A survey is given of the elegant physics of N -particle systems, both classicaland quantal, non-relativistic (NR) and relativistic, non-gravitational (SR) andgravitational (GR). Chapter 1 deals exclusively with NR systems; the corre-spondence between classical and quantal systems is highlighted and summarizedin two tables of Sec. 1.3. Chapter 2 generalizes Chapter 1 to the relativis-tic regime, including Maxwell’s theory of electromagnetism. Chapter 3 followsEinstein in allowing gravity to curve the spacetime arena; its Sec. 3.2 is devotedto the yet missing theory of elementary particles, which should determine theirproperties and interactions. If completed, it would replace QFT; promising isthe ‘metron’ approach.
Frontline physics has always appeared ‘crazy’ to the community of itsdays (Dyson 1992, p. 106). Still, a tenacious frontline problem has beenfor decades, and still is, a quantitative prediction of the properties of theelementary particles. In this survey of fundamental physics, I wish to fill agap in the modern textbook literature by demonstrating that most of ourknowledge in physics is coherent, free of arbitrariness, and based on sim-ple mathematical principles. There is much more universality and lawful-ness among its different branches: (1) classical and quantal, (2) Galileianand Lorentzian, (3) without and with strong gravity, than is often evident,which ought to be used as a guiding principle towards new arenas. Uni-versality and elegance have been reliable guides to new insight, whereby
1 Argelander Institute of Bonn University, Bonn, Germany; E-mail: [email protected]
new insight often asks for improved terminology. Chapters 2–4 of this sur-vey are hoped to demonstrate this principle in a systematic way, leadingstepwise from the non-relativistic motion of a free classical particle—Kepler’s laws—to the interaction of arbitrary matter at extreme densities,fieldstrengths, and temperatures, near and far from equilibrium. Physicsobeys rather uniform, elegant laws.
1. NONRELATIVISTIC SYSTEMS OF N PARTICLES
Physics deals with two types of non-relativistic N -particle systems:macroscopic ones, and microscopic ones. The dynamics of macroscopicparticles at non-relativistic speeds can be accurately described on the clas-sical 6N -dimensional phase space of their 3-d positions and momentawhenever their interaction Hamiltonian is given, whereby their initialconfiguration (state) must be properly prescribed; realistic systems (withdamping) are known to approach thermal equilibrium within estimabletimes, a subset of all steady-state solutions of Hamilton’s equations ofmotion.
This classical description breaks down on approach to the microscopic(or elementary-particle) regime, most notable in (i) interference phenom-ena during particle scattering, (ii) graininess in Compton scattering, (iii)an absence of damping in isolated atoms (which are stable in spite ofaccelerated motions of their electrons), even in (iv) various macroscopiclow-energy phenomena, like superfluidity and superconductivity, and in (v)the existence of diamagnetism. All such breakdowns of classical physicsoccur when the extents of particles shrink below those of their guiding deBroglie waves, of wavelength λ = h/mv, and are taken care of by quantummechanics. The transition from classical to quantal behaviour takes placeon the scale of large molecules: Myosin molecules, the motors in biology(composed of some 105 amino acids, i.e. of some 106 atoms), show nolonger time-symmetric (microscopic) behaviour.
In most textbooks, classical and quantal N -particle systems aredescribed by quite different formalisms—phase space vs Hilbert space—even though their mathematical properties are remarkably similar, and canbe formulated with almost identical-looking equations relating to corre-sponding types of experiment. Both of them deal with Lie–Hilbert alge-bras of observables and the evolution in time of their expectation val-ues, and can even be mapped onto each other isomorphically withinfirst order of � (by the Weyl–Wigner–Moyal map, W2M). They differ injust two properties: in the sharpness of the states in which ensembles
Fundamental Physics
(of one or more particles) can be prepared in space and time: delta-likevs. of extended support, and in the locality of their Lie product (whichinvolves derivatives of arbitrarily high order in the quantal case). AfterGottfried Falk, they are the only two mathematical realizations of thecanonical commutation relations [pa, qb] = δb
a for which the Lie prod-uct (on the lefthand side, LHS) satisfies Leibniz’ product rule, Eq. (6).Or, expressed somewhat more broadly: There are just two obvious waysto describe the physics of N -particle systems, viz. the classical, Newton–Lagrange–Hamiltonian way by functions on phase space, and the quan-tal, Born–Jordan–Schrodinger–Heisenberg way by operators on Hilbertspace.
In both cases, there are not only the analogous conserved quantities{energy, momentum, angular momentum, baryon number, lepton number,etc.} for particle collisions, but also the analogous descriptions of many-particle systems via their distribution functions on N -particle phase space,or statistical (von Neumann) operators on N -particle Hilbert space, respec-tively, whose projections on fewer (r ) dimensions are known as the (statis-tical) hierarchy of r -particle evolution equations for distribution {functions,operators}, and whose truncations give rise to the (time-asymmetric) kineticequations called after Boltzmann, Fokker–Planck, and Lenard–Balescu (forthe lowest-order cases r = 1, 2, 3, under simplifying assumptions on par-ticle interactions). From the kinetic equations, in turn, one can derive thefundamental thermo-hydrodynamic equations for fluid media, both gases andliquids, by insertion of the collision invariants into the equations of motionfor the expectation values; whereby those depending sensibly on microscopiccorrelations—like entropy—require coarse-graining in order to pass from thestrict, time-reversal invariant evolution to the probable (or realistic), time-asymmetric evolution. All these derivations of the basic equations of parti-cle and continuum dynamics can be performed, classically and quantally, inperfect analogy when the proper formalism is at hand, yielding the quantumdescription almost routinely once the classical description is known. The fol-lowing eight sections will show how to do it, starting from Kundt (1966), andattempting to incorporate additional insight by Pool (1966), Leaf (1968),De Groot and Suttorp (1972), Ehlers (1973, 1998), Fischer et al. (1998),Hasselmann (1998), and a number of unpublished lecture notes on statisti-cal mechanics.
1.1. Classical Description
The classical phase-space description of an N -particle system—charac-
terized by a (real) lagrangean l(qa,•
qa)—starts with observables a(pb, qb)
Kundt
on (real) 6N -d phase space, 1 ≤ a, b ≤ 3N , i.e. with real functions a(pb, qb)
of the canonical position variables qa and momentum variables,
pa := ∂ •qa
l(qb,•
qb) , which should be square-integrable on 6N -d Euclidean
space for forming expectation values<w, a> := ∫ ∫h−1dpdqw(p, q)a(p, q),
and infinitely differentiable for forming iterated Poisson brackets. (Warn-ing: in most of this chapter, we use small letters for classical quanti-ties, and capital letters for their quantum analogues). For simplicity, werestrict considerations to regular lagrangeans for which the matrix of sec-
ond partial derivatives of l w.r.t. the•
qa has maximal rank, i.e. for which
the velocities•
qb can be solved for the momenta pa , so that any (regu-
lar) function of the qa ,•
qa can be re-expressed as a function of the pb,qb. (The ‘canonical regularization’ of ‘singular’ lagrangean systems can befound, e.g., in Kundt (1966)). For economy of writing, the canonical vari-ables {pa, qb}will henceforth be denoted by {mα}, with 1 ≤ α, β ≤ 6N ,and a(pb, qb) =: a(m). This description has been entered in rows one andtwo of the long table in Sec. 1.3, subsequently called ‘Fuphy’ (:= ‘Funda-mental Physics’).
The statistical average—or (classical) expectation value < a > —of anobservable a(m) in a state described by a (real) distribution function, orprobability density, or state function w(m) ≥ 0 on phase space is obtainedby integrating over their product:
< a > := < w, a > , (1)
where braces < , > denote the scalar product on the (complex) Hilbertspace of square-integrable functions on phase space (with a∗(m) denotingthe complex conjugate of a(m) ):
< a, b > :=∫
dm a∗(m) b(m) , (2)
formed (for convenience) with the dimensionless measure
dm := h−3N3N∏
a=1
dpa dqa =: d3N p d3N q / h3N , (3)
and where w(m) is assumed normalized: <w, 1>= 1 ; h =: 2π� =10−26.178736 erg s is Planck’s constant, gleaned from microphysics. Notethat even the value of a(m) at a fixed phase-space point n can be repre-sented in the form of Eq. (1) in the limit of a delta distribution, i.e. for
Fundamental Physics
w(m) = δ(m −n) , which stands for the ‘pure’ classical state localized at n,and that the most frequently occurring observables, polynomials in m,are not square integrable at all. Mathematically more rigorously, there-fore, we should have spoken of a dual pair formed by the observables andthe state functionals, a(m) and w(m), for which the above phase spaceintegral (1) is defined. But their square-integrable subspaces lie dense inthe dual pair, which property will guarantee that all our future manipula-tions make sense, even for polynomials, and allow our language to be keptsimple.
The above three equations define a Hilbert-space structure on theclassical observables, cf. rows 3 to 5 of Fuphy, which are at the basisof classical statistical mechanics, and enter the derivation of thermo-hydrodynamics from first principles. Sec. 1.2 will show that quantum sta-tistical mechanics can be cast into the same form.
The evolution in time of any physical observable—or better of anyexpectation value—is generally formulated with the help of the Poissonbracket
[a, b] := εαβ ∂αa(m) ∂βb(m) with(εαβ
) :=(
03N 13N−13N 03N
)
, (4)
in which ε := (εαβ
) = −(εαβ) is the (skew) symplectic metric on 6N -dphase space, used also for describing canonical transformations. ThePoisson bracket is a Lie product on the observables, i.e. is linear inboth entries (factors), anti-metric: [b, a] = − [a, b] , and satisfies Jacobi’sidentity
[[a, b], c] + [[b, c], a] + [[c, a], b] = 0 , (5)
as a consequence of the skewness of ε, and of the permutability of par-tial derivatives. The Poisson bracket is a special Lie product in satisfyingthe product rule of Leibniz:
[ab, c] = a[b, c] + [a, c]b , (6)
by being linear in partial derivatives. As a special case, the canonical vari-ables {pa, qb} =: {mα} satisfy the classical analogue of the canonical com-mutation relations:
[mα,mβ ] = εαβ , (7)
i.e. their Poisson brackets vanish except for conjugated momenta and posi-tions, in which case they take the values ±1.
Kundt
As for ordinary vectors in 3-space, the above scalar and Lie productsatisfy the triple-product identity:
< a, [b, c] > = < [a, b], c > (8)
thanks to partial integration, because of the skewness of ε, and thepermutability of partial derivatives. With these properties at hand, Ham-ilton’s equations of motion for a hamiltonian h(m) and an observable a(m)in a state w(m) take either of the following two forms:
< a >• = < w, [h, a] > = < [w, h], a > , (9)
which are the classical analogues of the Heisenberg and Schrodingerpicture, respectively. We have thereby arrived at the Lie–Hilbert-algebrastructure of classical dynamics, completed by the entries in rows 6–10 ofFuphy.
For a later comparison with the corresponding quantal structure, wenow turn to the invariance group of classical mechanics, and to its faithfulunitary representation. A (twice differentiable, local) coordinate transfor-mation on phase space: mα′ = mα′
(mβ)—under which observables trans-form as scalars: a′(m′)= a(m)—is called ‘canonical’ iff it leaves the Poissonbrackets invariant, i.e. iff εα
′β ′ = εαβ holds, or: mα′,α ε
αβmβ ′,β = εαβ ,
or in matrix notation: M εMT = ε ; i.e. iff the transformation leavesthe symplectric metric ε invariant. As is well-known, canonical transforma-tions can be used to simplify the hamiltonian, and thereby the (integrationof the) equations of motion. They form the invariance (semi-) group ofthe Lie structure. The matrix equation for M shows that M has deter-minant ±1, implying that canonical transformations are volume preserv-ing; they thus leave the Hilbert-space structure invariant as well, i.e. arethe searched-for invariance group.
Evolution in time forms a 1-d subgroup of the canonical (semi-)group, generated by the hamiltonian. More generally, every differentiableobservable g(m) can be used as the generator of a 1-d subgroup, of groupparameter s ≥ 0, via:
mα(s) = e−s[g,...] mα(0) , (10)
in which the exponential operator e−s[g,...] stands for a Taylor expansionalong the ray spanned by the vector −gα := −εαβ g,β , (g,β := ∂βg) ,because of e−s[g,...] = e−sgα∂α , and because any contravariant vector gα
can be chosen locally as the unit vector in 1-direction: gαW= δα1 , by a
Fundamental Physics
suitable choice of coordinates, (where W stands for ‘without restriction ofgenerality’).
In particular, the subgroup (10) maps the point m := m(0) into thenew point m′ := m(1), and an analytic observable a(m) into a new observ-able:
a′(m) = a′(e[g,...]m′) = e[g,...] a(m) , (11)
whereby use has been made of the property a′(e[g,...] m′) = e[g,...] a′(m′)which holds for Taylor expansions, and of the transformation law a′(m′) =a(m) for scalars. In this way, a faithful representation has been obtainedof the observables g on the (pre-) Hilbert space spanned by the (analytic)observables: a→a′ = e[g,...]a , whose infinitesimal transformations −iΓ read:
[g, a] =: −iΓa with Γ = Γ+ , (12)
where the self-adjointness property Γ = Γ+ has yet to be shown; clearly,the map a→a′ is unitary iff the operator Γ := i[g, . . .] is selfadjoint. Nowthe ‘symmetric’ property of Γ means < Γa, b >=< a,Γb >, and hasbeen shown in (8) above. It remains to prove (for self-adjointness) that Γ
and Γ+ are both densely defined, equivalent to the emptiness of the nullspaces of the two operators Γ+ ± i1, which can be straight-forwardly veri-fied. With this, the representation (11) of the classical invariance group hasbeen shown to be unitary—in line with rows 11–13 of Fuphy—and ourfancy survey of classical mechanics has come to a preliminary end. It willbe resumed in Sec. 1.4, in parallel with its quantal treatment.
1.2. Quantal Description
Quantum mechanics starts with the replacement of the canonicalcoordinates p, q on phase space by self-adjoint operators P , Q on some(countably infinite) Hilbert space, the Hilbert space of (pure) state vec-tors, which must not be confused with the Hilbert space of (classical)observables even though all Hilbert spaces of equal dimension are isomor-phic. In the presence of N (microscopic) particles there are, correspond-ingly, 6N basic ‘variables’ {Pa , Qb} =: {Mα}, 1 ≤ a, b ≤ 3N , 1 ≤α, β ≤ 6N ,on which all (quantal) observables A, B depend as (mostly algebraic) func-tions, A = A+ = A(M). Iterated commutator formation will want A(M)to be an infinitely differentiable function, and expectation-value formationwill want the products of certain observables A, W to have a convergenttrace: | tr(W A) |< ∞ . Such restrictions of generality still leave us with
Kundt
dense subsets of the (quantal) Hilbert space of observables—in close cor-respondence with its classical analogue—so that by continuity, all subse-quently derived relationships hold whenever physically meaningful.
In perfect analogy to the classical description, our knowledge of aphysical ensemble, or ‘state’ can be expressed by the quantum-mechanicalexpectation value <A> which is a positive measure, or probability distribu-tion on the observables. As has most satisfactorily been shown by AndrewGleason (1957), each expectation value defines a distribution operator,or statistical operator, or state operator W —historically called ‘von Neu-mann’s density matrix’—such that < A > takes the form of its trace prod-uct with A:
< A > := � W, A � , (13)
where W is (self-adjoined and) non-negative, W = W + ≥ 0 , and normal-ized: � W, 1 � = 1, and where � W, A � stands for the trace product,which is a scalar product:
� A, B � := tr(A+B) (14)
on the (complex) Hilbert space of (Hilbert–Schmidt) operators A, B , i.e.is complex bilinear and non-negative: � A, A � ≥ 0. Equation (14) definesa Hilbert-space structure on the (quantal) observables. As for ordinary 3-vectors ψ , W ≥ 0 is defined by <ψ,Wψ > ≥ 0 for all ψ in the domainof definition of W , so that all eigen values ω of W (occurring in itsdiagonal-matrix form) are non-negative, whereupon � W, 1 � = 1 implies0 ≤ ω ≤ 1, and:
0 < W 2 ≤ W ≤ 1 (15)
holds, as a consequence of 0 < W and � W, 1 � = 1. This last inequalitydiffers from its classical analogue: Classical states violate w2 ≤ w when-ever approaching high confinement, (e.g., when approaching a delta distri-bution).
Quantal pure states can be defined by the above (middle) inequalityfor W turning into an equality: W 2 = W . For them, there is only a sin-gle eigen value ω different from zero (whence = 1), and W is a projectionoperator onto a 1-d ray: W = | ψ><ψ | ; when formulated in the positionrepresentation, ψ is known as Schrodinger’s wave function. Pure quan-tal states describe micro-states of maximal confinement; as is well known,they are less confined than pure macro-states. With this, we have coveredrows 1–5 in column 2 of Fuphy, together with a leap into rows 26, 27.
Central for a derivation of quantum dynamics is the (standardized)commutator of two observables A, B. It is the direct analogue of the
Fundamental Physics
Poisson bracket in classical dynamics; we define it as:
[A, B] := (i/�)(AB − B A) . (16)
In the literature, the factor i/� on the RHS tends to be omitted fromthe definition, i.e. spelled out explicitly, even though it is indispensablewhen we want the commutator product to map observables into (self-adjoint!) observables, and when we want the commutator between P andQ to be dimensionless; only a real, dimensionless c-number factor has sofar been arbitrary. (Notation is, of course, a matter of taste; but suitablenotation has occasionally helped getting new insight). The commutator isa Lie product: it is bilinear, and satisfies Jacobi’s identity [[A, B],C] +[[B,C], A]+[[C, A], B] = 0 because multiplication of operators is associa-tive, (the six different products of A, B,C occur with opposite signs each).In addition, it satisfies Leibniz’ product rule: [AB,C] = A[B,C]+[A,C]B,again because multiplication is associative.
With this notation, the fundamental formula of quantum mechanics—or Born–Jordan–Heisenberg’s canonical commutation relation—reads (Bornand Jordan 1925; Schweber 1994, p. 5):
[Mα,Mβ ] = εαβ1 , (17)
in which εαβ is the symplectic metric introduced in (4), and 1 stands forthe unit operator (or unit matrix) on Hilbert space. The relation saysthat all position operators and all momentum operators commute amongthemselves, and that only equal components of the position and momen-tum operators have non-vanishing commutator products, of value ±1.This relation took some 40 years to be discovered, primarily via the lawsof the photo effect, blackbody radiation, thermal motions, magnetism,low-temperature physics, and line spectra, and took its final shape by solv-ing the quantum versions of the Coulomb, oscillator, and rotator prob-lems, for a confrontation with atomic and molecular spectra. It introducesPlanck’s fundamental constant into physics—anticipated in Eqs. (3), (16)—and completes the introduction of a Lie–Hilbert-algebra structure on thequantal observables.
It is useful to realize that the commutator [P, A] acts like apartial differentiation of A w.r.t. Q : [P, A] = ∂Q A , and correspond-ingly: [A, Q] = ∂P A , thanks to Leibniz’ rule and (17), so that by induc-tion, commutators can be explicitly evaluated on the whole polynomialring generated by the canonical variables, like in the classical case for thePoisson brackets. And it is illuminating to know that after Falk (1955),a Lie product satisfying Leibniz’ rule on the polynomial ring generated
Kundt
by pairs of canonical variables with (7) essentially only admits the twoabove realizations: by Poisson brackets, or by commutators. No furthernew physics with similar structure is in sight.
It remains to be shown that the above Lie–Hilbert structure uniquelydetermines the dynamics of a quantized system, by again satisfying thetriple-product identity (8):
� A, [B,C] � = � [A, B],C � , (18)
this time via the cyclic-permutation invariance of the trace: tr(AC B) =tr(B AC). The equation of motion of an expectation value < A> can there-fore again be written in one of the following two forms, named after Hei-senberg and Schrodinger, respectively:
< A >• = � W, [H, A] � = � [W, H ], A � , (19)
in which the time dependence is thought to be described either by a timedepending observable A(t) in a fixed state W , or else by an evolving stateW (t) and a fixed observable A; both yield the same time dependence of< A > (t) . Rows 6–10 of Fuphy have thereby been covered.
For a quantitative comparison of this quantal description with itsclassical analogue, we are again interested in its invariance group, and ina faithful unitary representation thereof. This time one remembers thatselfadjoint operator algebras are invariant under unitary transformations(corresponding to isometric coordinate changes in their domains of defi-nition): A′ = U AU+ with U+ = U−1, because of U++ = U , and thatthe only other isometric correspondences are their anti-unitary analogues,which are unitary maps followed by a transition to the complex conjugate(Hilbert-space) vector. Clearly, both types of transformations leave the Lieproduct (16) invariant, and at the same time leave the scalar product (14)invariant, i.e. are isomorphisms of the Lie–Hilbert-algebra structure on theobservables.
Again, we can obtain a faithful unitary representation of the observ-ables G of a quantized system by considering its 1-d subgroups generatedby them—analogous to the 1-d evolution with time generated by the Ham-iltonian H—with s as the group parameter:
Mα(s) = e−i s
�G Mα(0) e
i s�
G = e−s[G,...] Mα(0) , (20)
and by passing to the implied map of the (analytic) observables which aretransported as scalars from M := M(0) to M ′ := M(1):
A′(M) = e[G,...] A(M) . (21)
Fundamental Physics
The reasoning here is identical to that around Eq. (11): The newobservables A′(M) are the images of the old observables A(M) under thetransformation e[G,...] which is only defined on their analytic subset (byacting like a generalized Taylor expansion), but by continuity on the wholeHilbert space of quantal observables as well. In order to see that this rep-resentation is unitary, we consider its infinitesimal transformations:
[G, A] =: −iΓA with Γ = Γ+ , (22)
and show as before that the linear operator Γ is both symmetrical anddensely defined, hence self-adjoint, as anticipated. Alternatively, the selfad-jointness of Γ can be gathered from the unitarity of e
i s�
G in (20). Thiscovers rows 11–13 of the quantal column of Fuphy.
An additional new structure of quantal particles (compared with clas-sical particles) is their spin angular momentum which George Uhlen-beck and Samuel Goudsmit proposed in 1925 (Pais 1986, p. 254, 277),and which Wolfgang Pauli taught us to describe (for single electrons) bya 3-vector of hermitian 2 × 2 matrices—known as their spin (operator)→σ —or, more generally (for systems of spinning, and revolving elementaryparticles), by a corresponding matrix representation of the rotation group,whose collinear magnetic moments enter into the Hamiltonian of such asystem when embedded in a magnetic field; see (98).
1.3. Classical-Quantal Correspondence
In Secs. 1.1 and 1.2 we have seen that the classical and quantaldescriptions of non-relativistic N -particle systems have quite similar struc-tures, in spite of their often very different mathematical building blocks,and different observed properties. The purpose of this section is to showthat the two structures can even be mapped onto each other almost iso-morphically—by the Weyl–Wigner–Moyal (W2M) map—so that a quan-titative comparison of their differences becomes possible, and a paralleltreatment is strongly suggested, mainly for statistical mechanics. We startby reviewing the results of the past and future sections in two longtables—called Fuphy I, II—which compare the description of a macro-scopic (classical) system with a microscopic (quantal) one. Fuphy I willalso contain the W2M map presented in this section, as well as a sketchyderivation of the ‘rest’ of non-relativistic physics whose explanations willbe the subject of the succeeding two sections. Two long tables, Fuphy I andFuphy II, will summarize our results on the following pages:
KundtB
uild
ing
Blo
ckC
lass
ical
Qua
ntal
1.P
hase
-Spa
ceV
aria
bles
(mα):=(p a,q
b),
{ 1≤
a,b
≤3N
1≤α,β
≤6N
}(M
α):=(P a,
Qb),
{ 1≤
a,b
≤3N
1≤α,β
≤6N
}
2.O
bser
vabl
ea(
m)
=sm
ooth
,re
alph
ase-
spac
efn
.A(M)=
smoo
th,
self
adj.
H.-
Sp.
op.
3.Sc
alar
Pro
duct
<a,
b>
:=∫
dm
a•(m)b(m)
,w
ith
�A,
B�
:=tr(A
• B)
dm
:=d
3Np
d3N
q(2π
�)3
N=
ph.-
sp.
mea
sure
4.St
ate
w(m)≥
0w
ith
<w,1>
=1
W>
0w
ith
�W,1
�=
1,
(⇒W
2≤
W)
5.E
xpec
tati
onV
alue
<a>
:=<w,a>
<A>
:=�
W,
A�
6.L
ieP
rodu
ct[a,
b]:=εαβ∂α
a∂β
b,
whe
re[A,
B]:=
(i/�)(
AB
−B
A)
( εαβ)
:=(
01
−10)
=−(εαβ)=:ε
7.C
anon
ical
Rel
atio
ns[mα,mβ]=
εαβ
[Mα,
Mβ]=
εαβ
1
8.L
eibn
izR
ule
[ab,c]
=a[b,c]
+[a,
c]b[A
B,C
]=A[B,C
]+[A,C
]B
9.T
ripl
eP
rodu
ct<w,[h,
a]>
=<
[w,
h],a
>�
W,[H,
A]�
=�
[W,
H],
A�
Fundamental PhysicsB
uild
ing
Blo
ckC
lass
ical
Qua
ntal
10.
Equ
atio
nof
Mot
ion
<a>
• =<w,[h,
a]>
n<
A>
• =�
W,[H,
A]�
=<
[w,
h],a
>=
�[W,
H],
A�
11.
Inva
rian
ceG
roup
( mα
′ ,α
)=:
M,
MεM
T=ε
A′ =
UA
U+ ,
U+
=U
−1,
whe
re
i.e.
cano
nica
ltr
ansf
orm
atio
nsU
={an
ti-}
unit
ary
oper
ator
12.
Uni
tary
Rep
rese
ntat
ion
mα(s)
=e−
s[g,...] m
α(0)
,an
d:Mα(s)
=e−
s[G,...]
Mα(0)
,an
d:a(
m)
=e[g
,...] a(m)
A(M)
=e[G
,...]
A(M)
13.
Infin
ites
imal
Tra
nsfo
rm.
[g,a]
=:−i
Γa
,Γ
=Γ
+[G,
A]=
:−iΓ
A,
Γ=
Γ+
14.
Can
onic
alT
rans
lati
ons
�i[m
α,v
n]! =
nαv
n,
⇒�
i[Mα,
Vn]! =
nαV
n,
⇒v
n=
e−i �εβγ
mβ
nγV
n=
e−i �εβγ
Mβ
nγ
15.
W2 M
-Map
a(m):=
��(m),
A�
,�(m):=
A:=
<�
,a>
,or
:
16.
�-O
pera
tor
∫d
ne
i �εβγ(M
β−m
β)nγ
=�
+ (m)=
=∫
dn
ei �εβγ(M
β−m
β)nγ
a(m)
∫d
re
i �pr
|q+
r 2><
q−
r 2|=
=e
−i� 2∂
Qa∂
P aa
Q(M)
=∫
ds
ei �
qs
|p−
s 2><
p+
s 2|
a(
� i∂
q,
� i∂
p
)e
i �(q
P+
pQ)| m
=0
17.
Pro
pert
ies
(W2 M
-Map
){re
al,
a(c α
mα)}
map
into
:...
{self
adjo
int
,a(
c αMα)}
δ(m
−n)
=��(m),�(n)�
�(n)=<�(m),δ(m
−n)>
KundtB
uild
ing
Blo
ckC
lass
ical
Qua
ntal
18.
Moy
alB
rack
et[[a,b]]
:=2 �
sin{
� 2εαβ
a ∂ αb ∂ β
}ab
[A,
B]
map
sun
der
W2 M
into
=[a,
b]+
O(�
2 )
19.
Syst
emP
rope
rtie
sχ
E(m)=
{ 0 1}fo
rm
{/∈ ∈} E
E=
E+
=E
2(=
proj
ecti
on)
20.
Pro
babi
lity
0≤<χ
E>
≤1
;<χ
E,χ
E′≥
00
≤<
E>
≤1
;�
E,
E´�
≥0
21.
Spec
tral
Dec
ompo
siti
ona
=∫α
de α
,w
ith:
A=
∫α
dEα
,w
ith:
e α=
e2 α≤
e αfo
rα
≤α
Eα
=E
2 α≤
Eα
forα
≤α
22.
Eig
enva
lues
sati
sfy
αm
in≤<
a>
≤α
max
αm
in≤<
A>
≤α
max
23.
Unc
erta
inty
�a
:=√<(a
−<
a>)2>
�A
:=√<(A−<
A>
1)2>
24.
Cov
aria
nce
c a,b
:=<(a
−<
a>)(
b−<
b>)>
c A,B
:=[<(A−<
A>)(
B−<
B>)>
+<(B
−<
B>)(
A−<
A>)]/
2
25.
Unc
erta
inty
Rel
atio
n�
a�
b≥|
c a,b
|�
A�
B≥
√c2 A,B
+(
� 2<
[A,
B]>)2
26.
Qua
ntum
Stat
e0<<w,w>
≤<w,1>
=1
,0<
W2
≤W
,�
W,1
�=
1,
|w(m)|≤
2,<w,w
′ >≥
0∀w
′�
W,
W′ �
≥0
for
all
stat
esW
′
27.
Pur
eQ
uant
umSt
ate
<w,w>
=1
W2
=W
;(⇐
⇒W
=|ψ
><ψ
|)
Fundamental PhysicsB
uild
ing
Blo
ckC
lass
ical
Qua
ntal
28.
Can
onic
alSt
ate
wN(m)=
e{α−β
hN(m)} ,
wit
h:W
N=
e{α−β
HN(M)}
,w
ith:
β:=
1/k
T,α
=:β
FN
β:=
1/k
T,α
=:β
FN
29.
Par
titi
onF
unct
ions
ZN
=1 N!∫
dm
e−β
hN
;an
d:Z
N=
∑ je−β
η(N)
j,
whe
re:
Z(x):=
∞ ∑ N=0
eβµ
NZ
N,
x:=
eβµ
η(N)
j=
∑ ιN(
j)ιηι,
N(
j)ι
={ 0,1,2,...
0,1
}
30.
The
rmal
Occ
up.N
s.<
Nκ>
=−
1 β∂ηκ
lnZ
N=
<Nκ>
=(eβ(ηκ−µ
)+ε)−
1=:
f κ,
=nλ
3 e−β
ηκ
,λ
:=h
√ 2πm
kTε
={0,
±1}
for
{M.-
B.,
F.-D
.,B
.-E
.}
31.
(Fre
e)E
nerg
yU
=∫
dmw
h,
F:=
−1 β
lnZ
NU
=∑wιηι
,K
:=−
1 βln
Z
32.
Ent
ropy
S=
kβ(U
−K
−µ
N)=
−S/k
=<
f,ln
f>
+ε<(1
−ε
f),
idea
lga
s=
Nk
{5 2
−ln(nλ
3 )}
ln(1
−ε
f)>,
f:=
Nw
33.
Hie
rarc
hyof
Stat
esw
r(m
1,...,
mr):=
Wr(M
1,...,
Mr):=
∫N ∏
j=r+
1dm
jw
N(m
1,...,
mN)
N Tr
j=r+
1WN(M
1,...,
MN)
34.
Nor
mal
izat
ion
<w
r,1>
=1
�W
r,1
�=
1
KundtF
uphy
II:
1.H
iera
rchy
d dtw
r(r):=
{∂ t+
H r+
U r}w
r=(N
−r)
∫d
6 mr+
1U′ r+
1wr+
1,
defs
.an
das
sum
ptns
.se
ete
xt
2.r≤
3d dtw
1(m):=
{∂ t+
→ p m∂
→ q}w
1(→ m)=
N∫
d6 m
2→ ϕ
21·∂
→ p1w
2(→ p
1,→ q
,→ p
2,→ q)
tru
ncat
ion,
f:=
Nw
→
3.T
runc
atio
nd dt
f(1)
=h−3
∫d
3p 2
d2 ω
′ |�
→ v 12
|σ(1,2
→1′,2′){
f(1′)
f(2′)−
f(1)
f(2)
},
j:=(→ p
j,→ q)
4.qu
anta
lre
plac
e:{f(1
′ )f(
2′)−
f(1)
f(2)
}by:
[∏
j=1,
2,1′,2
′f(j)
]{f ε(1)
f ε(2)−
f ε(1
′ )f ε(2
′ )}w
ith:
f ε:=
1 f−ε
f=
1/(
f ε+ε)
,ε
={ + −} 1
for{
Fer
mi–
Dir
acB
ose–
Ein
stei
n} ;(ε
=0
yiel
dsth
ecl
assi
cal
Max
w.–
Bol
tzm
.ca
se)
5.C
ons.
Law
s∂
t(n<
a>)+∂
→ q(n<
→ va>)−
n<
→ v∂
→ qa>
=n<∂
→ p(→ ϕ
a)>
,a
={m
,→ p
,m
→ v2 }
6.an
dE
qs.
ofSt
ate
d dt
lnµ
=−∂
iui=:
−Θ,
µ:=
mn
,→ u
:=<
→ p/m>
;p
=nk
T,
kT
:=m<(→ v
−→ u)2>/3
7.N
avie
r–St
okes
µd dtu
j=
nϕj−∂
kp
jk,
pjk
−δ
jkp
=−2ησ
jk,
→ q:=µ<(→ v
−→ u)(
→ v−
→ u)2>/2
=−κ
→ ∂k
T,
8.E
ntro
pyT(∂
ts+
ui ∂is)=
−(p
jk∂
ku
j+∂
iqi )
=2ησ
jkσ
jk+ζΘ
2+∂
j (κ∂
jkT)
,σ
jk:=∂(
juk)
−δ
jkΘ/3
9.R
elat
ivis
tic...
n=
A(T)
I11(
mc2
kT,α,ε)
,u/k
T=
A(T)
I21(
mc2
kT,α,ε)
,3p/k
T=
A(T)
I03(
mc2
kT,α,ε)
,
Fundamental Physics
10.
The
rmos
tati
cs:
s/
k=
A(T)(
I21+
1 3I03
−α
I11)
,A(T):=
w4π(
kT hc)3
,w
:={1,
2,2s
+1}
,{ F(V,T)
=U
−TS
G(p,
T)
=F
+pV
},
11.
Iab(ξ,α,ε):=
∞ ∫ ξ
dx
xa(x
2−ξ
2)b/2
ex−α
+ε
,{ ξ
→∞
:Iab(ξ,α,0)
→Γ(1
+b/2)
eα−ξξ
a+b/
2(1
+O(ξ
−1))
ξ→
0:I
11(ξ,0,ε)
→1 4(7
−ε)ζ(3)
,ζ(3)
=1.
202
},
and:
ξ→
0:{
I21(ξ,0,1),
I03(ξ,0,1)
→7π
4/12
0I2
1(ξ,0,−
1),
I03(ξ,0,−
1)→
π4/15
}.
12.
Max
wel
lE
qs.
F[ab,c
]=0
,F
ab,b
=4πρ
a,
F(a
b)=
0,ρ
a:=
en
ua
,T elm
ab=
−1 4π(F
acF
b .c
−1 4
gabF
cdF
cd)
13.
Rel
ativ
isti
cT
ab;b
=0
,T mat
ab=µ
ua
ub
+2
u(a
qb)+
p⊥g
ab+π
abw
ith:
{u
au
a=
−c2
,qa
ua
=0
⊥gab
ub
=0
=u
bπ
ab
}
14.
The
rmod
yn.
• µc2
+(µ
c2+
p)Θ
+pab
σab
+qa ,
a+
qa• u
a=
0,(b
ua
+ba) ,
a=
0,
b:=
{ bar
yon-
num
ber
dens
ity
}
µc• u
a+(ω
ab+σ
ab+
4Θ 3⊥g a
b)q
b+(p
c b;c+
• q b) ⊥
gb a=
nϕ
a,
{ ua;b
=ω
ab+σ
ab
+Θ 3
⊥g ab
−· uau
b
}
15.
Gen
eral
Rel
.R
ab−
R 2gab
=:G
ab=
8πG
c4T
ab,
=⇒T
ab;b
=0
;T
ab=
T mat
ab+
T elm
ab=
2δL/δg a
b
16.
Met
ron
Eqs
.(η
ab∂
a∂
b−ηαβ
kp α
kp β)
hp A
B=
qp A
B,
hA
B=
�∑p
hp A
B(x
C)e
ik
p αxα
,{ 1
≤a,
b≤
4,
5≤α,β
≤8
1≤
A,
B,C
≤8
} .
Kundt
How closely are classical and quantal physics related? A rigorous answerto this question is given by the W 2M-correspondence Φ, a linear mapof the classical Lie–Hilbert algebra of observables onto the quantal one,almost isomorphically, as summarized in rows 14–17 of Fuphy I. W2M isnot the only such map: The literature contains uncountably many similarmaps, with various properties, often based on its C∗ algebra structure (Fi-scher et al. 1998, and references therein). W2M is unique among them—up to unitary equivalence—by mapping the translations in classical phasespace onto the corresponding translations in operator space. Equivalently,it maps polynomials in the canonical variables onto the corresponding,completely symmetrized operator polynomials. It was introduced in 1928by Hermann Weyl, as a plausible rule to pass from the classical hamilto-nian to its operator equivalent, i.e. as a quantization prescription, and in1932 by Eugene Wigner for a phase-space representation of quantum statis-tics. In 1949, Jose Enrique Moyal highlighted its properties, and in 1966,James Pool provided rigorous proofs for many sketchy results of earlierinvestigations.
We introduce the correspondence Φ by its defining property of map-ping translations in phase space (m-space) linearly onto the correspond-ing translations in operator space (M-space), or, equivalently, as a linear,possibly isomorphic map of the Lie algebra of the canonical group ontothe Lie algebra of the (special) unitary group. Their infinitesimal gener-ators Γ form maximal abelian subsystems of self-adjoint observables, onaccount of the canonical commutation relations (17) which imply that alltheir commutators are constant, hence generate the identity transforma-tion (via (11), (21)). The corresponding simultaneous eigenvalue problemsread:
�i[mα, vn] = nαvn and �i[Mα, Vn] = nαVn (23)
respectively—whereby the factors � are inserted to endow the eigen valuesnα with the same physical dimensions as their corresponding mα—and arerespectively solved by the normalized (improper) simultaneous eigen vec-tors:
vn = e− i�εβγmβnγ and Vn = e− i
�εβγ Mβnγ , (24)
which are unique up to arbitrary phase factors ei f (n) , (with f (n) real).The searched-for map Φ thus maps mα onto Mα (by definition), and vn
onto ei f (n)Vn—whereby a non-vanishing f (n) corresponds to a (unitary)inner automorphism of the quantum algebra which does not modify theexpectation values, hence will be ignored in the sequel—and we end upwith the result:
Φ a(m) =∫
dn Vn < vn, a > =: < � , a > (25)
Fundamental Physics
by expanding a classical observable a(m) w.r.t. the basis vn , and trans-lating this expansion into the corresponding quantal expansion w.r.t. thebasis Vn , whereby:
�(m) =∫
dn Vn v∗n(m) =
∫dn e− i
�εαβ(Mα−mα)nβ = �+(m) . (26)
� is self-adjoint by construction (via unitary representations), � = �+;which can be checked formally by noting that the operator in the inte-grand changes under the adjoint map into its complex conjugate, but iseven restored after a simultaneous total reflection in (even-dimensional!)phase space, nα → −nα: � = ∫
dn ei�εβγ (Mβ−mβ)nγ .
The inverse map, Φ−1, can be obtained analogously by expandinga quantum observable A w.r.t. the basis Vn , and translating it into anexpansion w.r.t. the classical basis vn :
Φ−1 A =∫
dn vn � Vn, A � = � �(m), A � . (27)
This proves that Φ is one-to one. Φ is clearly linear, and maps thereal (square-integrable) phase-space functions a(m) onto the selfadjoint(Hilbert–Schmidt) quantum operators A , i.e. observables onto observ-ables. It even maps functions of arbitrary linear combinations of thecanonical variables, cαmα, onto the same functions of their quantumcorrespondents: Φa(cαmα) = a(cαMα) , because the canonical variablescommute as generators of 1-d (unitary) subgroups. As a special case, thisinsight tells us that all functions of the qa or pa alone transform into thesame functions of the canonical quantum variables Qa or Pa : Φa(qa) =a(Qa) , Φb(pb) = b(Pb) . But for more general (than linear) functions,this algebraic isomorphy no longer holds: Φ is not associative; the near-est property to this is that Φ conserves an H∗-algebra structure of theobservables (Pool, 1966). Still, δ(m − n) is mapped onto �(n) , (justify-ing its name): we have < �(m), δ(m − n) > = �(n) , by definition of theδ distribution, implying also � �(m),�(n) � = δ(m − n) for the inversemap. A direct proof of the last formula will follow soon.
Further properties of the W2M-map can be derived with the help ofmore explicit expressions for the �-operator. To this end, we need theBaker–Campbell–Hausdorff formula:
e(P+Q) = e− i�2 [P,Q] eQ eP = e
i�2 [P,Q] eP eQ (28)
which holds for two operators P , Q (whose powers have a dense commondomain of definition) whenever their commutator [P, Q]=:c/� , def. (16),
Kundt
commutes with them. Proof: one finds by partial differentiation ofe(P+Q)e−P e−Q w.r.t. P and Q that this operator product does not dependon them, hence is a constant, which can be calculated as the zero-orderterm of the left-ordered version of the Taylor expansion of e(P+Q), (inwhich all powers of Q stand to the left of all powers of P ). Recursively,one finds from (P + Q)k+1 = Q(P + Q)k + (P + Q)k P − kic(P + Q)k−1
that all odd-order terms vanish, and that for k = 2 j , c2 j = (2 j − 1)(2 j −3) . . . 3 · 1(−ic) j holds, so that the searched-for constant equals
∑∞k=0
ckk!
=∑∞
j=0(−ic) j
2 j j ! = e−ic/2 = e�
2i [P,Q] , as claimed. The 2nd equality followsby interchanging P and Q. Note that this formula does not hold, e.g., forthe rotator, for which the above expressions have no joint, dense domainof definition: The topology of the (translation-invariant) spectra of theposition operators matters.
From the Baker–Campbell–Hausdorff formula (28) we get for theintegrand of �(m):
e− i�εαβ(Mα−mα)nβ = e
i�
[(Pa−pa)ra−(Qa−qa)sa ] = ei�(Pa−pa)ra
e− i�(Qa−qa− ra
2 )sa .(29)
We use Dirac’s notation in the position representation for using the com-pleteness relation:
1 =∫
dq ′ | q ′ >< q ′ | , (30)
and the Taylor-series expansion:
ei�
P r | q > = | q − r > , (31)
apply the RHS of (29) to (30), apply (31), and integrate over n:
∫dn
∫dq ′e
i�(P−p)r e− i
�(Q−q− r
2 )s | q ′ >< q ′ | =∫
dr∫
ds∫
dq ′e− i�
[pr+(q ′−q− r2 )s] | q ′ − r >< q ′ | , (32)
whereby from here on, all coordinate indices have been omitted. Now anintegration over s yields δ(q ′ − q − r/2) , via Fourier completeness:
∫ds
he
i�
qs = δ(q) , (33)
Fundamental Physics
and subsequent integration over q ′ replaces it by q +r/2 , leaving us (afterrenaming r → −r ) with:
�(m) =∫
dr ei�
pr | q + r
2>< q − r
2| =
∫ds e
i�
qs | p − s
2>< p + s
2| ,
(34)
where the latter (momentum representation of �(m)) has been obtained inperfect analogy.
These formulae allow us to offer more explicit expressions for thephase-space representation Φ−1 of a quantum observable, and for itsinverse Φ. For instance, again in Dirac’s notation, the trace formula:
tr(| i >< j |) = < j | i > (35)
allows us to express Wigner’s (state) function w(m) (corresponding to thequantum state W ) in the form:
w(m) =∫
dr ei�
pr < q − r
2| W | q + r
2> . (36)
Boris Leaf (1968) has shown (up to a numerical error) that this integralcan be written as the difference of two non-negative functions w{c
(38)which implies that Wigner functions are bounded by 2:
| w(m) | ≤ 2 (39)
(because of 0 ≤ ws = 1 − w/2), and that they take negative values wher-ever ws(m) > wc(m) holds, in stark contrast to their classical counterparts.Such negative occupation (pseudo) densities do not always occur, however,as shown by the groundstate Wigner function of the harmonic oscillator:
w0(p, q) = 2 e−[(q/q0)2+(pq0/�)
2] . (40)
Like all Wigner functions of pure quantum states, w0 can be seen tosatisfy: 1 = < w0, 1 > = < w2
0, 1 >.
Kundt
In order to get useful expressions for Weyl’s quantization prescrip-tion Φ, we start again from (29), and write (25) more explicitly as:
Φ a =∫
dm∫
dn a(p, q) ei�(−pr+qs) e
i�(Pr−Qs) . (41)
For polynomials a(p, q), p can be replaced by i�∂r acting on the firstexponential factor, and correspondingly q by −i�∂s , or rather—after anintegration by parts—by respectively {−i�∂r , i�∂s}, this time acting on thesecond (operator) exponential. Integration over m can now be performed,yielding δ(p − r)δ(q − s), and subsequent integration over n yields:
Φ a = a(−i�∂p, i�∂q) ei�(Pp−Qq) |p=0=q . (42)
For polynomials a(m), this formula offers a handy evaluation of Φa; inparticular, Φa can be seen to be totally symmetric in P and Q.
Another handy formula for Φa can be derived from the Q-orderedanalogue of the RHS of (29), i.e. with all Qs to the left of all Ps:
Φ a =∫
dm∫
dn a(m) e−i�
rs2 e− i
�(Q−q)se
i�(P−p)r , (43)
in which r and s in the first exponential factor can be, respectively,replaced by {−i�∂P , i�∂Q}, whereupon this exponential operator can bemoved in front of the integrals, acting upon an expression which is clearlyQ-ordered, and agrees with a(m) when all the Mα are replaced by the mα,hence equals the Q-ordered operator aQ(M) :
Φ a = e−i�
2 ∂Q∂P aQ(M) . (44)
Φ has been constructed to conserve the scalar product of the two cor-responding Hilbert spaces, i.e. is a Hilbert-space isomorphism on the ob-servables. But under Φ−1, the commutator maps into the Poisson bracketonly within the first two lowest orders in � :
Φ−1[A, B] = 2�
sin{
�
2εαβ
a∂α
b∂β
}a b =: [[a, b]] = [a, b] + O(�2) , (45)
i.e. the W2M-map does not, in general, conserve the Lie-algebra structureat third (or higher) order, though it does so for the harmonic oscillator.When mapped into phase space, the commutator is of infinite order; it isnon-local. We meet the second deviation of quantal behaviour from classi-cal behaviour. Proofs of this non-trivial lack of isomorphism can be found
Fundamental Physics
in Moyal (1949), Kundt (1966), Pool (1966), or De Groot and Suttorp(1972), see also Schleich (2001); they make use of the above Eqs. (29),(31), and (33).
With this, the crucial rows 14–18 of Fuphy have been covered, whichcan be used for a rigorous phase-space description of quantum mechanics.Quantum physics differs from classical physics in exactly two ways: byits non-local Lie product, the Moyal bracket, and by the finite support(extent) of its states, i.e. of the Wigner functions.
1.4. Spectra and Uncertainties
With most of the mathematical tools at hand, we are now ready for amore complete comparison of quantum mechanics with classical mechan-ics. Clearly, quantum mechanics celebrated its immediate successes withthe interpretation of atomic and molecular spectra, in particular with theirdiscrete parts, the spectral lines.
The properties of a classical system can be described by the (closed)subdomains E of phase space or, equivalently, by their ‘characteristic’functions χE (m) which vanish outside of E , and take the value 1 withinE : χE (m) = 1 means that a system has its phase-space coordinates minside of E . Instead, the properties of a quantal system are described bythe (closed) subspaces of the Hilbert space of pure-state vectors ψ or,equivalently, by the (self-adjoint) orthogonal projection operators E ontothem, which satisfy E+ = E = E2. In suitable coordinates, their matri-ces are diagonal, with only 0 or 1 as eigen values; the number of 1s inthe diagonal counts the dimension of the subspace onto which E projects.Measurements can determine the probability < E > := � W, E � of find-ing a system with the property E when in a state W . It satisfies 0 ≤< E > ≤ 1, and 0 ≤ � E, E ′ � for any two properties E , E ′, as can beverified like the analogous inequalities for expectation values (because upto normalization, E is a special W ). Classical properties χE (m) satisfy thesame inequalities: 0 ≤< χE >≤ 1 , 0 ≤< χE , χE ′ > . Still, quantal proper-ties differ from classical properties by their famous greater ‘uncertainties’.
For both classical and quantal observables, there is the spectral theo-rem which says that they can be represented as {sums, integrals} over their{discrete, continuous} real eigen values α :
a =∫α deα , A =
∫α d Eα , (46)
where the spectral families {deα := χαdα , d Eα} consist of (differential)properties with {0 ≤ eα(m) ≤ eα′(m) ≤ 1, 0 ≤ Eα ≤ Eα′ ≤ 1} for −∞ ≤
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α ≤ α′ ≤ ∞, and where the integral stands symbolically for both dis-crete and continuous contributions. The {eα, Eα} are Heaviside’s Θ-func-tion on the observables {a − α, A − α1}. (In the quantal case, this spectralrepresentation holds more generally for ‘normal’ operators A, defined by[A, A+] = 0 , for which the eigen values α are in general complex). Inboth cases, the expectation values fall into the convex hull of the (set of)eigen values: {αmin ≤ <a> ≤ αmax, αmin ≤ <A> ≤ αmax}, whereby αminand/or αmax equal ∓∞ for unbounded observables.
Also the uncertainties �a and �A of observables {a, A}, or squareroots of their variances, are defined in the same way:
�A :=√< (A− < A > 1)2 > , (47)
with 1 standing for the unit operator (which can be omitted in the clas-sical case, and will even be henceforth omitted in the quantal case). Theyvanish iff the state W (entering the expectation value) is an eigen state ofA, as follows from the mutual orthogonality of properties Eα belongingto different eigenvalues: When, and only when a system is prepared in aneigen state of an observable is there no scatter of its values in repeatedmeasurements!
A difference shows up, however, between the classical and quantalvariances for repeated measurements of non-commuting observables A, B:we have:
�A �B ≥√
c2A,B +
(�
2< [A, B] >
)2(48)
for the uncertainty �B of a measurement of an observable B of a systemwhich has been prepared such that its observable A has uncertainty �A ,or vice versa. Here [A, B] is the commutator (16) of A and B, and cA,B
is the covariance of A and B :
cA,B := 12[< (A− < A >)(B− < B >)+ (B− < B >)(A− < A >) >] .
(49)The uncertainty relation (48) implies the more familiar form: �A�B ≥�
2 | < [A, B]> | whether or not cA,B vanishes. The proof of (48) is asimple consequence of the Cauchy–Schwarz inequality, applied to the (notstrictly positive) scalar product < A, B > := tr(W A+B) :
< A2 >< B2 > ≥ |< A, B >|2 = (� < A, B >)2 + (� < A, B >)2 , (50)
in which the RHS equals the radicand of (48) for vanishing expectationvalues < A >, < B > , by the very definitions of the real and imaginary
Fundamental Physics
parts of < A, B > , and of the commutator [A, B] . For non-vanishingexpectation values, the proof (50) can be repeated with A, B replaced byA− < A >, B− < B > , respectively.
The classical (correspondence of the) uncertainty relation can be han-dled analogously, with Cauchy–Schwarz applied to the scalar product(a, b) :=
∫dm w(m) a∗(m) b(m), leading to the simpler relation:
�a �b ≥| ca,b | with: ca,b :=< (a− < a >)(b− < b >) > . (51)
Here the scatter �a for successive measurements of a(m) and b(m) van-ishes iff at least one of the two observables is in a pure (classical) state.Quantum mechanics differs markedly through the occurrence of the com-mutator in the expected uncertainty.
In applications, it can be useful to introduce the scale-invariant corre-lation coefficient κA,B := cA,B/�A�B , which satisfies:
| κA,B | ≤√
1 −(
�
2< [A, B] >�A �B
)2 ≤ 1 , (52)
and which measures the degree of correlation of two observables; it van-ishes for conjugate variables with zero covariance, and assumes its maxi-mum in magnitude for observables with common eigen states. It plays arole in quantum logistics.
As already mentioned above, quantum states differ significantly fromtheir classical correspondents: Their maps under Φ−1—the so-called Wig-ner functions—satisfy 0 <w,w> ≤ <w, 1> = 1, hence are square-integrable, satisfy |w(m) | ≤ 2 according to (39), and can be negative insubdomains, properties that restrict the degree of confinement. The clean-est (quantum) states possible are the pure states, described by 1-d rays inHilbert space: W = | ψ ><ψ | , and can be characterized by being identi-cal to their square: W 2 = W ; their matrices have just one non-vanishingdiagonal element, which equals 1 because state operators are non-negative,0 < W , and because they are normalized: � W, 1 � := tr(W ) = 1 . Theirphase-space maps w(m) = Φ−1W under W2M are called pure Wignerfunctions, and satisfy 1 = <w, 1> = <w,w> because Φ−1 is a Hilbert-space isomorphism. They can be characterized by satisfying <w, v>≥ 0for all Wigner functions v , in consequence of the corresponding propertyof their images under Φ (Fischer et al. 1998). Like all Wigner functions,the pure ones also satisfy | w(m) |≤ 2 .
The formulae of this section have been selected as representative forthe key differences between classical and quantal dynamics. In Fuphy I,they occupy rows 19–27.
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1.5. Box Thermodynamics
This section will be restricted to the non-relativistic (NR) thermo-dynamics of closed, homogeneous systems. It will sketch the derivationof the fundamental equations of box thermodynamics from the distri-bution function wN (m1, . . . ,m N ) on 6N -dimensional phase space, or theN -particle distribution operator WN . A strictly parallel treatment is possi-ble of the classical and quantal case because the projections of N -particlesystems onto r -particle systems—via multiple integrations over their reducedphase-space distribution (classically), or via multiple ‘tracings’ of theirreduced distribution operator (quantally)—commute with the W2M map Φ,and all quantal calculations can be performed, in principle, with the reducedWigner functions; a proof of this commutability (due to Leschke 1970) fol-lows the formulae obtained above (36), whereby extra care is required tomake sure that also the Fermi–Dirac and Bose–Einstein symmetries areconserved. In this section, the use of small or capital letters will often besacrificed to the alternative textbook convention, starting with particle num-ber N and temperature T , and continuing with the thermodynamic (box)potentials F , U , S, G, etc.
Let us start, then, in our derivation of the thermodynamic equationsfor a NR N -particle system, with the assumption of the existence of a dis-tribution function wN (m1, . . . ,m N ) on 6N -dim phase space (in the macro-scopic case), or of a distribution operator WN on N -particle Hilbert space(in the microscopic case), and try to derive the basic equations of boxthermodynamics. A pragmatic approach restricts considerations to equilib-rium systems of distinguishable particles for which ln WN (or lnwN ) is aconstant of the motion, and is additive when subsystems are combined:0 = d
dt ln WN = ∂t ln WN , and: ln W1∪2 = ln W1 + ln W2 , hence dependsonly linearly on the additive constants of the motion [Grad theorem]. The
latter are particle number N , energy E (= H ), momentum→P , and angular
momentum→Q × →
P . When isotropy is postulated in addition, in the localsystem of rest, only the first two constants survive, and we have:
W = eα−βH(M) + γ N (53)
for the grand-canonical distribution (considering particle exchange with theenvirons), or:
WN = eα−βHN (M) (54)
for the canonical distribution describing systems whose particle number Nis considered fixed.
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The physical meaning of the constant β in (54) is revealed when wecalculate the mean kinetic energy < p2/2m >=: kT/2 of a 1-d (classical)particle, say, for a canonical distribution:
kT =< p ∂phN > = < p,−β−1∂p wN > = β−1 , (55)
where the scalar product allowed a permutation of factors, ∂pw = −βw∂phholds for (54), and an integration by parts yielded the result: β = 1/kT .More generally: for hamiltonians depending quadratically on the canoni-cal variables, the equipartition theorem yields an average energy of kT/2per degree of freedom.
Further insight is gained when we use the normalization condition for{wN , WN }: the definition 1 = < 1, eα−β hN > =: eαZ N yields the (canon-ical) partition integral {sum}:
e−α =: Z N ={ 1
N !∫
dm e−β hN
tr(e−βHN ) = ∑
jw j e−β η j
}
(56)
for{classical
quantal
}canonical N -particle systems, with energy eigen values η j of
statistical weights w j in the quantal case. Z N can, and has to be explicitlyevaluated for special systems. By definition, the energy α/β is called theHelmholtz free energy FN of the system:
FN := α/β = −kT ln Z N . (57)
For an ideal classical gas, the integral in Z N is the N th power of the3-d momentum-space integral Z1 = (V/h3)
∫d3 pe−β p2/2m = Vλ3 , where
λ := h /√
2πmkT (58)
is the thermal de Broglie wavelength of the particles, and where use hasbeen made of
∫dx e−x2 = √
π . The Stirling formula N ! = √2πN N+1/2
e−N (1 + ϑN/12N ) now yields for ln Z N and FN :
ln Z N = N [1 − ln(nλ3)] = −βFN . (59)
From these auxiliary formulae, all the other thermodynamic potentialsof equilibrium N -particle systems can be calculated, such as the internal
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energy U , the entropy S, and the Gibbs free enthalpy G N := U − T S + pV .For U we obtain (in the quantal case):
U :=< HN > = � HN , eα−β HN �= −eα∂β�1, e−β HN � = −∂β ln Z N
(60)
and for S :
S / k := − � WN , ln WN �= − < α − βHN >= (1 − β ∂β) ln Z N .(61)
In the case of a one-component ideal Maxwell–Boltzmann gas, insertion of(59) thus leads to the final expressions:
U = N32
kT, S = Nk[5
2− ln(nλ3)
]. (62)
And the average occupation numbers < N j > of the energy eigen value η j
follow from (56), (59), and Z N = Z N1 /N !, as:
< N j >= −β−1 ∂η j ln Z N = nλ3 e−β η j . (63)
Some extra effort is required for the statistics of quantal gases. Forthem, it is easier to work with the grand canonical distribution introducedin (53), whose partition function Z(x) is obtained analogously to (56) from
the normalization of W : 1 = tr(W ) = eα∞∑
N=0eγ N Z N =: eαZ(eγ ), as:
Z(x) :=∞∑
N=0
x N Z N with: K := α/β = −kT ln Z(eγ ) . (64)
In principle, once we know the grand canonical function Z(x), we canget its (canonical) coefficients via: Z N = Z (N )(0)/N ! But there is no need:Z(x) is easy to evaluate directly for (one-component) Fermi–Dirac andBose–Einstein gases, whose energy-occupation numbers N j run, respec-tively, through
{ 0 , 10,...,∞
}:
Z(x) =∑
N j
∞∏
j=1
x N j e−β N j η j =∞∏
j=1
∑
N j
(x e−β η j )N j =∞∏
j=1
(1 + ε e−β (η j −µ))ε
(65)
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for x = eγ , γ =: βµ , and ε = {+1−1
}. From here, the average thermal occu-
pation numbers < N j > of a quantum gas follow as:
< N j >= − 1β∂η j ln Z(eγ ) = (eβ (η j −µ)+ε)−1 =: f j for ε = {0, 1,−1} ,
(66)this time even for all three types of gases: Maxwell–Boltzmann, Fermi–Dirac, and Bose–Einstein, when ε runs through the values {0,+1,−1},respectively, cf. Eq. (63).
As a last effort of this review of box thermodynamics, let us calcu-late the thermodynamic potentials of a quantum gas. From the general def-inition (61)—with WN replaced by W —we recover the well-known grandcanonical expression for the entropy: S/k = −� W, ln W � = <−α+βH−γ N > = β(−K + U − µN ) , whence:
T S = U − K − µN = [−∂β + β−1(1 − µ∂µ)] ln Z , (67)
and from (65):
−S / k =< f, ln f > + ε < (1 − ε f ), ln(1 − ε f ) > with ε = {0,±1} ,
(68)
where f has been inserted for the mean occupation numbers (eβ (η j −µ) + ε)−1
obtained in (66) , cf. (53).Alternatively, from the first equality in (67), the Gibbs–Duhem relation
K = U − T S − µN = F − G = −pV (69)
for the Helmholtz free energy F and Gibbs free enthalpy G (introduced in(57) and below) can be obtained, in which g := (∂N G)p.T = (∂N U )S,V =µ is called the chemical potential. With this, rows 28–32 of Fuphy I havebeen covered.
1.6. Continuum Thermo-Hydrodynamics
This last section on NR mechanics is based on the statistical hierarchyof evolution equations for r -particle distribution functions wr (m1, . . . ,mr ),or distribution operators Wr (M1, . . . ,Mr ), in order to obtain the kineticequations (of Boltzmann, Fokker–Planck, and Lenard–Balescu) via trunca-tion, and therefrom the fundamental continuum equations (including thoseof Euler, and Navier–Stokes), as the balance equations (under collisions)
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of the conserved quantities: (rest) mass, momentum, and energy. As alreadymentioned at the beginning of the preceding section, a parallel treatmentof the classical and quantal problem is facilitated by the commutability ofthe transition between phase space and operator space (via Φ) with theprojection down the hierarchy.
Note that the basic laws of (both classical and quantal) physics aretime-reversal invariant—with the exception of a few decays of elementaryparticles—whereas the realistic evolution obeys the Second Law, the law ofgrowth of entropy, which deviates from the exact behaviour by averagingover all equivalent histories (due to different, but almost equivalent initialstates). The arrow of time enters into the fundamental equations throughthe truncation of the hierarchy, which replaces the exact (or ‘fine-grained’)temporal evolution by the probable (or ‘coarse-grained’) one, and becomesobvious through the diffusion and friction terms in the equations ofmotion which imply mixing and damping (for multi-component systems),and through the growth of the entropy function in all realistic processes.This entropic arrow of time agrees with that of microscopic (radioactive,K-meson, . . . ) decays, via the 4-momentum constraint for collisions, andwith the electromagnetic arrow of time defined by retarded (rather thanadvanced) potentials for antennae. Their relation to the cosmic arrow oftime (defined by expansion) has been emphatically discussed in the litera-ture; I do not see it. In any case, our equations make only (realistic) pre-dictions for the future, for sufficiently well-known initial states—not forthe past—i.e. for the time direction of increasing entropy.
The formulae of this section will all be formulated on phase space, ifnecessary by means of Wigner functions. Most of them will be too long tofit into the (two) columns of Fuphy I ; we shall thus transit to Fuphy II,and start with the generic r th-order hierarchy equation:
{∂t + Hr − Ur }wr = (N − r)∫
d6mr+1 U ′r+1wr+1 , 1 ≤ r ≤ N , (70)
for a classical N -particle system governed by interactions between pairsof particles only, of potential ϕ jk(
→q j − →
q k), in an external field of
potential ϕ(→p ,
→q ) with ∂→
p· ∂→
qϕ= 0 , in which the distribution functions
wr (m1, . . . ,mr ) have been assumed symmetrical in all their variables(without restriction of generality) because we are only interested in theexpectation values of (sum) observables which have this same (permuta-tion) symmetry:
wr (m1, . . . ,mr ) :=∫ N∏
j=r+1
dm j wN (m1, . . . ,m N ) , (71)
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and in which the kinetic and potential operators Hr , Ur , U ′r+1 are defined
as:
Hr :=r∑
i=1
( →pi
mi∂→
q i− ϕ
,→q i∂→
p i
), Ur := −
r−1∑
j<k , j=1
Φ jk ,
U ′r+1 :=
r∑
i=1
Φi(r+1) , (72)
with:Φ jk := ϕ
jk ,→q j∂→
p j+ ϕ
jk ,→q k∂→
p k. (73)
Note that the masses mi in (72) must not be confused with the canonicalvariables mα, and that the RHS of (70) vanishes for r = N , for which itis called the N -particle Liouville equation. Note, finally, that the r -particledistributions wr are (likewise) normalized:
< wr , 1 >= 1 . (74)
In order to truncate the hierarchy for a certain r � N , one rescalesthe coordinates
→q in units of r0/V 1/3 —where r0 is the (small) reach of
the pair interaction, and V the mean volume per particle—and rescales thepotential operators U in units of the (small) ratio α :=< E pot > / < Ekin >
formed for a typical interaction volume, and arrives at the (twiddled)rescaled hierarchy equations:
{∂t +∼
Hr −α∼
Ur }wr = ε α
∫d6mr+1
∼
U ′r+1wr+1 with: ε := n r3
0 , n := N/V ,
(75)which can be simplified (truncated) for short-range pair interactions: ε � 1and/or weak interaction forces: α� 1, in order to derive the well-knownkinetic equations of {Boltzmann, Fokker–Planck, Lenard–Balescu} for,respectively: {α ≈ 1,� 1,� 1}, {ε � 1,� 1,� 1}, and {αε� 1, � 1, � 1},or even those of {Liouville and Vlasov} under total neglect of the collisionterm on the RHS, for αε ≪ 1 , as special cases of {Boltzmann and Le-nard–Balescu}.
We illustrate the truncation for the case r = 1 in which LudwigBoltzmann factorized the 2-particle distribution w2(1, 2) before each col-lision, and integrated it through a collision under the assumptions of (i)ignorable triple collisions (because of ε � 1), (ii) essential dependence onthe relative coordinates of the two colliders only (because of ε � 1, α � 1),(iii) a reverse invariance of the collision cross section, σ(1, 2 → 1′, 2′) =
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σ(1′, 2′ → 1, 2), as a consequence of CPT invariance of the interaction,and (iv) ‘molecular chaos’: the incoming 2-particle distribution is alwaysuncorrelated: w2(1, 2) = w1(1) w2(2). Clearly, the time-reversability of theexact motions is destroyed by assumption (iv), which effects the transitionof the description of a system from its exact behaviour to its probable,macroscopic one. As a result, we get from (70):
d
dtw1(1) := {∂t +
→p
m∂→
q+ →ϕ ∂→
p} w1(
→m) = N
∫d6m2
→ϕ 21 · ∂→
p 1w2(1, 2)
trunc→(76)
d
dtf (1) = h−3
∫d3 p2d2ω′ | �→
v12 | σ(1, 2 → 1′, 2′){ f (1′) f (2′)− f (1) f (2)} ,(77)
j := (→p j ,
→q ), whereby the normalization of the r -particle distribution func-
tion has been (routinely) changed from wr to fr := Nrwr , with: f1 =: f ,and whereby the collision term on the RHS resulted from an approximateintegration of the 2nd equation of the hierarchy (with vanishing 3-particlecollision term), using center-of-mass and relative coordinates during thecollision.
We have thus arrived at one of the central equations of theoreticalphysics, the Boltzmann equation, but so far only for the (non-quantum)NQ, NR case. Its generalization to relativistic velocities will be the subjectof the next section. In this section, the so-far neglected quantal correspon-dent of the classical hierarchy ought to be handled. Instead, the literaturetends to treat the collisions by first-order quantum-mechanical perturba-tion theory, via an introduction of F.-D.- and B.-E.-statistics for the ingo-ing and outgoing particles. A more systematic approach should start withthe quantum prescription for describing the union of systems by the tensorproduct, Hr := H1⊗· · ·⊗H1, of their 1-particle Hilbert spaces H1, wherebyfor indistinguishable particles, only the subspaces of completely symmetrized(bosons: ε = −1), or completely antimetrized (fermions: ε = 1) products of1-particle states must be used. Correspondingly, the r -particle state opera-tor Wr is obtained from WN by partial tracing, and acts on this suitablysymmetric subspace Hε
r of the r -fold tensor product of 1-particle Hilbertspaces:
Wr (M1, . . . ,Mr ) := NT r
j=r+1WN (M1, . . . ,MN ) ; (78)
it will be assumed symmetrical in all its arguments—like in the classicalcase—because thermodynamics is only interested in the expectation valuesof (symmetric) sum variables. Like its classical correspondent, Wr is nor-malized:
� Wr , 1 � = 1 . (79)
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The quantal hierarchy can therefore be formulated in perfect analogy tothe classical one, as started in (70), and truncated for (sufficiently) weakparticle interactions of (sufficiently) short range. Already the most fre-quently used case of lowest order, Boltzmann’s, must now take care ofFermi–Dirac and/or Bose–Einstein statistics for indistinguishable collisionpartners when evaluating the integral in (76) via the 2-particle evolution.The well-known—though non-trivial — result of such a calculation for a1-component system generalizes the expression in curly brackets of (77) to:
{ f (1′) f (2′)− f (1) f (2)} →[ ∏
j=1,2,1′,2′f ( j)
]
{ fε(1) fε(2)− fε(1′) fε(2′)} with:
(80)fε := f −1 − ε, or: f = 1/( fε + ε) , where as before, ε = {0, 1,−1} holdsrespectively for {M.-B., F.-D., B.-E.} statistics.
The quantal version of the integrand in Boltzmann’s collision integralcontains the lengthy algebraic expression (80) formed from the 1-particledistribution function fε(m), which takes care of the proper statistics. Notethat for ε = 0 , it reproduces the classical expression (on the LHS). Alsoin the two new cases (ε = ±1), it guarantees the non-decrease of entropyfor solutions of the Boltzmann equation (for indistinguishable particles),because forming the time derivative of (68) yields a positive integrandmultiplied by d f
dt ln fε , and insertion of (77) for d fdt with the generalized
expression (80) leads to Boltzmann’s H-theorem: a d S/dt proportional to∫d3 p1
∫d3 p2[∏ f ( j)]{ fε(1) fε(2)− fε(1′) fε(2′)} ln fε(1) which can be seen
to be non-negative by writing it more symmetrically—letting the argumentof ln fε(1) run successively through {1, 2, 1′, 2′}—as:
d S
dt∼
∫d3 p1
∫d3 p2[
∏f ( j)] { fε(1) fε(2)− fε(1′) fε(2′)}1
4ln
fε(1) fε(2)fε(1′) fε(2′)
≥ 0 , (81)
the latter because {a − b} ln ab ≥ 0 holds for positive a/b, i.e. at least in
the classical approximation of non-negative f ( j) . (It is not clear to meat this time whether this result can be violated by certain Wigner func-tions, when they turn negative in subdomains. Such partially negative dis-tributions do occur for suitable potentials, see Hudson (1974), but may notoccur for simple systems near thermal equilibrium). Note that this growthof entropy, shown for solutions of the (truncated!) Boltzmann equation, isa result obtained for the probable (coarse-grained) evolution, in stark con-trast to the constancy of S for the exact, time-reversible evolution. Notealso that d S/dt vanishes iff fε(1) fε(2) = fε(1′) fε(2′) holds, in which caseln fε(m) is an additive constant of the motion, and fε(m) is canonical,
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cf. (53). These two results strengthen our confidence in the physical cor-rectness of the kinetic equations, in particular of (77) and (80). With this,we have covered the last two rows of Fuphy I, as well as the first four rowsof Fuphy II.
It remains to be shown that from the kinetic equations, all the fun-damental thermo-hydrodynamical equations can be derived, as the prop-agation equations for the collision invariants a(m). Again, we restrictdemonstrations to the Boltzmann equation, in this section still for the NRcase, either classical or quantal. The derivation starts with the equationof motion in the Schrodinger picture, Eq. (9), for the expectation value< a> := ∫
d6m a(m) w(m) = (nh3)−1∫
d3 p a(m) f (m) of a collisioninvariant a(m):
d
dt< a > = (nh3)−1
∫d3 p a(m)
d
dtf (m) , n := N/V . (82)
For a not explicitly time-dependent a(m), insertion of ddt f (m) from (77)
with (80), and three integrations by part yield the desired equation ofmotion:
∂t (n < a >)+ ∂→q(n <
→v a >)− n <
→v ∂→
qa > = n < ∂→
p(→ϕ a) > , (83)
because the collision integral on the RHS of (77) drops out, as it can bewritten in the symmetrized form (of its last factor, under the two momen-tum-space integrals, like for (81)):
→q ), whose second curly bracket vanishes for collision invariants
a(m).Equation (83) will now yield the searched-for five basic equations, by
letting a(m) run successively through the five collision invariants {m ,→p ,
(→p − →
u )2/m}, where m stands for the particles’ (rest) mass,→p for their
momentum vector, and the last entry for (twice) their kinetic energy;→u :=
<→v >. The five equations are the conservation of mass, the three Navier–
Stokes equations (generalizing the Euler equations to the presence of fric-tion), and an energy balance that takes care of non-ordered, randomized
Fundamental Physics
kinetic energy which is often called ‘heat equation’, but which deservesbeing called the entropy balance. With the notations:
n(q, t) :=∫
d3 p f (p, q, t) , µ := m n ,→u :=< →
p > /m ,
kT := m3< (
→v − →
u )2 > (85)
for the particle-number density n , mass density µ , bulk velocity→u , tem-
perature T , and:
→q := µ
2< (
→v − →
u )(→v − →
u )2 > , p jk := µ < (v j − u j )(vk − uk) > ,
for the heat-flow vector→q (not to be confused with the coordinate vector→
q ), and pressure tensor p jk , further by assuming momentum-independentforce fields
→ϕ =
→ϕ (
→q , t), and by passing again from the vector notation to
Ricci’s index notation, we get in a straight-forward manner the mass-con-servation law:
d
dtlnµ := (∂t + ui∂i ) lnµ = −Θ , Θ := ∂i u
i , (86)
with Θ := div→u = (local) expansion rate; further the Navier–Stokes equa-
tions :µ
d
dtu j = n ϕ j − ∂k p jk , (87)
which yield the acceleration of the mean flow→u as the sum of an external
acceleration n→ϕ and the divergence of the internal pressure tensor
↔p ; and
finally the energy balance:
32
nd
dtkT = − (∂i qi + p jk ∂ku j ) . (88)
The last equation describes the temperature evolution of a fluid sys-tem. It is not required for the description of its ordered motion, which isfully determined by its momentum balance (87). Instead, it describes thegeneration, and dissipation of disordered (kinetic) energy, which by boxthermodynamics is taken care of by the entropy function. And indeed, itcan be seen to exactly generalize (interpolate) the law of the growth ofentropy from box thermodynamics to continuum thermodynamics: Intro-duce the entropy per particle s through its defining (textbook) differentialrelation:
du =: T ds − p dV = T ds + p n−2dn . (89)
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Here u = 32 kT stands for the internal energy per particle, (not mean
velocity u!), cf. (62), whose insertion into (88) with (89) yields the entropybalance (for one-component systems):
Tds
dt= −(∂i qi + p jk ∂ku j − p Θ) . (90)
This growth of comoving entropy vanishes for (the adiabatic expansionof) ideal gases—as we shall soon see—and it describes quantitatively itsincreases due to inflowing heat, viscous shearing, and (chemical) mixing(for multicomponent systems). When Landau and Lifshitz in (VI, 1966,(49.5)) called it the ‘heat transport’ equation, they were unnecessarily shy.
In order to get the still missing constitution equations for the quan-tities p, qi and p jk , we must use the explicit shape of the distributionfunction f (m) which enters the equations of motion (83). This has beendone in the literature Cohen (1961), beginning with (canonical) localequilibrium distributions, and continued with first-order deviations there-from—by Chapman, Enskog and/or Hilbert—who recovered, after lengthycalculations, the ideal-gas equation of state, Fourier’s law of heat conduc-tion, and the viscosity tensor:
p = nkT , qi = −κ ∂i T , p jk − p δ jk = −2η σ jk , (91)
with the shear tensor σ jk defined by
σ jk := 12(∂ku j + ∂ j uk)− Θ
3δ jk , (92)
and with the following expressions for the dynamic-viscosity scalar η, andheat-conduction scalar κ , in terms of the (approximate, microscopic) col-lision cross section σ , and mean time between collisions τ :
η = nkT τ = 25
m κ =√
m kT
σ, κ = 5
2nkT
τ
m. (93)
With these relations, the growth law of entropy (90) can be written in theform:
nTds
dt= 2η σ jkσ jk + ζΘ2 + ∂ i (κ∂i kT ) (94)
whose (three) sources are (i) a (positive) production rate due to viscousshearing, plus (ii) another one due to bulk viscosity whose coefficient ζvanishes for simple gases in both the NR and ER limit, but can be shownto be positive for transrelativistic expansion velocities, and (iii) a conduc-tion term whose contribution to some arbitrary domain vanishes for zero
Fundamental Physics
net inflow (via Gauss’s theorem), but is positive for net inflow, and nega-tive for net outflow of heat, (i.e. cooling).
In principle, for media of higher density (like fluids), similar calcu-lations can be made to derive yet better approximations, in the form ofvirial expansions; there is no indication of a lack of reality in the basicequations. We have thus completed our coarse derivation of both classi-cal and quantal NR continuum dynamics from the statistical hierarchy ofr -particle evolution equations, represented by rows 5–8 of Fuphy II. Theirrelativistic generalizations will be the subject of the next chapter.
2. RELATIVISTIC SYSTEMS OF N PARTICLES
The preceding chapter was restricted to phenomena that proceedslowly compared with the speed of light, i.e. to non-relativistic (NR)dynamics. Maxwell’s electrodynamics and the special theory of relativityhave not yet been touched, nor have gravitational fields been consideredbeyond the Newtonian regime. It is the purpose of this chapter to showthat for spinless particles, there are no principal obstacles to an exten-sion of all our derivations into the relativistic regime: All that is requiredis familiarity with the existing literature plus perseverance. In particular,Boltzmann’s kinetic equation can be generalized to describe the probableevolution of a relativistic N -particle system—both classical and quantal—acted upon by electromagnetic and additional forces, and the basic thermo-hydrodynamical equations will be derived therefrom, often following Ehlers(1973).
We can calculate the spectrum of a hydrogen atom with an accuracyof 11 significant figures. Why then venture into quantum electrodynam-ics, and quantum field theory (QFT), with all their fundamental short-comings (Kundt 1966; Witten 2005)? It is our lack of exact knowledgehow to describe the elementary particles, i.e. how to calculate their masses,charges, spins, magnetic moments, and modes of interaction that keeps ussearching for the ‘theory of everything’. Is further quantization required,or should we aim at a joint boundary-value problem of an almost classi-cal theory for fields and particles? Will gravity be of importance? Gravityis an extremely weak force, Gm2
e/e2 = 10−42.6 times weaker for two elec-
trons than their mutual Coulomb repulsion. But it grows unlimitedly largewith decreasing separation, and gets comparable to several other forcesfor a Planck particle, the smallest (in volume) possible particle, of mass√
�c/G = 10−5 g, and size√
�G/c3 = 10−33 cm, whose de Broglie extentequals its minimal gravitational extent.
Kundt
For the common elementary particles, however, their measured massesare so small that gravity in their exteriors can at best be a minor pertur-bation. Still: after all the effort that has gone into general relativity, andits comparatively small importance for the solar system, and likewise onthe much larger scales of the observable universe, wouldn’t it be aestheticif it revealed its true importance on the very tiny scales of the constituentparticles?! Why does the gyromagnetic ratio of a spinning, charged blackhole equal that of the electron? We shall return to this line of thought inthe next and last chapter, after having thrown a closer look at Einstein’sgeneral theory of relativity, in Chapter 3.
2.1. Relativistic Thermo-Hydrodynamics
We return to the stationary distribution function w(m) of an N -par-ticle system at thermal equilibrium, and ask for its canonical shape, thistime without a restriction to non-relativistic particle velocities. Already inconnection with the entropy growth law (81) did we find that for one-component systems of indistinguishable particles, w should be replaced (asstated in (80)) by wε := w−1 − ε in the integrand of Boltzmann’s collisionintegral, with ε = ±1, in order to likewise cover the cases of Fermi–Diracand Bose–Einstein gases. (Note: − lnwε+1−ε2 is the derivative of the inte-grand in < s > w.r.t. w , see (68)). For such systems, the relativistic versionof Grad’s theorem tells that lnwε must be an additive collision invariant—in order for the (coarse-grained) entropy to be constant with time—hencemust be a linear function of 4-momentum: − lnwε = α+βa pa . Whence inthe local rest frame (of vanishing centre-of-mass 3-momentum):
w = 1e−α + β e + ε
, ε = {0 , ± 1} , (95)
with e := −cp4 standing for (the classical) energy, and β := 1/kT asbefore; note that Maxwell–Boltzmann statistics (for distinguishable parti-cles) are included for ε= 0. This is the famous textbook expression forthe canonical energy distribution in a (classical or quantal) gas, mostfamous for the electrons in a conductor, but now even valid in the rela-tivistic regime, as Planck’s law describing blackbody radiation (for α = 0,ε = −1).
As already mentioned at the end of Sec. 1.2, quantum mechan-ics requires a new description of the spin of a particle—or system ofparticles—whose associated magnetic moments interact with a non-van-ishing ambient magnetic field. For its relativistically correct description,
Fundamental Physics
Pauli’s NR 3-vector of spin matrices must be generalized (for a singleelectron) to Dirac’s SR spin 4-vector, i.e. from its rotation-covariant formto its Lorentz-covariant form. This non-trivial transition takes its depar-ture from Einstein’s SR expression for the energy e of a classical particle interms of its rest energy mc2 and 3-momentum
→p , also called 4-momentum
identity, or rather from its more general form in the simultaneous presenceof an electromagnetic field of 4-potential (ϕa) =: (ϕ0 , ϕα) :
0 = (mc)2−(e−eϕ)2/c2+(→p − e
c→ϕ )2 =
{(mc)2 + (pa − e
cϕa)(pa − e
cϕa)
−{γ0[γ a(pa − ecϕa)+ imc]}2
}
,
(96)whose last (non-trivial) version contains Dirac’s 4-spinors γ a which arecomplex 4×4 matrices satisfying the hermitian matrix identities
γ (aγ b) = gab ,�
iγ [αγ β] =: σαβ = εαβγ σγ , (97)
in which gab is the (contravariant, Minkowskian) 4-metric tensor, roundand square brackets around indices act as defined in (105), and in whichthe 3-vector σγ can be shown to be the direct sum of two (identical, her-mitian, 2×2) Pauli matrices.
From here, Dirac’s (slightly modified) expression for the energy oper-ator H := cPo of a quantal particle with spin is obtained by taking the(positive) square root of his version of (96)—a linear form (!) in pa—solv-ing it for po− e
cϕo , passing to its quantal version, squaring both sides ofthe equation (whereby [Pα,Φβ ] = Fαβ and Eqs. (97) are used), and subse-quently returning on both sides to their (positive) square roots:
H − eΦo = mc2
√
1 + 1m2c2
(→P − e
c
→Φ
)2
− 2e
m2c3→σ · →
B . (98)
Satisfactorily, this Hamiltonian is positive (by construction, at least fornot-too-strong electromagnetic fields), and tends to the Schroedinger–PauliHamiltonian in the limit of NR velocities. Its matrix form has nolonger a classical analogue, hence can be W 2 M-mapped routinely only asan injection, (one-to-one only for its expectation values), and therefore
defies a correspondent classical evaluation. Yet the identity∼Hd
∼H(
→q ) =
c2∼Pd
∼P(
→q ) holds like in the classical analogue for
∼H(
→q ) := H − e
c Φo(→q ),
(and correspondingly for∼P(
→q ) ), allowing to replace c2 p dp in the (clas-
sical) integrands by e de—though in general only in a→q -dependent man-
ner—and at least the case of a constant magnetic field→B can be evaluated
rigorously, (even without a subsequent integration over→q -space).
Kundt
Returning to (95), and assuming ϕa = 0 (to avoid the mentionedmatrix complification), we can now calculate the thermodynamic state vari-ables, i.e. the average {particle number per volume, kinetic energy per vol-ume, pressure, entropy per volume} for any gas or plasma with negligibleelectromagnetic field, as the canonical expectation values < a > for a ={1, e, (c
→p )2/3e, lnw + εwε ln(wwε)}. Starting with n , we get from (95):
n = h−3∫
d3 p w = 4π(hc)3
∫de e cp
e−α + β e + ε
= 4π(
kT
hc
)3 ∫dx x
√x2 − (mc2)2
ex − α + ε, (99)
because of: d3 p = 4πp2dp, and: e2−c2 p2 = (mc2)2, m being the rest mass.In the same vein, one gets the four generally valid integral expressions fora one-component gas (Kundt, 1971):
n = A(T ) I 11
(mc2
kT, α, ε
)
,u
kT= A(T ) I 21
(mc2
kT, α, ε
)
, (100)
3 pkT
= A(T ) I 03
(mc2
kT, α, ε
)
,s
k= A(T )
[
I 21 + 13
I 03 − α I 11]
,
(101)with:
A(T ) := w 4π(kT/hc)3 , I ab(ξ, α, ε) :=∫ ∞
ξ
dx xa (x2 − ξ2)b/2
ex −α + ε,
(102)in which w is the spin-statistical weight of the gas, equal to {1, 2, 2s + 1},respectively for particles of rest mass m, spin s, with: {m = 0 = s, m = 0 < s,ms > 0}. α, the only free parameter in above equations, is determined bythe density n through (100).
Of importance for the applications are the (extremely-relativistic) ERand NR limits of the integrals defined in (102). They can be evaluated inclosed form, with the results, for ξ → 0 (or: mc2 � kT ):
I 11(ξ, 0, ε) → 7 − ε
4ζ(3) ,
{I 21(ξ, 0, 1) and I 03(ξ, 0, 1) → 7π4/120I 21(ξ, 0,−1) and I 03(ξ, 0,−1) → π4/15
}
,
(103)
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ζ(3) = 1.202 ; and for ξ → ∞ (or: mc2 � kT ):
I ab(ξ, α, 0) → Γ
(
1 + b
2
)
eα − ξ ξa + b2 (1 + O(ξ−1)) . (104)
With them, box thermodynamics can be pursued in all regimes, as summa-rized in rows 9–11 of Fuphy II.
The most important force fields in physics are Maxwell’s electric field→E and magnetic field
→B , which special relativity combines (yet more ele-
gantly) into a skew tensor Fab = F[ab], with six independent components.Here braces around indices denote forming the antimetric part, whilstparentheses denote forming the symmetric part:
F[ab] := 12(Fab − Fba) , T(ab) := 1
2(Tab + Tba) . (105)
Note that from here on, the convention of Chapter 1 is finally given up, inwhich classical quantities (like T , Tab, Fab) were represented by small let-ters, in order not to deviate too grossly from the historical notation (evenwhen it is less systematic).
When expressed through Fab, Maxwell’s (eight) field equations read:
F[ab,c] = 0 , Fab,b = 4πρa , (106)
where ρa stands for the electric 4-current density ρua , with charge den-sity ρ := en, (dimensional) 4-velocity ua := cγ (βα, 1), Lorentz factor γ :=1/
√1 − β2, (dimensionless) 3-velocity
→β := →
u /cγ , and where ‘commas’denote partial differentiation: ∂αa =: a,α . The first, homogeneous set ofMaxwell’s equations is locally equivalent to the existence of a 4-potentialΦa :
Fab =: 2 Φ[b,a] , (107)
whose gauge freedom can be used to impose the Lorentz gauge Φa,a = 0,
whereupon the inhomogeneous set of field equations simplifies to:
� Φa := Φa,b
,b = −4πρa . (108)
This wave equation describes the (classical) far fields (Φa , or Fab) of amoving charge density ρ in a Lorentz-covariant manner, in harmony withspecial relativity (whose signals propagate in vacuum at the speed of light);no later improvements had to be performed on Maxwell’s original equa-tions, as opposed to Newton’s theory of gravitational interactions. A fieldFab exerts the force e
c Fabub onto a point charge e. The energy density,
Kundt
momentum density (Poynting flux), and tensions of a field Fab can be col-lected into Maxwell’s symmetric, trace-zero stress-energy-momentum tensor:
Telm
ab := 14π
(
− Fac Fb.c + 1
4gab Fcd Fcd
)
, T aa
elm= 0 , (109)
in which gab is the Minkowski metric, whose (tensor) divergence yields theforce density on distributed charged matter:
∂b Telm
ab = Fabρb , (110)
in generalization of the force law on a point charge mentioned above. Thisrather elegant formulation of electromagnetic dynamics will be extended tothe thermo-hydrodynamics of distributed matter.
The goal now is to relativistically generalize the hierarchy (of evolu-tion equations), its truncation, and evaluation of the temporal evolutionof its expectation values for the five basic collision invariants {m, pa}, ashas been done earlier for the NR case in (83). This non-trivial, thoughnowadays routine generalization from Galilean to Lorentzian (or evenEinsteinian) dynamics can e.g. be found in Ehlers (1971); it requires ageneralization from the 6-d Euclidean phase-space description to the 7-dtangent-bundle formulation over 4-d spacetime, for particles of fixed restmass m described by their 3-d hyperboloid in 4-d momentum space. Oncewe know that such a rigorous generalization has been achieved, it is notdifficult to read it off Eqs. (83), (86), (87) for the observables a ={e, pa},respectively, applied to a 1-component system:
ρa,a = 0 , ρa := e n ua , (111)
and:
∂b Tmat
ab = n ϕa = Fab ρb , Tmat
ab := n < pavb > = Tmat
(ab) . (112)
Here the first law describes the conservation of comoving baryon numberB, or rest mass m, or charge e (as formulated), depending on the forefac-tor in the flow vector (ρa) whose space-time divergence must vanish. Thesecond law—derived from the conservation of 4-momentum during colli-sions—equates the 4-divergence of the symmetric stress–energy–momentumtensor T ab, or second moment (on momentum space) of the distributionfunction, to the external (electromagnetic) 4-force density nϕa . For mul-ticomponent systems, each component contributes additively to T ab, and
Fundamental Physics
each component contributes additively its own 4-force density n jϕaj , and
only their sums balance. T ab is conserved, i.e. has vanishing 4-divergence,if and only if the sum
∑j n jϕ
aj of all the external force densities vanishes.
This relativistic description of continuum thermo-hydrodynamics dealswith the evolution of (positive linear combinations of) the symmetric, sec-ond-order expectation value < p(avb) >, from which four basic inequali-ties can be derived. Introducing the pressure p := 1
3 T αα , rest-mass densityµ0 := mn , and energy density µc2 := T44, one infers from the kinetic con-
ditions pa = mva , vava = −c2,→v = cγ
→β :
0 ≤ 3p ≤ 32
p +√(
32
p)2
+ (µ0c2)2 ≤ µ c2 ≤ µ0c2 + 3p . (113)
Here the first inequality, positivity of pressure, follows from <→v · →
v > ≥0; the second one is trivial, the third follows from the triangular inequal-ity
√a2 + b2 ≤ a + b, and the fourth from Schwarz’s inequality applied
to <vαvα >. The third inequality can also be gleaned from: I 21 ≥ I 03,Eq. (102).
It is even easier to see that the kinetic stress–energy–momentum ten-sor T ab defined in Eq. (112) is non-negative in the sense that T abvavb ≥ 0holds for all non-spacelike vectors va , i.e. whenever vava ≤ 0 , and thatits vanishing for a non-zero va can only occur for matter with zero restmass, i.e. for beamed radiation, in which case it takes the form T ab = lalb
with la lightlike, or ‘null’: lala = 0 . In all other cases, T ab can be broughtinto the form:
T ab = µ uaub + pab , with: uaua = −c2 , pabub := 0 < µ , (114)
in which ua is the mean dynamic velocity of the substratum, of (relativis-tic) mass density µ , (symmetric) pressure tensor pab, and (scalar) pressurep = 1
3 paa . For a vanishing pressure tensor, cosmologists talk of ‘dust’.
For a derivation of the relativistically generalized equations of motion(87), the matter tensor T ab will instead be referred to an often slightlydifferent (comoving) system of preferred velocity ua with a non-vanishingenergy–flux vector qa , in the form:
T ab = µ uaub + 2u(aqb) + p ⊥gab + πab, with: qaua := 0 =: πabub =: πaa ,
in which πab is the (symmetric) trace-removed pressure 3-tensor, and ⊥gab
is the 3-space metric projecting perpendicularly to ua . Whereas the equa-tions of type (111) guarantee conserved fluxes of electric charge, rest mass,
Kundt
baryon number, and so forth, including their mutual diffusions for multi-component systems, and are essentially in their final shape, Eqs. (112) cannow be cast into the following, more explicit shapes, by evaluating (112)for a 1-component system for a =α :
µc•
ua +[
ωab + σab + 4Θ
3 ⊥gab
]
qb + ⊥gba [(πc
b +p ⊥gcb),c + •
qb] = n ϕa ,
(115)and for a = 4 :
•µc2 + (µc2 + p) Θ + πabσab + qa
,a + qa •ua = 0 , (116)
which are readily recognized as the relativistic generalizations of (87) and(88). Note that the (relativistic) 3-momentum law (115) takes care of anasymmetric heat flow during local rotation (ω), shear (σ ), and expansion(Θ) as well as of a momentum transfer during changing stresses (π , p),and an evolving heat flow (
•q), and that energy conservation (116) takes
care of the energy transfer processes during expansion (Θ), viscous shear-ing (σ ), diverging heat flow (∂αqα), and heat flow through an accelerateddomain (qα
•uα). A comparison of this last equation with (88) and (90) will
tell the reader that here (again) we deal with a balance of the transfer ofordered kinetic energy into disordered one, i.e. with the (relativistic con-tinuum version of the) law of entropy.
With this we have covered rows 12–14 of Fuphy II, and have hope-fully convinced the reader that the physics of (conserved) N -particle sys-tems is well understood for all regimes, classically and quantally as wellas for non-relativistic and highly relativistic velocities of all constituents,both particles and fields. Missing is still an incorporation of (relativistic)gravity, and a quantitative prediction of the properties of the elementaryparticles, including their interactions.
3. GRAVITATING RELATIVISTIC SYSTEMS
At this point of the present survey, we still lack a consistent treat-ment of gravitational interactions at high velocities of the sources, and athigh mass concentrations. As is well known, here came Einstein’s 1915general theory of relativity to the rescue. It changed the Galilei invari-ance of Newton’s gravitation into the local Lorentz invariance of Max-well’s electrodynamics and of Minkowski’s kinematics, and thus achieveda consistent embedding of the physics in flat 4-d Minkowskian spacetimeinto that of curved 4-d Einsteinian spacetime, cf. Ehlers (1998). Guided
Fundamental Physics
by the (strong) principle of equivalence, the gravitational field equationsof Einstein & Hilbert have generalized our physical understanding of theuniverse from a fair approximation here and now, under solar-system con-ditions, to a convincing description throughout, with a possible insuf-ficiency at extremely high matter densities, as expected near a singularcosmic beginning, or inside of black holes (if such exist). After its sketchypresentation in the following section, we shall come back, in the last sec-tion of this survey, to the yet missing description of elementary particles.
3.1. General Relativistic Dynamics
The fundamental thermo-hydrodynamical equations of motion (115)cover a wide range of astrophysical phenomena, reaching from thelow-temperature physics of quantum fluids through the motion and com-bustion of magnetized and charged gases, fluids, and plasmas all the wayto nuclear detonations and radiation processes in their high- and low-den-sity environments. Yet they leave out the physics of the interiors of com-pact celestial bodies, like white dwarfs and neutron stars, or the centralregions of superclusters of galaxies which glow from radio frequencies toX-ray energies, and for whose motions and stability gravity is no longera minor perturbation force, achieving infall velocities up to 10% or moreof particle rest energies. For a quantitative description of processes nearcompact objects, Einstein’s general theory of relativity is in demand. Notonly has it a convincing mathematical structure, by its built-in coordi-nate covariance, and conformal (light cone) substructure which guaranteeslocal causality, but also its observational tests have confirmed it—com-pared with viable competing theories—not only by required corrections toplanetary motions and their light signals, but even by reaching and exceed-ing the 10−3 accuracy level for the kinematic fine structure of the motionsof a few close neutron-star binaries.
The transition from a special to a general relativistic treatment,i.e. from Minkowskian to normal-hyperbolic Riemannian (or Lorentzian)geometry, requires some additional technical knowledge, but its intuitivemeaning is not far from everyday experience: it generalizes the propertiesof the 2-d plane surface of a quiet ocean to that of its strong agitation,most notable during a tsunami. On small enough spatial scales, flat-spacegeometry is appropriate, but on scales comparable to the deformations,curvature effects become significant. Formally, this transition from flat tocurved (4-d) spacetime can be achieved by introducing a second-rank met-ric tensor gab, and by replacing (ordinary) partial derivatives by covari-ant ones (denoted by semicolons, instead of commata)—which transform
Kundt
tensors again into tensors (of higher rank)—and by replacing their com-mutability by (second-order) correction terms in the form of Riemann’scurvature tensor Rabcd . The components of the Riemann tensor—or ratherof the trace-reversed version Gab of its first contraction, the Ricci tensorRab—should then balance those of the stress–energy–momentum tensor Tab
of (all) the field-generating substratum, and one arrives at Einstein’s cele-brated fundamental equations of general relativity:
Rab − R
2gab =: Gab = 8πG
c4Tab , R := Ra
a , (117)
in which Bianchi’s identities guarantee that the ‘Einstein tensor’ Gab hasvanishing divergence:
Gab;b = 0 = T ab
;b . (118)
Note that all the curvature tensors have dimension length−2; they measuresquares of inverse curvature radii. The factor 8πG/c4 in front of T ab—in which G stands for Newton’s gravitational coupling constant—convertsenergy densities into squared curvatures; it is obtained by assuring New-ton’s law in the limit of weak gravity. By this property, the field equationsimply that in the absence of additional forces, their sources move accord-ing to the generalized law of free fall, along geodesics of spacetime. When-ever additional forces are present, like electromagnetic, their stresses areto be (additively) included into T ab; they then guarantee that the substra-tum moves in agreement with (generally relativistic) continuum thermo-hydrodynamics, in direct generalization of Eqs. (115).
Once we accept this generalization of Minkowski’s 4-d flat spacetimeto Einstein’s (mildly) curved spacetime, as the correct incorporation ofgravity into the fundamental laws of physics, we can easily update all theequations of the preceding chapter by replacing partial derivatives by par-tial covariant derivatives, formally by replacing commata by semicolons. Inother words: incorporation of gravity does not destroy any of the fore-going descriptions; it only requires their smooth modification on lengthscales exceeding those of the distances to the interfering additional (heavy)bodies in the universe, beyond the solar system. At the same time, it intro-duces new observable effects, by predicting the presence of gravitationalwaves (of spin 2) from rapidly changing sources, like stellar explosions orclose binary revolutions, and by the fact that heavy masses act on lightrays like achromatic lenses, yielding additional cues on the dynamics ofdistant celestial objects.
There is also a recipe for how to get the correct stress–energy–momentum tensor T ab in (117): Wanted is the Lagrange function, kinetic
Fundamental Physics
minus potential energy in classical electrodynamics, L = L(gab, µ,Φa; xa):
L = − c4
16πGR + 1
16πFab Fab + µc2 − Φa ρa , (119)
whose variational derivative w.r.t. the metric yields the above field equa-tions, with:
T ab := 2δLδgab
, T ab = Tmat
ab + Telm
ab , (120)
where L is the (Lagrange) density belonging to L : L := √gL , and
T ab := √gT ab the tensor density belonging to T ab. This covers row 15
of Fuphy II .
3.2. The Elementary Particles
This last section of the present survey ventures into unknown scien-tific territory: how to incorporate the elementary particles? What are theirmasses, charges, spins, magnetic moments, and modes of interaction? Whydo elementary particles of the same kind have identical properties suchthat they satisfy Fermi–Dirac statistics, or Bose–Einstein statistics, indepen-dent of their histories? Why are some of them stable, apparently for unlim-ited times, whereas others decay, with measured (half-life) ages? And whatare their decay products? Of course, there is not complete ignorance: quan-tum chromodynamics supplies partial answers to these questions, based onthe semi-empirical ‘standard model’ which describes all the ≥ 36 elemen-tary particles by a ‘gauge-invariant’ Lagrangean, a sum over 24 spinori-al fermion terms, 18 quarks and six leptons (Kane 1987); but a completeprediction of their spectra, from first principles, is not in sight.
Historically, most of the interactions between photons and electronshave been calculated by quantum electrodynamics (QED). This has beenachieved in two ways, whereby the simpler way calculated their collisioncross sections (only) with the methods of Sec. 2.1, i.e. from the relativ-istic version of quantum dynamics, see Penzlin (1963), Bjorken and Drell(1964). In conversation with Freeman Dyson, Paul Adrien Maurice Diraconce expressed his attitude towards the more sophisticated way of QEDby: “I might have thought that the new ideas were correct if they hadnot been so ugly” (Dyson 1992, p. 306); it involves the subtraction ofinfinities, and ignores certain mathematical inconsistencies (Kundt, 1966).Barut (1988) pursued the first approach, and reported even better predic-tions than QED for certain processes. QED has then been generalised toQuantum Field Theory (QFT), throughout the past few decades, with some
Kundt
ordering insight (into a classification of particles and their interactions,including the standard model, and into the behaviour of matter near phasetransitions), but no convincing overall solution of the elementary-parti-cle problem has ever emerged from it, nor a predicted table of numbers(cf. Shipsey 2005; Wilczek 2005). Then came string theory and its evolvingvariants, but none of them has satisfied its early expectations either, as wasrecently summarized by Edward Witten (2005), cf. Ellis (2006). It appearsthat a different approach to this last unsolved problem of microphysics iswanted. Does it have to involve ‘branes’ of all sorts of dimensions, evap-orating 5-d mini black holes, and/or non-commutative, discrete geometry?This survey tries to get away without such non-intuitive generalizations.
At this stage of the discussion, the question may be raised of whetherQFT is really an expected theory, or even a consistent theory. Thereappears to be a distinct “no”: within the framework of QFT, WilliamUnruh has calculated the radiation emitted by a uniformly acceleratedcharge, in interaction with the ‘fluctuations of the vacuum’, whereasthe classical Abraham–Laue–Lorentz–Dirac equation of motion for samecharge yields vanishing radiation, (Kundt 2005, p. 46). If the former resultwas realistic, it would contradict the validity of Fermi statistics: Two elec-trons lying at rest on two tables in different floors of the same sky scraperwould radiate differently, being permanently accelerated by the Earth’sgravitational field of different strength, and would therefore have differentrest masses after some time. Only the classical answer is consistent withthe validity of Fermi statistics.
Should not at least the gravitational field be quantized? In the (obser-vational) absence of exploding mini black holes, there is not a single gravi-tational experiment whose evaluation would require quantization, not evena contrived thought experiment. For instance, all proposed emitters ofgravitational waves do so in the extremely classical limit of large quan-tum numbers. Quantized gravitation would be a theory without testablepredictions.
Physics deals exclusively with the stable states of classical measure-ment devices: Whatever instrument is used for a measurement, its readoffrequires an objective, irreversible state of at least one of its pointers. Quan-tum mechanics can therefore only serve to relate the results of classicalmeasurements, whilst the language of physics must remain classical. Whythen try to describe all of physics in quantum language? Not even JohnBell’s inequalities for certain (entanglement) experiments with elementaryparticles can convey this message, because their crucial assumption is cau-sality, the existence of an arrow of time (de Beauregard 1976; Hasselmann1998). Instead, all fundamental equations of physics are time reversible,both classical and quantal. The atomic electrons do not radiate because
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their accelerations should be described with half-advanced, half-retardedpotentials, as done by Wheeler and Feynman (1945, 1949), for which emis-sion and absorption of photons cancel, and Bell-type experiments cannotbe used to transfer superluminal signals because they work in the quasi-stationary, time-symmetric regime.
Once we watch out for a (parameter-free) description of the elemen-tary particles, a convincing platform should be relativistic microphysics,as surveyed in Chapter 2. Note that the spectrum of the hydrogen atomcan already be approximated by the NR Bohr–Sommerfeld quantizationof the electron’s elliptical motion around the positively charged protonat its centre, and that finestructure and hyperfinestructure corrections havebeen obtained respectively by treating the electron’s motion relativistically(for an infinitely heavy nucleus), and by not ignoring the proton’s magneticmoment. A realistic treatment of the (tiny) nuclear motion has not beenachieved to my knowledge; instead, Willis Lamb calculated a small levelshift via QED methods. Moreover, there is Dirac’s (SR) 4-d spin vector, ingeneralization of Pauli’s (NR) 3-d spin vector, which has stood the test ofmany QED-type calculations. Apparently, we are not all that far from asatisfying description of the H atom and its constituents, within relativis-tic microphysics.
But we want more: a derivation of the properties, and interactions, ofall the elementary particles. For this goal, a linear force field (like Max-well’s) is unlikely to yield a boundary-value problem with discrete solu-tions, as needed to predict the mass spectra, etc. of free particles; gravityought to be included. We have argued above that the gravitational forcesof the elementary particles are tiny, at least on the scale of their de Brog-lie extents (h/mv), but that they grow comparable with other forces forPlanck particles, and that they may play a role in their core domains asthey grow like r−2. A successful approach towards their description maytherefore have to include gravity, i.e. start from Einstein’s general theory ofrelativity (GR): Why then not replace the phenomenological description ofpoint particles in Eqs. (119), (120) by their de Broglie-type wave functions,rather than by delta-function-type phenomenological terms?! Any soliton-type solution of such an ansatz would automatically guarantee their cor-rect dynamics, via ∇bT ab = 0 . It would be a triumph for Einstein’s elegantrelativity theory if it could also be applied to the fundamental particleproblem.
The only promising approach of this kind that I am aware of is KlausHasselmann’s 8-d metron theory (Hasselmann 1997, 1998), whose basic
Kundt
boundary-value equations take the form:
(ηab∂a∂b − ηαβ k pα k p
β ) h pAB = q p
AB , h AB = �∑
p
h pAB(x
C ) ei k pα xα
(121)
for the complex, normalized components h AB(xC ) of the first-order devia-tions of the 8-d bundle metric over 4-d spacetime from their Minkowski-an values, with 1 ≤ a, b ≤ 4 , 5 ≤ α, β ≤ 8 , 1 ≤ A, B,C ≤ 8 , in whichindividual wave components (partons, numbered by p) are electromagneticand generalized de Broglie–Dirac-type guiding fields of the sought-for ele-mentary particles, and q p
AB are quadratic and higher-order functions ofthem which play the role of wave guides, all determined by the equationsof Ricci flatness, RAB = 0. More in detail, the source terms q p
AB arise fromthe nonlinear (8-d, interaction) terms in RAB and replace the phenomeno-logical energy–momentum terms in the 4-d Einstein equations for matter.The waves h p
AB are locally trapped in their wave guides which are them-selves generated by their non-linear interaction; their far fields representthe standard gravitational and electromagnetic fields of pointlike particlesplus an additional Dirac field that acts as a de Broglie-type guiding field,and is the cause of the wave-like interference properties of the particles.
Indeed, once we consider the gravitational fields of the elementaryparticles essential, there is an important result from black-hole struc-ture: their regular far fields, for non-vanishing charge and spin, dictatethe gyromagnetic ratio g = 2 of a Kerr–Newman metric , i.e. magneticmoment/spin = e/m , (instead of g = 1, as for orbital angular momentum),(Heusler 1996; Kundt 2005 p. 109). This is the gyromagnetic ratio of thelowest-mass stable elementary particle, the electron.
Returning to metron theory, Hasselmann follows Theodor Kaluza(1921) and Oskar Klein (1926), cf. Witten (1981), in passing to higherdimensions in order to incorporate the (spinorial) wave amplitudes of theelementary particles into a higher-dimensional metric, which he postulates(again) to be Ricci-flat, RAB = 0, like for Einstein’s vacuum fields. In moremodern language, he considers the 8-d bundle of fundamental interactionsover 4-d spacetime, the metric of whose fibres measures the local pres-ence of elementary particles, both fermions and bosons, forcing them tomove along (8-d) geodesics. (Note that the metric structure of any man-ifold is determined by its projective structure—viz. the set of all its geo-desics—up to a conformal factor, which in the present case is fixed bythe 4-d spacetime metric). This ansatz is generally covariant, includes the
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12-dimensional fundamental group of the standard model, U (1)× SU (2)×SU (3), on the space spanned by the wave numbers as its (gauge) symme-try group, and reproduces Maxwell’s electrodynamics, and Dirac’s relativ-istic spinor equation, together with additional systems of nonlinear equa-tions (121) which result from an incorporation of the wave amplitudes ofthe standard model into the 8-d bundle metric. It describes the dynamicson 4-d spacetime via the kinematics on 8-d bundle space.
This new system of 36 nonlinear parameter-free, second-order tensorequations in eight dimensions describes the local presence, and interac-tions of the fundamental particles in a deterministic manner—via theirguiding de Broglie–Dirac waves—already at first order beyond flatness,with their correct dynamics guaranteed by Einstein’s equations (117) hold-ing on 4-d spacetime. They are not easy to solve: Hasselmann andHasselmann (2005). Their eigen solutions may well answer the elementary-particle problem. To me, this approach looks like the natural continuationof what has been surveyed in Fuphy.
4. SUMMARY
This survey of contemporary fundamental physics has started—inChapter 1—with the dual structure of macro- and microphysics, whosedynamics are described by Lie–Hilbert algebras of {phase-space functions,Hilbert-space operators}, and whose almost isomorphy is quantified bythe W2M map. It has then continued—in Chapter 2—into the relativis-tic regime by switching from Galileian to Lorentzian dynamics, in whichmomentum and energy fuse into 4-momentum, and in Chapter 3 intothe regime of general relativity which faces the dominance of gravity forlarge mass-to-length ratios (in units of c2/G), whose arena is the 7-dphase-space bundle over 4-d spacetime. At all three levels, the governingthermo-magneto-hydrodynamical continuum equations of motion have beenobtained from (canonical) phase-space representations of the concernedN -particle systems by projection (N → r ) down the hierarchy, by theirstochastic truncation, and by their evaluation for the (≥5) collision in-variants. Unexplored (in Section 3.2) has remained a suggestive extensionof Einstein’s theory to microphysics, whereby its phenomenological stress–energy–momentum tensor for the sources is replaced by the guiding wavefields of the elementary particles.
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ACKNOWLEDGEMENTS
My sincere thanks go to all those who have helped me reach thisinsight, both teachers and students, or who have encouraged me to col-lect and expose it, throughout some 50 years of roaming through bothbasic and applied physics. They are many. They have convinced me thatthere is a deep and cogent consistency and harmony in physics, deliveredmost notably to many of us by the work of Albert Einstein, which maysometimes be less evident to the younger masters of the craft. Two ofmy friends have been particularly influential to this survey, throughout thedecades: Klaus Hasselmann and Hajo Leschke; to them go my particularthanks. Additional warm thanks go to Gunter Lay for knowledgeable andpatient help with the electronic data handling.
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