g)−1)T , V ∗). In van der Waerden notation, one uses upperindices and writes

ψα → ((Ω−1)T )αβψβ

Writing elements of the dual as row vectors, our example above of a par-ticular Ω acts by (

ψ1 ψ2

)→(ψ1 ψ2

)eiθ2σ3

Note that the matrix ε gives an isomorphism of representations betweenSL and S∗L, given in index notation by

ψα = εαβψβ

where

εαβ =

(0 1−1 0

)• SR: This is the complex conjugate representation to SL, with Ω ∈ SL(2,C)

acting on ψR ∈ SR byψR → ΩψR

The van der Waerden notation uses a separate set of dotted indices forthese, writing this as

ψα → Ωβ

αψβ

367

Another common notation among physicists puts a bar over the ψ todenote that the vector is in this representation, but we’ll reserve thatnotation for complex conjugation. The Ω corresponding to a rotationabout the z-axis acts as (

ψ1

ψ2

)→ ei

θ2σ3

(ψ1

ψ2

)• S∗R: This is the dual representation to SR, with Ω ∈ SL(2,C) acting onψ∗R ∈ S∗R by

ψ∗R → (Ω−1

)Tψ∗R

and the index notation uses raised dotted indices

ψα → ((Ω−1

)T )αβψβ

Our standard example of a Ω acts by(ψ1 ψ2

)→(ψ1 ψ2

)e−i

θ2σ3

Another copy of ε

εαβ =

(0 1−1 0

)gives the isomorphism of SR and S∗R as representations, by

ψα = εαβψβ

Restricting to the SU(2) subgroup of SL(2,C), all these representationsare unitary, and equivalent. As SL(2,C) representations, they are not unitary,and while the representations are equivalent to their duals, SL and SR areinequivalent.

36.2 Dirac γ matrices and Cliff(3, 1)

In our discussion of the fermionic version of the harmonic oscillator, we definedthe Clifford algebra Cliff(r, s) and found that elements quadratic in its genera-tors gave a basis for the Lie algebra of so(r, s) = spin(r, s). Exponentiating thesegave an explicit construction of the group Spin(r, s). We can apply that generaltheory to the case of Cliff(3, 1) and this will give us explicitly the representations( 1

2 , 0) and (0, 12 ).

If we complexify our R4, then its Clifford algebra becomes just the algebraof 4 by 4 complex matrices

Cliff(3, 1)⊗C = Cliff(4,C) = M(4,C)

We will represent elements of Cliff(3, 1) as such 4 by 4 matrices, but shouldkeep in mind that we are working in the complexification of the Clifford algebra

368

that corresponds to the Lorentz group, so there is some sort of condition onthe matrices that should be kept track of. There are several different choices ofhow to explicitly represent these matrices, and for different purposes, differentones are most convenient. The one we will begin with and mostly use is some-times called the chiral or Weyl representation, and is the most convenient fordiscussing massless charged particles. We will try and follow the conventionsused for this representation in [66].

Digression. Note that the Aj and Bj we constructed using the lj and kj werealso complex 4 by 4 matrices, but they were acting on complex vectors (the com-plexification of the vector representation ( 1

2 ,12 )). Now we want 4 by 4 matrices

for something different, putting together the spinor representations ( 12 , 0) and

(0, 12 ).

Writing 4 by 4 matrices in 2 by 2 block form and using the Pauli matricesσj we assign the following matrices to Clifford algebra generators

γ0 = −i(

0 11 0

), γ1 = −i

(0 σ1

−σ1 0

), γ2 = −i

(0 σ2

−σ2 0

), γ3 = −i

(0 σ3

−σ3 0

)One can easily check that these satisfy the Clifford algebra relations for gener-ators of Cliff(1, 3): they anti-commute with each other and

γ20 = −1, γ2

1 = γ22 = γ2

3 = 1

The quadratic Clifford algebra elements − 12γjγk for j < k satisfy the com-

mutation relations of so(3, 1). These are explicitly

−1

2γ1γ2 = − i

2

(σ3 00 σ3

), −1

2γ1γ3 = − i

2

(σ2 00 σ2

), −1

2γ2γ3 = − i

2

(σ1 00 σ1

)and

−1

2γ0γ1 =

1

2

(−σ1 0

0 σ1

), −1

2γ0γ2 =

1

2

(−σ2 0

0 σ2

), −1

2γ0γ3 =

1

2

(−σ3 0

0 σ3

)They provide a representation (π′,C4) of the Lie algebra so(3, 1) with

π′(l1) = −1

2γ2γ3, π

′(l2) = −1

2γ1γ3, π

′(l3) = −1

2γ1γ2

and

π′(k1) = −1

2γ0γ1, π

′(k2) = −1

2γ0γ2, π

′(k3) = −1

2γ0γ3

Note that the π′(lj) are skew-adjoint, since this representation of the so(3) ⊂so(3, 1) sub-algebra is unitary. The π′(kj) are self-adjoint and this representa-tion π′ of so(3, 1) is not unitary.

On the two commuting SL(2,C) subalgebras of so(3, 1)⊗C with bases

Aj =1

2(lj + ikj), Bj =

1

2(lj − ikj)

369

this representation is

π′(A1) = − i2

(σ1 00 0

), π′(A2) = − i

2

(σ2 00 0

), π′(A3) = − i

2

(σ3 00 0

)and

π′(B1) = − i2

(0 00 σ1

), π′(B2) = − i

2

(0 00 σ2

), π′(B3) = − i

2

(0 00 σ3

)We see explicitly that the action of the quadratic elements of the Clifford

algebra on the spinor representation C4 is reducible, decomposing as the directsum SL ⊕ S∗R of two inequivalent representations on C2

Ψ =

(ψLψ∗R

)with complex conjugation (interchange of Aj and Bj) relating the sl(2,C) ac-tions on the components. The Aj act just on SL, the Bj just on S∗R. Analternative standard notation to the two-component van der Waerden notationis to use the four components of C4 with the action of the γ matrices. Therelation between the two notations is given by

Ψα ↔(ψαφα

)where the index α on the left takes values 1, 2, 3, 4 and the indices α, α on theright each take values 1, 2.

An important element of the Clifford algebra is constructed by multiplyingall of the basis elements together. Physicists traditionally multiply this by i tomake it self-adjoint and define

γ5 = iγ0γ1γ2γ3 =

(−1 00 1

)This can be used to produce projection operators from the Dirac spinors ontothe left and right-handed Weyl spinors

1

2(1− γ5)Ψ = ψL,

1

2(1 + γ5)Ψ = ψ∗R

There are two other commonly used representations of the Clifford algebrarelations, related to the one above by a change of basis. The Dirac representationis useful to describe massive charged particles, especially in the non-relativisticlimit. Generators are given by

γD0 = −i(

1 00 −1

), γD1 = −i

(0 σ1

−σ1 0

)

γD2 = −i(

0 σ2

−σ2 0

), γD3 = −i

(0 σ3

−σ3 0

)370

and the projection operators for Weyl spinors are no longer diagonal, since

γD5 =

(0 11 0

)A third representation, the Majorana representation, is given by (now no

longer writing in 2 by 2 block form, but as 4 by 4 matrices)

γM0 =

0 0 0 −10 0 1 00 −1 0 01 0 0 0

, γM1 =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

γM2 =

0 0 0 10 0 −1 00 −1 0 01 0 0 0

, γM3 =

0 −1 0 0−1 0 0 00 0 0 −10 0 −1 0

with

γM5 = i

0 −1 0 01 0 0 00 0 0 10 0 −1 0

The importance of the Majorana representation is that it shows the interestingpossibility of having (in signature (3, 1)) a spinor representation on a real vectorspace R4, since one sees that the Clifford algebra matrices can be chosen to bereal. One has

γ0γ1γ2γ3 =

0 −1 0 01 0 0 00 0 0 10 0 −1 0

and

(γ0γ1γ2γ3)2 = −1

The Majorana spinor representation is on SM = R4, with γ0γ1γ2γ3 a realoperator on this space with square −1, so it provides a complex structure onSM . Recall that a complex structure on a real vector space gives a splitting ofthe complexification of the real vector space into a sum of two complex vectorspaces, related by complex conjugation. In this case this corresponds to

SM ⊗C = SL ⊕ S∗R

the fact that complexifying Majorana spinors gives the two kinds of Weylspinors.

371


Most quantum field theory textbook have extensive discussions of spinor rep-resentations of the Lorentz group and gamma matrices, although most use theopposite convention for the signature of the Minkowski metric. Typical exam-ples are Peskin-Schroeder [42] and Quantum Field Theory in a Nutshell by TonyZee, see Chapter II.3 and Appendix E [71].

372

Chapter 37

The Poincare Group and itsRepresentations

In the previous chapter we saw that one can take the semi-direct product ofspatial translations and rotations and that the resulting group has infinite-dimensional unitary representations on the state space of a quantum free parti-cle. The free particle Hamiltonian plays the role of a Casimir operator: to getirreducible representations one fixes the eigenvalue of the Hamiltonian (the en-ergy), and then the representation is on the space of solutions to the Schrodingerequation with this energy. This is a non-relativistic procedure, treating time andspace (and correspondingly the Hamiltonian and the momenta) differently. Fora relativistic analog, we will use instead the semi-direct product of space-timetranslations and Lorentz transformations. Irreducible representations of thisgroup will be labeled by a continuous parameter (the mass) and a discrete pa-rameter (the spin or helicity), and these will correspond to possible relativisticelementary particles.

In the non-relativistic case, the representation occurred as a space of solu-tions to a differential equation, the Schrodinger equation. There is an analogousdescription of the irreducible Poincare group representations as spaces of solu-tions of relativistic wave equations, but we will put off that story until succeedingchapters.

37.1 The Poincare group and its Lie algebra

Definition (Poincare group). The Poincare group is the semi-direct product

P = R4 o SO(3, 1)

with double-coverP = R4 o SL(2,C)

The action of SO(3, 1) or SL(2,C) on R4 is the action of the Lorentz group onMinkowski space.

373

We will refer to both of these groups as the “Poincare group”, meaning bythis the double-cover only when we need it because spinor representations of theLorentz group are involved. The two groups have the same Lie algebra, so thedistinction is not needed in discussions that only need the Lie algebra. Elementsof the group P will be written as pairs (a,Λ), with a ∈ R4 and Λ ∈ SO(3, 1).The group law is

(a1,Λ1)(a2,Λ2) = (a1 + Λ1a2,Λ1Λ2)

The Lie algebra LieP = LieP has dimension 10, with basis

t0, t1, t2, t3, l1, l2, l3, k1, k2, k3

where the first four elements are a basis of the Lie algebra of the translationgroup, and the next six are a basis of so(3, 1), with the lj giving the subgroup ofspatial rotations, the kj the boosts. We already know the commutation relationsfor the translation subgroup, which is commutative so

[tj , tk] = 0

We have seen that the commutation relations for so(3, 1) are

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

[k1, k2] = −l3, [k3, k1] = −l2, [k2, k3] = −l1and that the commutation relations between the lj and kj correspond to thefact that the kj transform as a vector under spatial rotations, so for examplecommuting the kj with l1 gives an infinitesimal rotation about the 1-axis and

[l1, k1] = 0, [l1, k2] = k3, [l1, k3] = −k2

The Poincare group is a semi-direct product group of the sort discussed inchapter 15 and it can be represented as a group of 5 by 5 matrices in much thesame way as elements of the Euclidean group E(3) could be represented by 4by 4 matrices (see chapter 17). Writing out this isomorphism explicitly for abasis of the Lie algebra, we have

l1 ↔

0 0 0 0 00 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 0

l2 ↔

0 0 0 0 00 0 0 1 00 0 0 0 00 −1 0 0 00 0 0 0 0

l3 ↔

0 0 0 0 00 0 −1 0 00 1 0 0 00 0 0 0 00 0 0 0 0

k1 ↔

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

k2 ↔

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

k3 ↔

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

374

t0 ↔

0 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

t1 ↔

0 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 0

t2 ↔

0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

t3 ↔

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 1

We can use this explicit matrix representation to compute the commutators

of the infinitesimal translations tj with the infinitesimal rotations and boosts(lj , kj). t0 commutes with the lj and t1, t2, t3 transform as a vector underrotations, For instance, for infinitesimal rotations about the 1-axis

[l1, t1] = 0, [l1, t2] = t3, [l1, t3] = −t2

with similar relations for the other axes.

For boosts one has

[kj , t0] = tj , [kj , tj ] = t0, [kj , tk] = 0 if j 6= k, k 6= 0

Note that infinitesimal boosts do not commute with infinitesimal time transla-tion, so after quantization boost will not commute with the Hamiltonian andthus are not the sort of symmetries which act on spaces of energy eigenstates,preserving the energy.

37.2 Representations of the Poincare group

We want to find unitary irreducible representations of the Poincare group. Thesewill be infinite dimensional, so given by operators π(g) on a Hilbert space H,which will have an interpretation as a single-particle relativistic quantum statespace. The standard physics notation for the operators giving the representationis U(a,Λ), with the U emphasizing their unitarity. To classify these representa-tions, we recall from chapter 15 that irreducible representations of semi-directproducts N oK are associated with pairs of a K-orbit Oα in the space N andan irreducible representation of the corresponding little group Kα.

For the Poincare group, N = R4 is the space of characters (one-dimensionalrepresentations) of the translation group of Minkowski space. These are labeledby an element p = (p0, p1, p2, p3) that has a physical interpretation as the energy-momentum vector of the state such that π(x) (for x in the translation groupN) acts as multiplication by

ei(−p0x0+p1x1+p2x2+p3x3)

375

Equivalently, the p0, p1, p2, p3 are the eigenvalues of the energy and momentumoperators

P0 = −iπ′(t0), P1 = iπ′(t1), P2 = iπ′(t2), P3 = iπ′(t3)

that give the representation of the translation part of the Poincare group Liealgebra on the states.

The Lorentz group acts on this R4 by

p→ Λp

and, restricting attention to the p0 − p3 plane, the picture of the orbits lookslike this

Unlike the Euclidean group case, here there are several different kinds oforbits Oα. We’ll examine them and the corresponding stabilizer groups Kα

each in turn, and see what can be said about the associated representations.One way to understand the equations describing these orbits is to note thatthe different orbits correspond to different eigenvalues of the Poincare groupCasimir operator

P 2 = −P 20 + P 2

1 + P 22 + P 2

3

376

This operator commutes with all the generators of the Lie algebra of the Poincaregroup, so by Schur’s lemma it must act as a scalar times the identity on anirreducible representation (recall that the same phenomenon occurs for SU(2)representations, which can be characterized by the eigenvalue j(j+1) of the Cas-mir operator J2 for SU(2)). At a point p = (p0, p1, p2, p3) in energy-momentumspace, the Pj operators are diagonalized and P 2 will act by the scalar

−p20 + p2

1 + p22 + p2

3

which can be positive, negative, or zero, so given by m2,−m2, 0 for variousm. The value of the scalar will be the same everywhere on the orbit, so inenergy-momentum space orbits will satisfy one of the three equations

−p20 + p2

1 + p22 + p2

3 =

−m2

m2

0

Note that in this chapter we are just classifying Poincare group representa-tions, not actually constructing them. It is possible to construct these represen-tations using the data we will find that classifies them, but this would requireintroducing some techniques (for so-called “induced representations”) that gobeyond the scope of this course. In later chapters we will explicitly constructthese representations in certain specific cases as solutions to certain relativisticwave equations.

37.2.1 Positive energy time-like orbits

One way to get negative values −m2 of the Casimir P 2 is to take the vectorp = (m, 0, 0, 0), m > 0 and generate an orbit Om,0,0,0 by acting on it withthe Lorentz group. This will be the upper, positive energy, hyperboloid of thehyperboloid of two sheets

−p20 + p2

1 + p22 + p2

3 = −m2

so

p0 =√p2

1 + p22 + p2

3 +m2

The stabilizer group of Km,0,0,0 is the subgroup of SO(3, 1) of elements ofthe form (

1 00 Ω

)where Ω ∈ SO(3), so Km,0,0,0 = SO(3). Irreducible representations are classi-fied by the spin. For spin 0, points on the hyperboloid can be identified withpositive energy solutions to a wave equation called the Klein-Gordon equationand functions on the hyperboloid both correspond to the space of all solutionsof this equation and carry an irreducible representation of the Poincare group.

377

In the next chapter we will study the Klein-Gordon equation, as well as thequantization of the space of its solutions by quantum field theory methods.

We will later study the case of spin 12 , where one must use the double cover

SU(2) of SO(3). The Poincare group representation will be on functions onthe orbit that take values in two copies of the spinor representation of SU(2).These will correspond to solutions of a wave equation called the massive Diracequation.

For choices of higher spin representations of the stabilizer group, one canagain find appropriate wave equations and construct Poincare group represen-tations on their space of solutions, but we will not enter into this topic.

37.2.2 Negative energy time-like orbits

Starting instead with the energy-momentum vector p = (−m, 0, 0, 0), m > 0,the orbit O−m,0,0,0 one gets is the lower, negative energy component of thehyperboloid

−p20 + p2

1 + p22 + p2

3 = −m2

satisfying

p0 = −√p2

1 + p22 + p2

3 +m2

Again, one has the same stabilizer group K−m,0,0,0 = SO(3) and the same con-stuctions of wave equations of various spins and Poincare group representationson their solution spaces as in the positive energy case. Since negative energieslead to unstable, unphysical theories, we will see that these representations aretreated differently under quantization, corresponding physically not to particles,but to anti-particles.

37.2.3 Space-like orbits

One can get positive values m2 of the Casimir P 2 by considering the orbitO0,0,0,m of the vector p = (0, 0, 0,m). This is a hyperboloid of one sheet,satisfying the equation

−p20 + p2

1 + p22 + p2

3 = m2

It is not too difficult to see that the stabilizer group of the orbit is K0,0,0,m =SO(2, 1). This is isomorphic to the group SL(2,R), and it has no finite-dimensional unitary representations. These orbits correspond physically to“tachyons”, particles that move faster than the speed of light, and there isno known way to consistently incorporate them in a conventional theory.

37.2.4 The zero orbit

The simplest case where the Casimir P 2 is zero is the trivial case of a pointp = (0, 0, 0, 0). This is invariant under the full Lorentz group, so the orbitO0,0,0,0 is just a single point and the stabilizer group K0,0,0,0 is the entire Lorentz

378

group SO(3, 1). For each finite-dimensional representation of SO(3, 1), one getsa corresponding finite dimensional representation of the Poincare group, withtranslations acting trivially. These representations are not unitary, so not usablefor our purposes.

37.2.5 Positive energy null orbits

One has P 2 = 0 not only for the zero-vector in momentum space, but for athree-dimensional set of energy-momentum vectors, called the null-cone. Bythe term “cone” one means that if a vector is in the space, so are all productsof the vector times a positive number. Vectors p = (p0, p1, p2, p3) are called“light-like” or “null” when they satisfy

|p|2 = −p20 + p2

1 + p22 + p2

3 = 0

One such vector is p = (1, 0, 0, 1) and the orbit of the vector under the actionof the Lorentz group will be the upper half of the full null-cone, the half withenergy p0 > 0, satisfying

p0 =√p2

1 + p22 + p2

3

The stabilizer group K1,0,0,1 of p = (1, 0, 0, 1) includes rotations about thex3 axis, but also boosts in the other two directions. It is isomorphic to theEuclidean group E(2). Recall that this is a semi-direct product group, and ithas two sorts of irreducible representations

• Representations such that the two translations act trivially. These areirreducible representations of SO(2), so one-dimensional and characterizedby an integer n (half-integers when one uses the Poincare group doublecover).

• Infinite dimensional irreducible representations on a space of functions ona circle of radius r

The first of these two gives irreducible representations of the Poincare groupon certain functions on the positive energy null-cone, labeled by the integer n,which is called the “helicity” of the representation. We will in later chaptersconsider the cases n = 0 (massless scalars, wave-equation the Klein-Gordonequation), n = ± 1

2 (Weyl spinors, wave equation the Weyl equation), and n =±1 (photons, wave equation the Maxwell equations).

The second sort of representation of E(2) gives representations of the Poincaregroup known as “continuous spin” representations, but these seem not to cor-respond to any known physical phenomena.

37.2.6 Negative energy null orbits

Looking instead at the orbit of p = (−1, 0, 0, 1), one gets the negative energypart of the null-cone. As with the time-like hyperboloids of non-zero massm, these will correspond to anti-particles instead of particles, with the sameclassification as in the positive energy case.

379


The Poincare group and its Lie algebra is discussed in pretty much any quantumfield theory textbook. Weinberg [66] (Chapter 2) has some discussion of therepresentations of the Poincare group on single particle state spaces that we haveclassified here. Folland [20] (Chapter 4.4) and Berndt [8] (Chapter 7.5) discussthe actual construction of these representations using the induced representationmethods that we have chosen not to try and explain here.

380

Chapter 38

The Klein-Gordon Equationand Scalar Quantum Fields

In the non-relativistic case we found that it was possible to build a quantumtheory describing arbitrary numbers of particles by “second quantization” of thestandard quantum theory of a free particle. This was done by taking as classicalphase space the space of solutions to the free particle Schrodinger equation, aspace which carries a unitary representation of the Euclidean group E(3). This isan infinite dimensional space of functions (the space of solutions can be identifiedwith the space of initial conditions, which is the space of wave-functions at afixed time), but one can quantize it using analogous methods to the case ofthe finite-dimensional harmonic oscillator (annihilation and creation operators).After such quantization we get a quantum field theory, with a state space thatdescribes an arbitrary number of particles. Such a state space provides a unitaryrepresentation of the E(3) group and we saw how to construct the momentumand angular momentum operators that generate it.

To make the same sort of construction for relativistic systems, we want tostart with an irreducible unitary representation not of E(3), but of the Poincaregroup P. In the last chapter we saw that such things were classified by orbitsOα of the Lorentz group on momentum space, together with a choice of rep-resentation of the stabilizer group Kα of the orbit. The simplest case will bethe orbits Om,0,0,0, and the choice of the trivial spin-zero representation of thestabilizer group Km,0,0,0 = SO(3). These orbits are characterized by a positivereal number m, and are hyperboloids in energy-momentum space. Points onthese orbits correspond to solutions of a relativistic analog of the Schrodingerrepresentation, the Klein-Gordon equation, so we will begin by studying thisequation and its solutions.

381

38.1 The Klein-Gordon equation and its solu-tions

Recall that a condition characterizing the orbit in momentum space that wewant to study was that the Casimir operator P 2 of the Poincare group acts onthe representation corresponding to the orbit as the scalar m2. So, we have theoperator equation

P 2 = −P 20 + P 2

1 + P 22 + P 2

3 = −m2

characterizing the Poincare group representation we are interested in. Interpret-ing the Pj as the standard differentiation operators −i ∂

∂xjon a space of wave-

functions, generating the infinitesimal action of the translation group on suchwave-functions, we get the following differential equation for wave-functions:

Definition (Klein-Gordon equation). The Klein-Gordon equation is the second-order partial differential equation

(− ∂2

∂t2+

∂2

∂x21

+∂2

∂x22

+∂2

∂x23

)φ = m2φ

or

(− ∂2

∂t2+ ∆−m2)φ = 0

for functions φ(x) on Minkowski space (which may be real or complex valued).

This equation is the simplest Lorentz-invariant wave equation to try, andhistorically was the one Schrodinger first tried (he then realized it could notaccount for atomic spectra and instead used the non-relativistic equation thatbears his name). Taking Fourier transforms

φ(p) =1

(2π)2

∫d4xe−i(−p0x0+p·x)φ(x)

the Klein-Gordon equation becomes

(p20 − p2

1 − p22 − p2

3 −m2)φ(p) = 0

Solutions to this will be functions φ(p) that are non-zero only on the hyperboloid

p20 − p2

1 − p22 − p2

3 −m2 = 0

in energy-momentum space R4. This hyperboloid has two components, withpositive and negative energy

p0 = ±ωp

where

ωp =√p2

1 + p22 + p2

3 +m2

Ignoring one dimension these look like

382

In the non-relativistic case, a continuous basis of solutions of the Schrodingerequation labeled by p ∈ R3 was given by the functions

eip·xe−i|p|22m t

with a general solution a superposition of these with coefficients ψ(p) given bythe Fourier inversion formula

ψ(x, t) =1

(2π)3/2

∫R3

ψ(p)eip·xe−i|p|22m td3x

The complex values ψ(p) gave coordinates on our single-particle space H1, andwe had actions on this of the group of time translations (generated by the Hamil-tonian) and the Euclidean group E(3) (generated by momentum and angularmomentum).

In the relativistic case we want to study the corresponding single-particlespace H1 of solutions to the Klein-Gordon equation, but parametrized in a waythat makes clear the action of the Poincare group on this space. Coordinates onthe space of such solutions will now be given by complex-valued functions φ(p) onthe energy-momentum space R4, supported on the two-component hyperboloid(here p = (p0,p)). The Fourier inversion formula giving a general solution interms of these coordinates will be

φ(x, t) =1

(2π)3/2

∫M4

δ(p20 − ω2

p)φ(p)ei(p·x−p0t)d4p

383

with the integral over the 3d hyperboloid expressed as a 4d integral over R4

with a delta-function on the hyperboloid in the argument.The delta function distribution with argument a function f(x) depends only

on the zeros of f , and if f ′ 6= 0 at such zeros, one has

δ(f(x)) =∑

xj :f(xj)=0

δ(f ′(xj)(x− xj)) =1

|f ′(xj)|δ(x− xj)

For each p, one can apply this to the case of the function of p0 given by

f = p20 − ω2

p

on R4, and usingd

dp0(p2

0 − ω2p) = 2p0 = ±2ωp

one finds

φ(x, t) =1

(2π)3/2

∫M4

1

2ωp(δ(p0 − ωp) + δ(p0 + ωp))φ(p)ei(p·x−p0t)dp0d

3p

=1

(2π)3/2

∫R3

(φ+(p)e−iωpt + φ−(p)eiωpt)eip·xd3p

2ωp

Here

φ+(p) = φ(ωp,p), φ−(p) = φ(−ωp,p)

are the values of φ on the positive and negative energy hyperboloids. We seethat instead of thinking of the Fourier transforms of solutions as taking valueson energy-momentum hyperboloids, we can think of them as taking values juston the space R3 of momenta (just as in the non-relativistic case), but we dohave to use both positive and negative energy Fourier components, and to geta Lorentz invariant measure need to use

d3p

2ωp

instead of d3p.A general complex-valued solution to the Klein-Gordon equation will be

given by the two complex-valued functions φ+, φ−, but we can impose thecondition that the solution be real-valued, in which case one can check that thepair of functions must satisfy the condition

φ−(p) = φ+(−p)

Real-valued solutions of the Klein-Gordon equation thus correspond to arbitrarycomplex-valued functions φ+ defined on the positive energy hyperboloid, which

fixes the value of the other function φ−.

384

38.2 Classical relativistic scalar field theory

We would like to set up the Hamiltonian formalism, finding a phase space H1

and a Hamiltonian function h on it such that Hamilton’s equations will give usthe Klein-Gordon equation as equation of motion. Such a phase space will bean infinite-dimensional function space and the Hamiltonian will be a functional.We will here blithely ignore the analytic difficulties of working with such spaces,and use physicist’s methods, with formulas that can be given a legitimate inter-pretation by being more careful and using distributions. Note that now we willtake the fields φ to be real-valued, this is the so-called real scalar field.

Since the Klein-Gordon equation is second order in time, solutions will beparametrized by initial data which, unlike the non-relativistic case now requiresthe specification at t = 0 of not one, but two functions,

φ(x) = φ(x, 0), φ(x) =∂

∂tφ(x, t)|t=0

the values of the field and its first time derivative.We will take as our phase space H1 the space of pairs of functions (φ, π),

with coordinates φ(x), π(x) and Poisson brackets

φ(x), π(x′) = δ(x− x′), φ(x), φ(x′) = π(x), π(x′) = 0

We want to get the Klein-Gordon equation for φ(x, t) as the following pair offirst order equations

∂

∂tφ = π,

∂

∂tπ = (∆−m2)φ

which together imply∂2

∂t2φ = (∆−m2)φ

To get these as equations of motion, we just need to find a Hamiltonianfunction h on the phase space H1 such that

∂

∂tφ =φ, h = π

∂

∂tπ =π, h = (∆−m2)φ

One can check that two choices of Hamiltonian function that will have thisproperty are

h =

∫R3

H(x)d3x

where

H =1

2(π2 − φ∆φ+m2φ2) or H =

1

2(π2 + (∇φ)2 +m2φ2)

where the two different integrands H(x) are related (as in the non-relativisticcase) by integration by parts so these just differ by boundary terms that areassumed to vanish.

385

One could instead have taken as starting point the Lagrangian formalism,with an action

S =

∫M4

L d4x

where

L =1

2((∂

∂tφ)2 − (∇φ)2 −m2φ2)

This action is a functional now of fields on Minkowski space M4 and is Lorentzinvariant. The Euler-Lagrange equations give as equation of motion the Klein-Gordon equation

(2−m2)φ = 0

One recovers the Hamiltonian formalism by seeing that the canonical momentumfor φ is

π =∂L∂φ

= φ

and the Hamiltonian density is

H = πφ− L =1

2(π2 + (∇φ)2 +m2φ2)

Besides the position-space Hamiltonian formalism, we would like to have onefor the momentum space components of the field, since for a free field it is thesethat will decouple into an infinite collection of harmonic oscillators. For a realsolution to the Klein-Gordon equation we have

φ(x, t) =1

(2π)3/2

∫R3

(φ+(p)e−iωpt + φ+(−p)eiωpt)eip·xd3p

2ωp

=1

(2π)3/2

∫R3

(φ+(p)e−iωpteip·x + φ+(p)eiωpte−ip·x)d3p

2ωp

where we have used the symmetry of the integration over p to integrate over−p instead of p.

We can choose a new way of normalizing Fourier coefficients, one that reflectsthe fact that the Lorentz-invariant notion is that of integrating over the energy-momentum hyperboloid rather than momentum space

α(p) =φ+(p)√

2ωp

and in terms of these we have

φ(x, t) =1

(2π)3/2

∫R3

(α(p)e−iωpteip·x + α(p)eiωpte−ip·x)d3p√2ωp

386

The α(p), α(p) will have the same sort of Poisson bracket relations as thez, z for a single harmonic oscillator, or the α(p), α(p) Fourier coefficients in thecase of the non-relativistic field:

α(p), α(p′) = −iδ3(p− p′), α(p), α(p′)) = α(p), α(p′)) = 0

To see this, one can compute the Poisson brackets for the fields as follows. Wehave

π(x) =∂

∂tφ(x, t)|t=0 =

1

(2π)3/2

∫R3

(−iωp)(α(p)eip·x − α(p)e−ip·x)d3p√2ωp

and

φ(x) =1

(2π)3/2

∫R3

(α(p)eip·x + α(p)e−ip·x)d3p√2ωp

so

φ(x), π(x′) =1

2(2π)3

∫R3×R3

(α(p), iα(p′)ei(p·x−p′x′)

− iα(p), α(p′)ei(−p·x+p′x′))d3pd3p′

=1

2(2π)3

∫R3×R3

δ3(p− p′)(ei(p·x−p′x′) + ei(−p·x+p′x′))d3pd3p′

=1

2(2π)3

∫R3

(eip·(x−x′) + e−ip·(x−x

′))d3p

=δ3(x− x′)

As in the non-relativistic case, one really should work elements of H∗1 of theform (for appropriately chosen class of functions f, g)

φ(f) + π(g) =

∫R3

(f(x)φ(x) + g(x)π(x))d3x

getting Poisson bracket relations

φ(f1) + π(g1), φ(f2) + π(g2) =

∫R3

(f1(x)g2(x)− f2(x)g1(x))d3x

This is just the infinite-dimensional analog of the Poisson bracket of two linearcombinations of the qj , pj , with the right-hand side the symplectic form Ω onH∗1.

38.3 The complex structure on the space of Klein-Gordon solutions

Recall from chapter 20 that if we intend to quantize a classical phase space Mby the Bargmann-Fock method, we need to choose a complex structure J onthat phase space (or on the dual phase space M = M∗). Then

M⊗C =M+J ⊕M

−J

387

where M+J is the +i eigenspace of J , M−J the −i eigenspace. The quantum

state space will be the space of polynomials on the dual of M+J . The choice of

J corresponds to a choice of distinguished state |0〉J ∈ H, the Bargmann-Fockstate given by the constant polynomial function 1.

In the non-relativistic quantum field theory case we saw that basis elementsofM could be taken to be either the linear functionals ψ(x) and their conjugatesψ(x) or, Fourier transforming, the linear functionals α(p) and their conjugatesα(p). These coordinates are not real-valued, but complex-valued, and as a resultM came with a distinguished natural complex structure J , which is +i on theψ(x) or the α(p), and −i on their conjugates.

In the relativistic scalar field theory, we must do something very different.The solutions to the Klein-Gordon equation we are considering are real-valued,not complex-valued functions, and give a real phase space M to be quantized(what happens when we consider a theory with configuration space complex val-ued fields will be discussed in chapter 39). When we complexify and look at thespaceM⊗C, it naturally decomposes as a representation of the Poincare groupinto two pieces: M+, the complex functions on the positive energy hyperboloidand M−, the complex functions on the negative energy hyperboloid. More ex-plicitly, we can decompose a complexified solution φ(x, t) of the Klein-Gordonequation as φ = φ+ + φ−, where

φ+(x, t) =1

(2π)3/2

∫R3

α(p)e−iωpteip·xd3p√2ωp

and

φ−(x, t) =1

(2π)3/2

∫R3

α(p)eiωpte−ip·xd3p√2ωp

We will take as complex structure the operator J that is +i on positiveenergy wave-functions and −i on negative energy wavefunctions. Complexifiedclassical fields in M+ get quantized as annihilation operators, those in M−as creation operators. Since conjugation interchanges M+ and M−, non-zeroreal-valued classical fields have components in both M+ and M− since theyare their own conjugates.

One motivation for this particular choice of J is that it leads to a state spacewith states of non-negative energy. Theories with states of arbitrarily negativeenergy are considered undesirable since they will tend to have no stable vacuumstate (since any supposed vacuum state could potentially decay into states oflarge positive and large negative energy, while preserving total energy). To seethe mechanism for non-negative energy, first consider again the non-relativisticcase, where the Hamiltonian is (for d = 1)

h =1

2m

∫ +∞

−∞| ddxψ(x)|2dx =

∫ ∞−∞

p2

2m|α(p)|2dp

This is positive definite, either on M+ (the ψ) or M− (the ψ). Quantizationtakes

h→ H =

∫ ∞−∞

p2

2ma†(p)a(p)dp

388

which is an operator with positive eigenvalues (this is in normal-ordered form,but the non-normal-ordered version is still positive, although adding an infinitepositive constant).

WARNING: HAVEN’T FINISHED REWRITING REST OF THIS SEC-TION.

The Hamiltonian function h is the quadratic polynomial function of thecoordinates φ(x), π(x)

h =

∫R3

1

2(π2 + (∇φ)2 +m2φ2)d3x

and by laborious calculation one can substitute the above expressions for φ, πin terms of α(p), α(p) to find h as a quadratic polynomial in these coordinateson the momentum space fields. A quicker way to find the correct expression isto use the fact that different momentum components of the field decouple, andwe know the time-dependence of such components, so just need to find the righth that generates this.

If, as in the non-relativistic case, we interpret φ as a single-particle wave-function, Hamilton’s equation of motion says

φ, h =∂

∂tφ

and applying this to the component of φ+ with momentum p, we just getmultiplication by −iωp. The energy of such a wave-function would be ωp, theeigenvalue of i ∂∂t . These are called “positive frequency” or “positive energy”wave-functions. In the case of momentum components of φ−, the eigenvalue is−ωp, and one has “negative frequency” or “negative energy” wave-functions.

An expression for h in terms of momentum space field coordinates that willhave the right Poisson brackets on φ+, φ− is

h =

∫R3

ωpα(p)α(p)d3p

and this is the same expression one could have gotten by a long direct calcula-tion.

In the non-relativistic case, the eigenvalues of the action of i ∂∂t on the

wave-functions ψ were non-negative ( |p|2

2m ) so the single-particle states had non-negative energy. Here we find instead eigenvalues ±ωp of both signs, so single-particle states can have arbitrarily negative energies. This makes a physicallysensible interpretation of H1 as a space of wavefunctions describing a singlerelativistic particle difficult if not impossible. We will however see in the nextsection that there is a way to quantize this H1 as a phase space, getting asensible multi-particle theory with a stable ground state.

38.4 Quantization of the real scalar field

Given the description we have found in momentum space of a real scalar fieldsatisfying the Klein-Gordon equation, it is clear that one can proceed to quan-

389

tize the theory in exactly the same way as was done with the non-relativisticSchrodinger equation, taking momentum components of fields to operators byreplacing

α(p)→ a(p), α(p)→ a†(p)

where a(p), a†(p) are operator valued distributions satisfying the commutationrelations

[a(p), a†(p′)] = δ3(p− p′)

For the Hamiltonian we take the normal-ordered form

H =

∫R3

ωpa†(p)a(p)d3p

Starting with a vacuum state |0〉, by applying creation operators one can createarbitary positive energy multiparticle states of free relativistic particles withsingle-particle states having the energy momentum relation

E(p) = ωp =√|p|2 +m2

If we try and consider not momentum eigenstates, but position eigenstates,in the non-relativistic case we could define a position space complex-valued fieldoperator by

ψ(x) =1

(2π)3/2

∫R3

a(p)eip·xd3p

which has an interpretation as an annhilation operator for a particle localizedat x. Solving the dynamics of the theory gave the time-dependence of the fieldoperator

ψ(x, t) =1

(2π)3/2

∫R3

a(p)e−ip2

2m teip·xd3p

For the relativistic case we must do something somewhat different, defining

Definition (Real scalar quantum field). The real scalar quantum field operatorsare the operator-valued distributions defined by

φ(x) =1

(2π)3/2

∫R3

(a(p)eip·x + a†(p)e−ip·x)d3p√2ωp

(38.1)

π(x) =1

(2π)3/2

∫R3

(−iωp)(a(p)eip·x − a†(p)e−ip·x)d3p√2ωp

(38.2)

By essentially the same computation as for Poisson brackets, one can com-pute commutators, finding

[φ(x), π(x′)] = iδ3(x− x′), [φ(x), φ(x′)] = [π(x), π(x′)] = 0

These can be interpreted as the relations of a unitary representation of a Heisen-berg Lie algebra, now the infinite dimensional Lie algebra corresponding to thephase space H1 of solutions of the Klein-Gordon equation.

390

The Hamiltonian operator will be quadratic in the field operators and canbe chosen to be

H =

∫R3

1

2: (π(x)2 + (∇φ(x))2 +m2φ(x)2) : d3x

This operator is normal ordered, and a computation (see for instance [10]) showsthat in terms of momentum space operators this is just

H =

∫R3

ωpa†(p)a(p)d3p

the Hamiltonian operator discussed earlier.The dynamical equations of the quantum field theory are now

∂

∂tφ = [φ,−iH] = π

∂

∂tπ = [π,−iH] = (∆−m2)φ

which have as solution the following equation for the time-dependent field op-erator:

φ(x, t) =1

(2π)3/2

∫R3

(a(p)e−iωpteip·x + a†(p)eiωpte−ip·x)d3p√2ωp

This is a superposition of annihilation operators for momentum eigenstatesof positive energy and creation operators for momentum eigenstates of negativeenergy. Note that, unlike the non-relativistic case, here the quantum field oper-ator is self-adjoint. In the next chapter we will see what happens in the case ofa complex scalar quantum field, where the operator and its adjoint are distinct.

It is a characteristic feature of relativistic field theory that what one is quan-tizing is not just a space of positive energy wave-functions, but a space thatincludes both positive and negative energy wave-functions, assigning creationoperators to one sign of the energy, annihilation operators to the other, and bythis mechanism getting a Hamiltonian operator with spectrum bounded below.In more complicated quantum field theories there will be other operators (forinstance, a charge operator) that can distinguish between “particle” states thatcorrespond to wave-functions of positive energy and “anti-particle” states thatcorrespond to wave-functions of negative energy. The real scalar field case israther special in that there are no such operators and one says that here “aparticle is its own anti-particle.”

38.5 The propagator

As explained in the non-relativistic case, in quantum field theory explicitlydealing with states and their time-dependence is awkward, so we work in theHeisenberg picture, expressing everything in terms of a fixed, unchangeable

391

state |0〉 and time-dependent operators. For the free scalar field theory, we haveexplicitly solved for the time-dependence of the field operators. A basic quantityneeded for describing the propagation of quanta of a quantum field theory isthe propagator:

Definition (Green’s function or propagator, scalar field theory). The Green’sfunction or propagator for a scalar field theory is the amplitude, for t > t′

G(x, t,x′, t′) = 〈0|φ(x, t)φ(x′, t′)|0〉

By translation invariance, the propagator will only depend on t − t′ andx − x′, so we can just evaluate the case (x′, t′) = (0, 0), using the formula forthe time dependent field to get

G(x, t,0, 0) =1

(2π)3

∫R3×R3

〈0|(a(p)e−iωpteip·x + a†(p)eiωpte−ip·x)

(a(p′) + a†(p′))|0〉 d3p√2ωp

d3p′√2ωp′

=

∫R3×R3

δ3(p− p′)e−iωpteip·xd3p√2ωp

d3p′√2ωp′

=

∫R3

e−iωpteip·xd3p

2ωp

For t > 0, this gives the amplitude for propagation of a particle in time tfrom the origin to the point x.

Plan to expand this section. Compare to non-relativistic propagator. Com-pute commutator of fields at arbitrary space-time separations, show that com-mutator of fields at space-like separations vanishes.

38.6 Fermionic scalars

Explain nature of problems. Non-positivity of the Hamiltonian. Commutatorat space-like separations does not vanish.


Pretty much every quantum field theory textbook has a treatment of the rela-tivistic scalar field with more details than here, and significantly more physicalmotivation. A good example with some detailed versions of the calculationsdone here is chapter 5 of [10]. See Folland [20], chapter 5 for a mathematicallymore careful treatment of the distributional nature of the scalar field operators.

392

Chapter 39

Symmetries and RelativisticScalar Quantum Fields

Just as for non-relativistic quantum fields, the theory of free relativistic scalarquantum fields starts by taking as phase space an infinite dimensional space ofsolutions of an equation of motion. Quantization of this phase space involvesconstructing field operators which provide a representation of the correspondingHeisenberg Lie algebra, by an infinite dimensional version of the Bargmann-Fockconstruction. The equation of motion has its own representation-theoreticalsignificance: it is an eigenvalue equation for a Casimir operator of a group ofspace-time symmetries, picking out an irreducible representation of that group.In this case the Casimir operator is the Klein-Gordon operator, and the space-time symmetry group is the Poincare group. The Poincare group acts on thephase space of solutions to the Klein-Gordon equation, preserving the Poissonbracket. One can thus use the same methods as in the finite-dimensional caseto get a representation of the Poincare group by intertwining operators forthe Heisenberg Lie algebra representation (that representation is given by thefield operators). These methods give a representation of the Lie algebra of thePoincare group in terms of quadratic combinations of the field operators.

We’ll begin with the case of an even simpler group action on the phase space,that coming from an “internal symmetry” one gets if one takes multi-componentscalar fields, with an orthogonal group or unitary group acting on the real orcomplex vector space in which the classical fields take their values.

39.1 Internal symmetries

The real scalar field theory of chapter 38 lacks one feature of the non-relativistictheory, which is an action of the group U(1) by phase changes on complexfields. This is needed to provide a notion of “charge” and allow the introductionof electromagnetism into the theory. In the real scalar field theory there isno distinction between states describing particles and states describing anti-

393

particles. To get a theory with such a distinction we need to introduce fieldswith more components. Two possibilities are to consider real fields with mcomponents, in which case we will have a theory with SO(m) symmetry, andU(1) = SO(2) the m = 2 special case, or to consider complex fields with mcomponents, in which case we have theories with U(m) symmetry, and m = 1the U(1) special case.

39.1.1 SO(m) symmetry and real scalar fields

Taking as single particle phase spaceH1 the space of pairs φ1, φ2 of real solutionsto the Klein-Gordon equation, elements g(θ) of the group SO(2) will act on thedual phase space H∗1 of coordinates on such solutions by(

φ1(x)φ2(x)

)→ g(θ) ·

(φ1(x)φ2(x)

)=


)(φ1(x)φ2(x)

)(π1(x)π2(x)

)→ g(θ) ·

(π1(x)π2(x)

)=


)(π1(x)π2(x)

)Here φ1(x), φ2(x), π1(x), π2(x) are the coordinates for initial values at t = 0 of aKlein-Gordon solution. The Fourier transforms of solutions behave in the samemanner.

This group action on H1 breaks up into a direct sum of an infinite number(one for each value of x) of identical cases of rotations in a configuration spaceplane, as discussed in section 16.3.2. We will use the calculation there, where wefound that for a basis element L of the Lie algebra of SO(2) the correspondingquadratic function on the phase space with coordinates q1, q2, p1, p2 was

µL = q1p2 − q2p1

For the case here, we just take

q1, q2, p1, p2 → φ1(x), φ2(x), π1(x), π2(x)

To get a quadratic functional on the fields that will have the desired Poissonbracket with the fields for each value of x, we need to just integrate the analogof µL over R3. We will denote the result by Q, since it is an observable thatwill have a physical interpretation as electric charge when this theory is coupledto the electromagnetic field (see chapter 40):

Q =

∫R3

(π2(x)φ1(x)− π1(x)φ2(x))d3x

One can use the field Poisson bracket relations

φj(x), πk(x′) = δjkδ(x− x′)

to check that

Q,(φ1(x)φ2(x)

) =

(−φ2(x)φ1(x)

), Q,

(π1(x)π2(x)

) =

(−π2(x)π1(x)

)394

Quantization of the classical field theory gives us a unitary representation Uof SO(2), with

U ′(L) = −iQ = −i∫R3

(π2(x)φ1(x)− π1(x)φ2(x))d3x

The operator

U(θ) = e−iθQ

will act by conjugation on the fields:

U(θ)

(φ1(x)

φ2(x)

)U(θ)−1 =


)(φ1(x)

φ2(x)

)

U(θ)

(π1(x)π2(x)

)U(θ)−1 =


)(π1(x)π2(x)

)It will also give a representation of SO(2) on states, with the state space de-

composing into sectors each labeled by the integer eigenvalue of the operator Q(which will be called the “charge” of the state).

Using the definitions of φ and π (38.1 and 38.2) one can compute Q in termsof annihilation and creation operators, with the result

Q = i

∫R3

(a†2(p)a1(p)− a†1(p)a2(p))d3p (39.1)

One expects that since the time evolution action on the classical field spacecommutes with the SO(2) action, the operator Q should commute with the

Hamiltonian operator H. This can readily be checked by computing [H, Q]using

H =

∫R3

ωp(a†1(p)a1(p) + a†2(p)a2(p))d3p

Note that the vacuum state |0〉 is an eigenvector for Q and H with eigenvalue

0: it has zero energy and zero charge. States a†1(p)|0〉 and a†2(p)|0〉 are eigen-

vectors of H with eigenvalue and thus energy ωp, but these are not eigenvectors

of Q, so do not have a well-defined charge.All of this can be generalized to the case of m > 2 real scalar fields, with a

larger group SO(m) now acting instead of the group SO(2). The Lie algebra isnow multi-dimensional, with a basis the elementary anti-symmetric matrices εjk,with j, k = 1, 2, · · · ,m and j < k, which correspond to infinitesimal rotations inthe j − k planes. Group elements can be constructed by multiplying rotationseθεjk in different planes. Instead of a single operator Q, we get multiple operators

−iQjk = −i∫R3

(πk(x)φj(x)− πj(x)φk(x))d3x

and conjugation by

Ujk(θ) = e−iθQjk

395

rotates the field operators in the j − k plane. These also provide unitary oper-ators on the state space, and, taking appropriate products of them, a unitaryrepresentation of the full group SO(m) on the state space. The Qjk commutewith the Hamiltonian, so the energy eigenstates of the theory break up into ir-reducible representations of SO(m) (a subject we haven’t discussed for m > 3).

39.1.2 U(1) symmetry and complex scalar fields

Instead of describing a scalar field system with SO(2) symmetry using a pairφ1, φ2 of real fields, it is more covenient to identify the R2 that the fields takevalues in with C, and work with complex scalar fields and a U(1) symmetry.This will allow us to work with a set of annihilation and creation operatorsfor particle states with a definite value of the charge observable. Note that wewere already forced to introduce a complex structure (given by the splitting ofcomplexified solutions of the Klein-Gordon equation into positive and negativeenergy solutions) as part of the Bargmann-Fock quantization. This is a secondand independent source of complex numbers in the theory.

We express a pair of real-valued fields as complex-valued fields using

φ =1√2

(φ1 + iφ2), π =1√2

(π1 − iπ2)

These can be thought of as initial-value data parametrizing complex solutionsof the Klein-Gordon equation, giving a phase space that is infinite-dimensional,with four real dimensions for each value of x. Instead of

φ1(x), φ1(x), π1(x), π2(x)

we can think of the complex-valued fields and their complex conjugates

φ(x), φ(x), π(x), π(x)

as providing a real basis of the coordinates on phase space.The Poisson bracket relations on such complex fields will be

φ(x), φ(x′) = π(x), π(x′) = φ(x), π(x′) = φ(x), π(x′) = 0

φ(x), π(x′) = φ(x), π(x′) = δ(x− x′)

and the classical Hamiltonian is

h =

∫R3

(|π|2 + |∇φ|2 +m2|φ|2)d3x

Note that introducing complex fields in a theory like this with field equationsthat are second-order in time means that for each x we have a phase spacewith two complex dimensions (φ(x) and π(x)). Using Bargmann-Fock methodsrequires complexifying one’s phase space, which is a bit confusing here sincethe phase space is already is given in terms of complex fields. We can howeverproceed to find the operator that generates the U(1) symmetry as follows.

396

In terms of complex fields, the SO(2) transformations on the pair φ1, φ2 ofreal fields become U(1) phase transformations, with Q now given by

Q = −i∫R3

(π(x)φ(x)− π(x)φ(x))d3x

satisfying

Q,φ(x) = iφ(x), Q,φ(x) = −iφ(x)

Quantization of the classical field theory gives a representation of the infinitedimensional Heisenberg algebra with commutation relations

[φ(x), φ(x′)] = [π(x), π(x′)] = [φ†(x), φ†(x′)] = [π†(x), π†(x′)] = 0

[φ(x), π(x′)] = [φ†(x), π†(x′)] = iδ3(x− x′)

Quantization of the quadratic functional Q of the fields is done with the normal-ordering prescription, to get

Q = −i∫R3

: (π(x)φ(x)− π†(x)φ†(x)) : d3x

Taking L = i as a basis element for u(1), one gets a unitary representationU of U(1) using

U ′(L) = −iQ

and

U(θ) = e−iθQ

U acts by conjugation on the fields:

U(θ)φU(θ)−1 = e−iθφ, U(θ)φ†U(θ)−1 = eiθφ†

U(θ)πU(θ)−1 = eiθπ, U(θ)π†U(θ)−1 = e−iθπ†

It will also give a representation of U(1) on states, with the state space de-composing into sectors each labeled by the integer eigenvalue of the operatorQ.

In the Bargmann-Fock quantization of this theory, we can express quantumfields now in terms of a different set of two annihilation and creation operators

a(p) =1√2

(a1(p) + ia2(p)), a†(p) =1√2

(a†1(p)− ia†2(p))

b(p) =1√2

(a1(p)− ia2(p)), b†(p) =1√2

(a†1(p) + ia†2(p))

The only non-zero commutation relations between these operators will be

[a(p), a†(p′)] = δ(p− p′), [b(p), b†(p′)] = δ(p− p′)

397

so we see that we have, for each p, two independent sets of standard annihilationand creation operators, which will act on a tensor product of two standardharmonic oscillator state spaces. The states created and annihilated by thea†(p) and a(p) operators will have an interpretation as particles of momentump, whereas those created and annihilated by the b†(p) and b(p) operators willbe anti-particles of momentum p. The vacuum state will satisfy

a(p)|0〉 = b(p)|0〉 = 0

Using these creation and annihilation operators, the definition of the complexfield operators is

Definition (Complex scalar quantum field). The complex scalar quantum fieldoperators are the operator-valued distributions defined by

φ(x) =1

(2π)3/2

∫R3

(a(p)eip·x + b†(p)e−ip·x)d3p√2ωp

φ†(x) =1

(2π)3/2

∫R3

(b(p)eip·x + a†(p)e−ip·x)d3p√2ωp

π(x) =1

(2π)3/2

∫R3

(−iωp)(a(p)eip·x − b†(p)e−ip·x)d3p√2ωp

π†(x) =1

(2π)3/2

∫R3

(−iωp)(b(p)eip·x − a†(p)e−ip·x)d3p√2ωp

These operators provide a representation of the infinite-dimensional Heisen-berg algebra given by the linear functions on the phase space of solutions tothe complexified Klein-Gordon equation. This representation will be on a statespace describing both particles and anti-particles. The commutation relationsare

[φ(x), φ(x′)] = [π(x), π(x′)] = [φ†(x), φ†(x′)] = [π†(x), π†(x′)] = 0

[φ†(x), π(x′)] = [φ(x), π†(x′)] = iδ3(x− x′)

The Hamiltonian operator will be

H =

∫R3

: (π†(x)π(x) + (∇φ†(x))(∇φ(x)) +m2φ†(x)φ(x)) : d3x

=

∫R3

ωp(a†(p)a(p) + b†(p)b(p))d3p

Note that the classical solutions to the Klein-Gordon equation have bothpositive and negative energy, but the quantization is chosen so that negativeenergy solutions correspond to anti-particle annihilation and creation operators,and all states of the quantum theory have non-negative energy.

398

39.2 Poincare symmetry and scalar fields

Momentum and energy operators, angular momentum operators. Discuss actionof Lorentz boosts.

The Poincare group action on the coordinates φ(p) on H1 will be given by

U(a,Λ)φ(p) = e−ipaφ(Λ−1p)


399

400

Chapter 40

U(1) Gauge Symmetry andCoupling to theElectromagnetic Field

We have now constructed both relativistic and non-relativistic quantum fieldtheories for free scalar particles. In the non-relativistic case we had to usecomplex valued fields, and found that the theory came with an action of a U(1)group, the group of phase transformations on the fields. In the relativistic casereal-valued fields could be used, but if we took complex-valued ones (or usedpairs of real-valued fields), again there was an action of a U(1) group of phasetransformations. This is the simplest example of a so-called “internal symmetry”and it is reflected in the existence of an operator Q called the “charge”.

In this chapter we’ll see to to go beyond the theory of free particles byintroducing classical electromagnetic forces acting on quantized particles. Wewill see that this can be done using the U(1) group action, with Q now havingthe interpetation of “electric charge”: the strength of the coupling between theparticle and the electric field. This requires a new sort of space-time dependentfield, called the “vector potential”, and by using this field one finds quantumtheories with a large, infinite-dimensional group of symmetries, a group calledthe “gauge group”. In this chapter we’ll study this new symmetry and seehow to use it and the vector potential to get quantum field theories describingparticles interacting with electromagnetic fields. In the next chapter we’ll go onto study the question of how to quantize the vector potential field, leading to aquantum field-theoretic description of photons.

40.1 U(1) gauge symmetry

In sections 34.1.1 and 39.1 we saw that the existence of a U(1) group actionby overall phase transformations on the field values led to the existence of an

401

operator with certain commutation relations with the field operators, actingwith integral eigenvalues on the space of states. Instead of just multiplyingfields by a constant phase eiϕ, one can imagine multiplying by a phase thatvaries with the coordinates x, so

ψ(x)→ eiϕ(x)ψ(x)

(so ϕ will be a function, taking values in R/2π). By doing this, we are makinga huge group of transformations act on the theory. Elements of this group arecalled gauge transformations:

Definition (Gauge group). The group G of functions on R4 with values in theunit circle U(1), with group law given by point-wise multiplication

eiϕ1(x) · eiϕ2(x) = ei(ϕ1(x)+ϕ2(x))

is called the U(1) gauge group, or group of U(1) gauge transformations.

This is an infinite-dimensional group, and new methods are needed to studyits representations (although we will mainly be interested in invariant states, sojust the trivial representation of the group).

Terms in the Hamiltonian that just involve ψ(x)ψ(x) will be invariant underthe group G, but terms with derivatives such as

|∇ψ|2

will not, since whenψ → eiϕ(x)ψ(x)

one has the inhomogeneous behavior

∂µψ(x)→ ∂µ(eiϕ(x)ψ(x)) = eiϕ(x)(i∂µϕ(x) + ∂µ)ψ(x)

To deal with this problem, one introduces a new degree of freedom

Definition (Connection or vector potential). A U(1) connection (mathemati-cian’s terminology) or vector potential (physicist’s terminology) is a function Aon space-time R4 taking values in R4, with its components denoted

Aµ(x)

and such that the gauge group G acts on the space of U(1) connections by

Aµ(x)→ Aµ(x) + i∂µϕ(x)

With this new object one can define a new sort of derivative which will havehomogeneous transformation properties

Definition (Covariant derivative). Given a connection A, the associated co-variant derivative in the µ direction is the operator

(DA)µ = ∂µ − iAµ(x)

402

Note that under a gauge tranformation, one has

(DA)µψ → eiϕ(x)(DA)µψ

and terms in a Hamiltonian such as

3∑j=1

((DA)jψ)((DA)jψ)

will be invariant under the infinite-dimensional group G.Write out Hamiltonian version of Schrodinger and KG, coupled to a vector

potential

Digression. Explain the path-integral formalism, weighting of paths. Lagrangianform, just minimal coupling.

40.2 Electric and magnetic fields

While the connection A is the fundamental geometrical quantity needed to con-struct theories with gauge symmetry, one often wants to work instead withquantities derived from A which describe the information contained in A thatdoes not change when one acts by a gauge transformation. To a mathemati-cian, this is the curvature of a connection, to a physicist, it is the field strengthsderived from a vector potential.

Digression. If one is familiar with differential forms, the definition of the cur-vature F of a connection A is most simply made by thinking of A as an elementof Ω(R4), the space of 1-forms on space-time R4. Then the curvature of A issimply the 2-form F = dA, where d is the de Rham differential. The gaugegroup acts on connections by

A→ A+ dϕ

The curvature or field strength of a vector potential is defined by:

Definition (Curvature or field strength). To a connection Aµ(x) one can as-sociate the curvature

Fµν = ∂µAν − ∂νAµThis is a set of functions on R4 depending on two indices µ, ν that can eachtake values 0, 1, 2, 3.

The 6 independent components of Fµν are often written in terms of twovectors E,B with components

Ej = −F0j = −∂Aj∂t

+∂A0

∂xjor E = −∂A

∂t+∇A0

Bj = εjkl(∂Al∂xk− ∂Ak∂xl

) or B = ∇×A

E is called the electric field, B the magnetic field.

403

40.3 The Pauli-Schrodinger equation in an elec-tromagnetic field

The Pauli-Schrodinger equation (29.1) describes a free spin-half non-relativisticquantum particle. One can couple it to a vector potential by the “minimalcoupling” prescription of replacing derivatives by covariant derivatives, withthe result

i(∂

∂t− iA0)

(ψ1(q)ψ2(q)

)= − 1

2m(σ · (∇− iA))2

(ψ1(q)ψ2(q)

)Give examples of an external magnetic field, and of a Coulomb potential.

40.4 Non-abelian gauge symmetry


404

Chapter 41

Quantization of theElectromagnetic Field: thePhoton

Understanding the classical field theory of coupled scalar fields and vector po-tentials is rather difficult, with the quantized theory even more so, due to thefact that the Hamiltonian is no longer quadratic in the field variables. If onesimplifies the problem by ignoring the scalar fields and just considering thevector potentials, one does get a theory with quadratic Hamiltonian that canbe readily understood and quantized. The classical equations of motion arethe Maxwell equation in a vacuum, with solutions electromagnetic waves. Thequantization will be a relativistic theory of free, massless particles of helicity±1, the photons.

To get a sensible, unitary theory of photons, one must take into accountthe infinite dimensional gauge group G that acts on the classical phase space ofsolutions to the Maxwell equations. We will see that there are various ways ofdoing this, each with its own subtleties.

41.1 Maxwell’s equations

First using differential forms:

dF = 0, d ∗ F = 0

Then in components.

Show that gauge transform of a solution is a solution, gauge group acts onthe solution space.

405

41.2 Hamiltonian formalism for electromagneticfields

Equations in Hamiltonian form. Hamiltonian is E2 +B2.First problem: data at fixed t does not give a unique solution. Deal with

this by going to temporal gauge A0 = 0.Second problem: no Gauss’s law. Have remaining symmetry under time-

independent gauge transformations. Compute moment map for time-independentgauge transformation.

41.3 Quantization

Two general philosophies: impose constraints on states, or on the space onequantizes.

41.4 Field operators for the vector potential

Relate to the helicity one irreps of Poincare.


406

Chapter 42

The Dirac Equation andSpin-1/2 Fields

The space of solutions to the Klein-Gordon equation gives an irreducible repre-sentation of the Poincare group corresponding to a relativistic particle of massm and spin zero. Elementary matter particles (quarks and leptons) are spin 1/2particles, and we would like to have a relativistic wave equation that describesthem, suitable for building a quantum field theory. This is provided by a re-markable construction that uses the Clifford algebra and its action on spinorsto find a square root of the Klein-Gordon equation. The result, the Dirac equa-tion, requires that our fields take not just real or complex values, but values in aspinor ( 1

2 , 0)⊕ (0, 12 ) representation of the Lorentz group. In the massless case,

the Dirac equation decouples into two separate Weyl equations, for left-handed(( 1

2 , 0) representation of the Lorentz group) and right-handed ((0, 12 ) represen-

tation) Weyl spinor fields. As with the scalar fields discussed earlier, for nowwe are just considering free quantum fields, which describe non-interacting par-ticles. In following chapters we will see how to couple these fields to U(1) gaugefields, allowing a description of electromagnetic particle interactions.

42.1 The Dirac and Weyl Equations

Recall from our discussion of supersymmetric quantum mechanics that we foundthat in n-dimensions the operator

n∑j=1

P 2j

has a square root, given byn∑j=1

γjPj

407

where the γj are generators of the Clifford algebra Cliff(n). The same thingis true for Minkowski space, where one takes the Clifford algebra Cliff(3, 1)which is generated by elements γ0, γ1, γ2, γ3 satisfying

γ20 = −1, γ2

1 = γ22 = γ2

3 = +1, γjγk + γkγj = 0 forj 6= k

which we have seen can be realized in various ways an an algebra of 4 by 4matrices.

We have seen that −P 20 +P 2

1 +P 22 +P 2

3 is a Casimir operator for the Poincaregroup, and that we get irreducible representations of the Poincare group by usingthe condition that this acts as a scalar −m2

−P 20 + P 2

1 + P 22 + P 2

3 = m2

Using the Clifford algebra generators, we find that this Casimir operator has asquare root

±(−γ0P0 + γ1P1 + γ2P2 + γ3P3)

so one could instead look for solutions to

±(−γ0P0 + γ1P1 + γ2P2 + γ3P3) = im

If the operators Pj continue to act on functions φ(x) on Minkowski space as−i ∂

∂xj, then this square root of the Casimir acts on the tensor product of the

spinor space S ' C4 and a space of functions on Minkowski space. We willdenote such objects by Ψ; they are C4-valued functions on Minkowski space.Taking the negative square root, we have

Definition (Dirac operator and the Dirac equation). The Dirac operator is theoperator

/D = −γ0∂

∂x0+ γ1

∂

∂x1+ γ2

∂

∂x2+ γ3

∂

∂x3

and the Dirac equation is the equation

/DΨ = mΨ

This rather simple looking equation contains an immense amount of non-trivial mathematics and physics, providing a spectacularly successful descriptionof the behavior of physical elementary particles. Writing it more explicitlyrequires a choice of a specific representation of the γj as complex matrices, andwe will use the chiral or Weyl representation, here written as 2 by 2 blocks

γ0 = −i(

0 11 0

), γ1 = −i

(0 σ1

−σ1 0

), γ2 = −i

(0 σ2

−σ2 0

), γ3 = −i

(0 σ3

−σ3 0

)which act on

Ψ =

(ψLψR

)408

(recall that using γ-matrices this way, ψL transforms under the Lorentz groupas the SL representation, ψR as the dual of the SR representation).

The Dirac equation is then(0 ( ∂

∂x0− σ · ∇)

( ∂∂x0

+ σ · ∇) 0

)(ψLψR

)= m

(ψLψR

)or, in components

(∂

∂x0− σ · ∇)ψR = mψL

(∂

∂x0+ σ · ∇)ψL = mψR

In the case that m = 0, these equations decouple and we get

Definition (Weyl equations). The Weyl wave equations for two-componentspinors are

(∂

∂x0− σ · ∇)ψR = 0, (

∂

∂x0+ σ · ∇)ψL = 0

To find solutions to these equations we Fourier transform, using

ψ(x0,x) =1

(2π)2

∫d4p ei(−p0x0+p·x)ψ(p0,p)

and see that the Weyl equations are

(p0 + σ · p)ψR = 0

(p0 − σ · p)ψL = 0

Since(p0 + σ · p)(p0 − σ · p) = p2

0 − (σ · p)2 = p20 − |p|2

both ψR and ψL satisfy(p2

0 − |p|2)ψ = 0

so are functions with support on the positive (p0 = |p|) and negative (p0 = −|p|)energy null-cone. These are Fourier transforms of solutions to the masslessKlein-Gordon equation

(− ∂2

∂x20

+∂2

∂x21

+∂2

∂x22

+∂2

∂x23

)ψ = 0

Recall that our general analysis of irreducible representations of the Poincaregroup showed that we expected to find such representations by looking at func-tions on the positive and negative energy null-cones, with values in representa-tions of SO(2), the group of rotations preserving the vector p. Acting on thespin-1/2 representation, the generator of this group is given by the followingoperator:

409

Definition (Helicity). The operator

h =σ · p|p|

is called the helicity operator. It has eigenvalues ± 12 , and its eigenstates are

said to have helicity ± 12 . States with helicity + 1

2 are called “left-handed”, thosewith helicity − 1

2 are called “right-handed”.

We see that a continuous basis of solutions to the Weyl equation for ψL isgiven by the positive energy (p0 = |p|) solutions

uL(p)ei(−p0x0+p·x)

where uL ∈ C2 satisfies

huL = +1

2uL

so the helicity is + 12 , and negative energy (p0 = −|p|) solutions

uL(p)ei(−p0x0+p·x)

with helicity − 12 . After quantization, this wave equation gives a field theory

describing masslessleft-handed particles and right-handed anti-particles. TheWeyl equation for ψR will give a description of massless right-handed particlesand left-handed anti-particles.

Show Lorentz covariance.Canonical formalism Hamiltonian, LagrangianSeparate sub-sections for the Dirac and Majorana cases

42.2 Quantum Fields

Start by recalling fermionic harmonic oscillatorWrite down expressions for the quantum fields in the various cases

42.3 Symmetries

Write down formulas for Poincare and internal symmetry actions.

42.4 The propagator


Maggiore

410

Chapter 43

An Introduction to theStandard Model

43.1 Non-Abelian gauge fields

Explain non-abelian case.Note no longer quadratic.Asymptotic freedom.Why these groups and couplings?

43.2 Fundamental fermions

mention the possibility of a right-handed neutrinoSO(10) spinor.Problem, why these representations? Why three generations

43.3 Spontaneous symmetry breaking

Explain general idea with U(1) case. Mention superconductors.Higgs potential.Mass terms. Kobayashi-Maskawa.


411

412

Chapter 44

Further Topics

There’s a long list of topics that should be covered in a quantum mechanicscourse that aren’t discussed here due to lack of time in the class and lack ofenergy of the author. Two important ones are:

• Scattering theory. Here one studies solutions to Schrodinger’s equationthat in the far past and future correspond to free-particle solutions, witha localized interaction with a potential occuring at finite time. This isexactly the situation analyzed experimentally through the study of scat-tering processes. Use of the representation theory of the Euclidean group,the semi-direct product of rotations and translations in R3 provides in-sight into this problem and the various functions that occur, includingspherical Bessel functions.

• Perturbation methods. Rarely can one find exact solutions to quantummechanical problems, so one needs to have at hand an array of approxima-tion techniques. The most important is perturbation theory, the study ofhow to construct series expansions about exact solutions. This techniquecan be applied to a wide variety of situations, as long as the system inquestion is not too dramatically of a different nature than one for whichan exact solution exists.

More advanced topics where representation theory is important:

• Conformal geometry. For theories with massless particles, conformal groupas extension of Poincare group. Twistors.

• Infinite dimensional non-abelian groups: loop groups and affine Lie alge-bras, the Virasoro algebra. Conformal quantum field theories.

413

414

Appendix A

Conventions

I’ve attempted to stay close to the conventions used in the physics literature,leading to the choices listed here. Units have been chosen so that ~ = 1.

To get from the self-adjoint operators used by physicists as generators ofsymmetries, multiply by −i to get a skew-adjoint operator in a unitary repre-sentation of the Lie algebra, for example

• The Lie bracket on the space of functions on phase space M is given bythe Poisson bracket, determined by

q, p = 1

Quantization takes 1, q, p to self-adjoint operators 1, Q, P . To make thisa unitary representation of the Heisenberg Lie algebra h3, multiply theself-adjoint operators by −i, so they satisfy

[−iQ,−iP ] = −i1, or [Q,P ] = i1

In other words, our quantization map is the unitary representation of h3

that satisfies

Γ′(q) = −iQ, Γ′(p) = −iP, Γ′(1) = −i1

• The classical expressions for angular momentum quadratic in qj , pj , forexample

l1 = q2p3 − q3p2

under quantization go to the self-adjoint operator

L1 = Q2P3 −Q3P2

and −iL1 will be the skew-adjoint operator giving a unitary representationof the Lie algebra so(3). The three such operators will satisfy the Liebracket relations of so(3), for instance

[−iL1,−iL2] = −iL3

415

For the spin 12 representation, the self-adjoint operators are Sj =

σj2 ,

the Xj = −iσj2 give the Lie algebra representation. Unlike the integerspin representations, this representation does not come from the bosonicquantization map Γ.

Given a unitary Lie algebra representation π′(X), the unitary group actionon states is given by

|ψ〉 → π(eθX)|ψ〉 = eθπ′(X)|ψ〉

Instead of considering the action on states, one can consider the action onoperators by conjugation

O → O(θ) = e−θπ′(X)Oeθπ

′(X)

or the infinitesimal version of this

d

dθO(θ) = [O, π′(X)]

If a group G acts on a space M , the representation one gets on functions onM is given by

π(g)(f(x)) = f(g−1 · x)

Examples include

• Space translation (q → q + a). On states one has

|ψ〉 → e−iaP |ψ〉

which in the Schrodinger representation is

e−ia(−i ddq )ψ(q) = e−addqψ(q) = ψ(q − a)

So, the Lie algebra action is given by the operator −iP = − ddq . On

operators one hasO(a) = eiaPOe−iaP

or infinitesimallyd

daO(a) = [O,−iP ]

• Time translation (t → t − a). The convention for the Hamiltonian H isopposite that for the momentum P , with the Schrodinger equation sayingthat

−iH =d

dt

On states, time evolution is translation in the positive time direction, sostates evolve as

|ψ(t)〉 = e−itH |ψ(0)〉

416

Operators in the Heisenberg picture satisfy

O(t) = eitHOe−itH

or infinitesimallyd

dtO(t) = [O,−iH]

which is the quantization of the Poisson bracket relation in Hamiltonianmechanics

d

dtf = f, h

Conventions for special relativity.Conventions for representations on field operators.Conventions for anti-commuting variables. for unitary and odd super Lie

algebra actions.

417

418

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