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Acta Physica Austriaca, Supp!. XVIII, 111-151 (1977)© by Springer-Verlag 1977

CLASSICAL LIMIT OF QUANTUM ELECTRODYNAMICS+

by

I. BIALYNICKI-BIRULAInst. of Theoretical Physics

Warsaw, Polandand

Department of PhysicsUniv. of Pittsburgh, Pittsburgh,PA.

CONTENTS

1. Introduction

2. Classical limit of nonrelativistic quantum mechanics2.1. Hydrodynamic formulation2.2. Weyl-Wigner-Moyal transforms

3. Classical limit of relativistic quantum mechanics

4. Classical limit of the quantum theory of the EM field4.1. Classical limit in terms of photon number states4.2. Classical limit in terms of coherent states4.3. Classical limit and the low-frequency limit

+ Lecture given at XVI.lnternationale Universitatswochen fUrKernphysik,Schladming,Austria,February 24-March 5, 1977.

,... wu .._.

112 \~

1. INTRODUCTION

The relationship between quantum and classicalelectrodynamics is a complex subject, with many aspects,not all of which are at present well understood. Clearly,it can not be a simple problem, since we do not even knowwhat is the true content of the two theories which we aresupposed to compare. Quantum electrodynamics exists nowas a perturbation theory, with strong indications thatthe perturbation series is at best only asymptotic.Classical electrodynamics is not in a much better shape,since to my knowledge, there is no systematic approxima-tion scheme that would be guaranteed to give finite re-sults in each step and which would have at the same timethe properties of relativistic invariance, causality, po-sitive definitness of energy and indubitable physical in-terpretation. A very good historical review of the funda-mental problems of classical electrodynamics can be foundin a recent article by Rohrlich [1].

,,

These statements apply to quantum electrodynamicsand classical electrodynamics understood as completetheories of the electromagnetic field and interactingcharged particles. In various truncated versions of electro-dynamics, the relationship between the quantum and theclassical theory is easier to uncover. There are twoextreme cases in which the relationship can be fully un-derstood both physically and mathematically. One is thenonrelativistic mechanics of charged particles moving inan external electromagnetic field and the other is theMaxwell theory of the electromagnetic field interactingwith external sources. I believe that these two greatlysimplified models do help us to understand some aspects I

• II c,

!

!\.

II 1\

113

of the relationship between the fullfledged quantumand classical theories. It seems, however, that thereis much more to that relationship that can be learnedfrom these two simple models and it is quite possiblethat we are still far from understanding the very natureof this relationship.

One may question the need for this understandingarguing that after all it is QED, not its classicalcounterpart, which is the correct theory and that lackof a direct link between the quantum and the classicaldescription only emphasizes this fact. Such a pragmaticapproach is easily refuted, however, by observing thatQED like any other quantum theory, in the words of Bohm[2], "presupposes a classical level and the correctnessof classical concepts describing this level". Acceptingthis proposition, we should intensify rather than abandonthe search for the classical level of QED, especially inview of all the successes of this theory.

It was the fusion of quanta and relativity thatled to the emergence of QED and it is precisely the com-bination of quantum effects (controlled by the Planckconstant n) and relativistic effects (controlled by thevelocity of light c) that makes the classical limit ofQED so difficult to reach. The only available calculationaltool - perturbation theory - means an expansion in powersof the dimensionless fine structure constant a = e2/nc.In this perturbative framework the simple "n ->- 0" ruledoes not work since it implies a ->- 00. Such statements as:"It is well known that if one makes tr tend to zero in anyformula of the quantum theory, one obtains the corres-ponding classical theory formula" [3] or "In the limit,when tr + 0, the laws of quantum mechanics must reduce

114

to those of classical mechanics" [4] are just notapplicable to QED, because it is very unlikely thatQED in its present form can be extended to arbitrarilylarge values of a.

Not all the problems with the classical limit ofQED arise in connection with higher order terms in a.The electron spin is also a source of complications inthe classical limit. Contrary to the widespread opinion,which originated with Pauli [5], the effects of spin donot necessarilly disappear in the classical limit. Thisproblem was first discussed by de Broglie [6] and laterwas further clarified by Rubinov and Keller [7], who haveshown that the spin may affect the trajectory in theclassical limit if the motion of a particle is studiedover large distances (of the order of l/fi). This isprecisely what we are dealing with in all scatteringproblems.

In the absence of exact solutions of the relativ-istic scattering problems we can not prove that the spininfluences the cross section in the. classical limit, butthere are strong indications that this indeed happens.For example, the ratio of the differential cross sectiondo (1/2) for a spin 1/2 particle to that of a spinlessparticle do(O) in the Born approximation is

do (1/2)do (0)

v a 21 - (- sin -)c 2 ( 1)

The decrease in the differential cross section in thebackward direction, predicted by this formula for spin1/2 particles, is clearly seen in experiments on the ..

-'

115

scattering of relativistic electrons on nuclei. However,we can not draw a definite conclusion from here that thespin influences the scattering in the classical limit,because in wave mechanics the regions of validity of theBorn approximation and the classical approximation ingeneral do not overlap.

We will return to these problems after a systematicanalysis of the classical limit in the nonrelativistictheory.

2. CLASSICAL LIMIT OF NONRELATIVISTICQUANTUM MECHANICS

Historically, the first example of the classicallimit of a quantum theory was, of course, the transitionfrom the Planck law for the black body radiation to theclassical Raleigh-Jeans formula. In the words of Planckhimself [8] "for a vanishingly small value of the quantumof action, fi, the general formula

E)..c2n 1-5- ---cn).. exp(k)..T)-l

(2 )

degenerates into Raleigh's formula"

E).. ckTo ( 3)

~This first example exhibits very clearly one importantfeature of the classical limit. Namely, we should becomparing the same physical quantity (the spectral energy

116

density) calculated according to two different theories.It is only in this restricted sense that we will use theterm classical limit, always having in mind a connectionbetween the same concept described by the two theories.

The second example of the classical limit is thecorrespondence principle of Bohr, which played a crucialrole in the discovery of quantum mechanics. According tothis principle [9] "in the region of high quantum numbersthe intensities of spectral lines become asymptoticallyequal to the intensity of radiation computed classicallyfrom the corresponding Fourier amplitude". Again we havethe same situation as before that the same physicalcharacteristic (light intensity) derived from the twotheories are compared.

After this historical introduction let us turn tothe study of the classical limit of quantum mechanics.

Several approaches have been developed to investi-gate this limit, which differ in the choice of the phy-sical characteristics that are to be compared betweenthe quantum and the classical theory. One natural possi-bility is to choose as those characteristics the tra-jectories of particles. In view of the statistical natureof quantum mechanics, we must compare average trajectoriesevaluated in this theory with classical trajectories. Thisproblem was investigated for the first time by Ehrenfest[10] in the early days of quantum mechanics. Startingfrom the Schrodinger equation he derived what is nowcalled the Ehrenfest theorem. We will discuss the Ehren-fest theorem here with the help of the hydrodynamic for-mulation, which by itself represents a convenient toolto explain some aspects of the classical limit.

\'

117

2.1. HydrodynamiC formulation

In the hydrodynamic formulation, which was dis-covered by Madelung [11], we replace the complex wavefunction by the set of four real hydrodynamic-like va-

+riables: the probability density p and the velocity v+ +defined through the probability current, j = pv. For a

charged particle moving in an external EM field we ob-tain the following relation between the real amplitudeR and the phase S/fi of the wave function and the hydro-dynamic variables

p R2 (4 )

+v

-1 e +m (V S - - A)c (5)

+The Schrodinger equation rewritten in terms of p and vreads

d tP -V (p~) (6a)

mdt(pvk)+

ep (E + ~ x B)k - Vi Tik ' (6b)

where the stress tensor Tik is

Tikfi2

- p 4m ViVk £np + mp vivk . (7)

There are more solutions to Eqs. (6) than to the Schrodin-ger equation, since we have replaced two real functionsby four. Only those solutions to Eqs. (6) correspond tosolutions of the original Schrodinger equation whichsatisfy the following quantization condition

\

118

f d"t (mY'E

-* e-*x v + c B) 2rrnn (8)

for every surface E. If this does not hold we can not-*construct ~ from p and v. It is sufficient to impose

-*the condition (8) on initial values of v to have itsatisfied at all times [12] .

The hydrodynamic variables satisfying Eqs. (6),unlike the wave functions obeying the Schrodingerequation, can be directly compared with the corres-ponding statistical classical concepts. It is worthwhilenoticing what becomes of the quantization condition (8)in the classical limit. The correct way of taking thislimit comes with the recognition that Eq. (8) is equi-valent to one of the Bohr-Sommerfeld quantization con-ditions of the old quantum theory. In the limit, whenn -* 0, the r.h.s. of (8) can be made equal to anything,because the separation between the allowed values getssmaller with the decrease of n. In the classical limit

-*there is no restriction on the velocity v.

The Ehrenfest theorem can be obtained from Eqs. (6)by first multiplying Eq. -*(6a) by r and integrating itover the whole space and next integrating Eq. (6b). Theresults can be written in the form

d -*dt<r>-*<v> (9a)

d -*m dt<v>-*-* v -*

e«E + - x B»c (9b)

where the brackets denote the standard quantum-mechanicalaverages (i.e. integrals with p over the whole space).

/ ..\'

119

There is no direct contribution to the Ehrenfest equation(9b) from the stress tensor because only the divergenceof this tensor enters Eq. (6b). This is why the Ehrenfestequations have a purely classical look; the only placewhere the Planck constant appears in the hydrodynamicequation is in the first term of this tensor. We may notehere in passing that the Ehrenfest equations are remarkablystable; they do not change under nonlinear modificationsof the Schrodinger equation [13].

The Ehrenfest equations do not have the same contentas the classical equations of motion. The average position<r> and the average velocity <v> vary with time in a diffe-rent fashion than their classical counterparts because theaverage force is, in general, not equal to the force eva-luated at the average position and average velocity

-*-* v -* -*-*< (E + - x B» 1- E«r»c

-*<v> -*-*+ -- x B«r»c

The above inequality becomes less and less pronounced whenthe probability density shrinks to the Dirac 0 function,but this can not happen uniformly for all times due tothe spreading of wave packets. This spreading is causedby the so called quantum pressure term i.e. the n-depen-dent term in the stress tensor. In the classical limit,when n -* 0, the spreading disappears and the hydrodynamicequations (6) assume the form of the equations of motionfor a statistical ensemble of classical charged particles(charged dust) moving in an external EM field, eachparticle following its classical trajectory.

120 121

2.2. Weyl - Wigner - Moyal transformsoperators W(q,p) are often made simpler with the helpof the following orthogonality and completeness re-lations

One can go one step further in imitating classicalstatistical mechanics and introduce in quantum mechanicsthe counterpart of the distribution function f(t,~,t).This method goes back to Weyl [14] and Wigner [15] andwas fully developed by Moyal [16].

Tr{W(q,p)Wt (q',p')} 2TIn8 (q-q')8 (p_p') (l2a)

The classical distribution function f(t,~,t) gene-+ralizes the probability density p (r,t). It describes

the statistical distribution of both the position andthe velocity. In quantum mechanics the uncertaintyprinciple precludes the possibility of having authenticdistribution function of t and ~, but the Weyl- Wigner -Moyal transform (WWM transform) is as close an approxi-mation to the distribution function as we can get underthe circumstances.

fdr (q,p) (1/I11wt(q,P)1/I2)(1/I3Iw(q,P)1/I4)=(1/I111/l4)(1/1311/12).(12b)

In proving these relations one repetitiously uses theBaker-Hausdorff formula

W(q ,p)

i i infiqP - irPQ ~. pe e""l\l- (13)

In the simplest case of a system with only onedegree of freedom, the WWM transform 0 (q,p) of a ope-rator (j is defined by the formula

or its equivalents and the completeness relations forthe (generalized) eigenfunctions of q and p, for whichwe adopt the following normalization conventions

o (q',p') fdr (q,p) Tr{W(q,p) (j }e-~(qpl-pql)fdqlq><ql f~ Ip><pl1 = 2TIfi

!qprre

(14a)

(10)<qlp> (14b)

where

W(q,p)irr (qP-pQ)

e (11) With the help of Eq. (12b) we can invert the formula (10)and obtain a representation for 0 in terms of its WWMtransform

dr(q,p) = (2TIl'l:)-1dq dp,and Q and P are the position andmomentum operators. The definition (10) may be appliedto dynamical operators and to the density operators.

i-fi (qp I -pq I )

(j =f ar (q,p) f ar (q I , p')W (q,p)e 0 (q I ,p') . (15)

Calculations involving the family of unitaryThe WWM transforms of dynamical operators (also known as

122

Weyl representations of these operators) coincide withthe expressions for the corresponding classical dynami-cal variables, provided we always fully symmetrize pro-ducts of Q and P when we construct quantum operators.

The WWM transforms of density matrices (known asWigner functions) are the counterparts of classicaldistribution functions. In order to justify this claimwe will express in terms of WWM transforms the averagevalue <A> of the dynamical operator A evaluated in thestate described by the density operator P,

<A> Tr {Ap} j dr (q,p)A(q,p)p (q,p) (16)

This formula is obtained from (15) and (12a). It hasthe same form as in the classical theory. Despite thegeneral validity of (16), the function p (q,p) is nota genuine distribution function: even though it is realand normalized it is nonnegative only for a very re-stricted class of states.

To study the motion of charged particles in anEM field, we will introduce a gauge invariant versionof the WWM transform [17] which employs the kineticmomentum K rather than the canonical momentum P,

K = mV = P - ~ A .c(17)

In this case

i (-+- -'"k' "'"k -+-')-- r. -·r-+- "'" J -+- "'" -+- -+- np(r',k')= dr(r,k)Tr{W(r,k)p}e (18)

"

123

Xi~'l,and

i -+- -+- -+- -+-ii(r·K-k.R)e

-+- -+-W(r,k) (19)

In particular, for a pure state we obtain

+ + 1 7+ + J 3 + ~ * + ~ i7 7 e 7" + X +p(r,k)= d ~1jJ(r+i)1jJ(r-i)exp{-fix·[K+cJdctA(r+i-ctR.)]}.(20)

o

In the derivation of this formula the following generali-zation of the Baker-Hausdorff formula (13) was used

i+ + i + "'"fir.p ihr.k . 1W(~,k)=e e exp{-R[k'R+~ J dctr'A(R-ct~)]} .

o(21)

The line integral in Eq. (20) guarantees the invarianceof p (~,k) under the simultaneous gauge transformationsof the wave function and of the vector potential.

The classical limit can not be directly taken onthe formula (20), because of the presence of the wavefunctions in it. However, as we shall show now, theequation of motion for p (~,k,t) does have a simple limit,when n -+- O. To this end we calculate the time derivativeof p (~,k,t), assuming that the time evolution of thedensity operator is determined by the von Neumann-Liouville equation,

-1(il'i) Tr{ (WH - HW)p (t)} .Cl t Tr{Wp (t)} (22)

For simplicity, we have assumed that the EM field is

124

time independent. Otherwise we would have to include anadditional term in Eq. (22): the time derivative of W.

The calculation of the r.h.s. of (22) is madesimple with the help of the following identities:

fi 1 + +(- I V'k + '2 r) W = WR

n 1 + +(- I V'k - '2 r) W = RW

11 1+ e f + + n 1+ + +[-V' + -k + - daarxB(-~V' + -r - ar)]W = WKi r 2 c ~ k 20

(23a)

(23b)

(24a)

[ ~V'~ r

1+ e 1 + + n 1+ +-k - - J da(l-a)rxB(- 7V' + -r-ar)]W2 c ~ k 2o

+KW , (24b)

which can be derived with the help of (21). We also noticethat under the integral in formula (18) we can make thereplacements

11 + (25a)- I Ilk+ r' ,

1;! 11 ++ k' , (25b)~ r

+ 1'1: (26a)r + - I V'k' ,

+ 11 (26b)k + 7 V'r' .~

From Eqs. (18) and (22-26) we obtain in the limit, whenn + 0 ,

'l'Since the Hamiltonian is time independent and the Poissonbracket is invariant under canonical transformation, inparticular under the time translation, we can write

125

++ -±+ v-±+ ++[at + V'V' + e(~(r)+-xe(r»'V' ]p (r,v,t)rev o , (27)

where we have used the velocity instead of kinetic mo-mentum k. All quantum corrections can be obtained byexpanding relations (23-24) in which the substitutions(25-26) are made, into powers of n. These correctionsare small for slowly varying fields and they disappearcompletely for homogeneous fields.

Eq. (22) determines the time evolution of thestates of charged particles. Similar equations can bewritten for all dynamical variables:

-(i11)-l Tr {(WH - HW)A(t)} •at Tr {WAtt)} (28)

Since this differs from Eq. (22) only in the sign ofthe time derivative, we can immediately write downfrom (27) the classical limit of Eq. (28),

+ + + + +[v'V'r + F(r) 'V'v]ACL(r,v,t)a A + +t CL(r,v,t) (29)

+ +where F(r) is the Lorentz force. The differential ope-rator in the square bracket is the Liouville operatori.e. the Poisson bracket with the Hamiltonian:

{ACL' H} .at ACL (30)

126

Eq. (29) in the form

~ + + + + + +a tACL (r,v ,t)=[v (t) ."r (t)+F (r(t))."v (t)]ACL (r,v,t), (31)

provided r(t) and t(t) obey the equations of classicaldynamics. The solution of this equation, therefore, canbe written in the form

A (->-->-CL r,v,t)->- ->-

ACL(r(t), v(t),t 0) ( 32)

In order to obtain the quantum analogy of the particle->- ->-

trajectory we must specify A(r,v,t) assuming it to be->-the WWM transform of the position operator R(t),

i ->- ->- ->-->-->- ->- ->- ->- ->- ->- ->- -fi(mr·v'-k.r')

R(r',v',t)=Jdr(r,k)Tr{W(r,k)R(t)}e . (33)

According to Eq. (31), this function together with itstime derivative (velocity) obeys in the classical limitthe classical equations of motion.

One can generalize the description in terms of WWMtransforms to the case of many particles moving in anarbitrary (also time dependent) external field or eveninteracting mutually, provided the interaction is des-cribed by a potential term in the Hamiltonian. Thus, inthe nonrelativistic quantum mechanics we have a completeunderstanding of the classical limit. Quantum mechanicalcounterparts of classical dynamical variables exist andthey tend to their classical limits, when n ->- o.

'i

,...

127

Difficulties are encountered, when we try togeneralize this approach to a relativistic theory andto include field quantization.

3. CLASSICAL LIMIT OF RELATIVISTIC QUANTUM MECHANICS

Relativistic quantum mechanics of a single particlemoving in an external field is not a selfconsistent theory.Creation and annihilation of particles caused by externalfields force us to abandon the quantum mechanical des-cription, whose main characteristic is the fixed numberof particles. To deal with this new situation we mustintroduce a collection of wave functions, each describingone component of the state vector and this is usuallydone in the field-theoretic framework. It is only in thespecial case of static fields that we may consider oneparticle at a time, since in most static fields creationand annihilation of particles do not take place. At firstone may think that these processes can never occur instatic fields because of the energy conservation. How-ever, as was shown by Schwinger [18], even in a verysimple case of a uniform electric field one finds con-tinuous pair creation from the vacuum.

We will begin our discussion with the simpler caseof spinless particles. Assuming that the creation andannihilation of particles do not occur we can consideronly those solutions of the Klein-Gordon equation

e ~ 2 2-A ) - m c ]~(x)c[(ina - §:A ) (Hia u

~ c ~o , ( 34)

128

which describe particles. The time dependence of such1· t . 1 .t . f i (- iwtso utlons con alns on y POSl lve requenc es e ,

w > 0). It turns out that we can easily generalize themethod of WWM transforms to the relativistic situationby taking the "square root" of the Klein-Gordon equation

il'idtlP(x) rz:» h+ [c/m c + (-:-'V- 1

(35)~A) 2 + e<jlH (x)c

and hence by considering only one particle or one anti-particle at a time. Following the same procedure as thatdescribed previously for static fields we can obtain inthe classical limit the relativistic analog of Eq. (27) .What is lacking in the relativistic case even for thestatic fields as compared to the nonrelativistic caseis the clear physical interpretation of basic dynamicalvariables. As was shown by Newton and Wigner [19], theposition operator must be modified in the relativistictheory and itceases to be identical with the argument ~of the external field. In other words, the center of masscan not be identified with the center of charge. That iswhy the WWM does not have an unambiguous physical inter-pretation, even though it can be formally applied to thetheory based on Eq. (35). We can shed some light on theseproblems by considering again the hydrodynamic formulationand deriving from it the relativistic analog of the Ehren-fest theorem.

The hydrodynamic formulation of the relativistictheory based on the Klein-Gordon equation is obtainedin the same manner as in the nonrelativistic case,except that we introduce the velocity in the form of

\ )

129

1.~

~t"t

ca four-vector v~ and scale the density p by a factorof 2. To obtain the hydrodynamic formulation there isno need to exclude time dependent EM fields.

c The equations of motion for the hydrodynamic va-riables read

d (o v~)u

o , (36a)

d T~vu e fV"C;p vA (36b)

where

p = 2R2 , (37a)

-1Cd S+~A)v = - mu u c u

and

2~V n; a~ (lv ~np u vT = - p 4m + mp v v

(37b)

(37c)

Eqs. (36) can also be written as the equations for thepropagation Of P and v~ along the stream lines,

v~a pu

-p a v~u

(38a)

uv -,vA

p tr2fA v +- d (p-l/2C] pl/2)p 2m2 A 4

emc (38b)

~ The quantization condition for the velocity takes on nowalso a relativistic form [20]:

l

130 131

f ~v edo [m(8 v - a v) + -f ] = 2rrttnE ~ v v ~ c ~v .

.}

and apply to this equation the methods developed forthe Klein-Gordon equation. In particular, we may de-compose into the phase factor exp(iS/n) and the re-maining spinor field u and obtain in the limit, whenn + 0, the classical equations:

(39)

In the classical limit the equations of motion (36) or(38) become identical with those for a charged dustmoving in an external field and the quantization con-condition disappears.

a (u u v~) = 0 , (43a)From Eqs. (36) we can obtain by integration the u

relativistic analog of the Ehrenfest theorem:v~a vA = ~ f vP . (43b)

u mc APf 3 + 0at d r r P v Ic f d3r +Pv (40a)

f 3 + 0at m d r PV v Ic f 3 + 1+ +d r ep (E + -vXB)c (40b)

However, since the classical limit resulting from thedecomposition

1jJ= e(i/n)S u (44)As in the nonrelativistic case, the Planck constant doesnot appear explicitely in these equations. However, theappearance of factors vOlc on the l.h.s. is a clear in-dication of the fact that the center of charge does notcoincide with the center of mass.

is essentially the same as obtained by Pauli, the ob-jections raised by de Broglie [6,7] are valid here.

[Y~(ina - ~A ) + mc]1jJu c ~ o (41)

The problem is basically that of uniqueness of theclassical limit. We may obtain different limits, makingdifferent assumptions about the asymptotic behaviour,when IT + 0, of various physical variables. For example,if we hold the size of the spatial domain, to which theparticle is confined, fixed while taking the limit, thenwe obtain the result of Pauli. If this size increases asl/n, then small corrections to particle trajectories ofthe order of fi, which are due to the effects of themagnetic moment, get multiplied by l/tt and contributein the classical limit.

Things become more complicated for the Diracequation. The difference between the center of chargeand the center of mass is even more pronounced in thiscase owing to the presence of Zitterbewegung.

It is true that we can rewritte the Dirac equation

in the "squared" form

[. e . u e u 2 2 en u v(~na --A ) (~na --A ) -m c - -0 f U~ c ~ c 2c ~v'1 •

I believe that the pair creation processes mayalso contribute in the classical limit if we only knewhow to calculate contributions from pair creation summed

o (42)

132

over all orders of perturbation theory with respect tothe external field.

In the absence of the exact solutions of theKlein-Gordon and Dirac equations that would exhibitpair creation in time-varying external field, we mustleave this problem unresolved.

4. CLASSICAL LIMIT OF THE QUANTUM THEORY OF THE EM FIELD

The principal difference between the classicallimit of the quantum theory of the EM field and of thequantum mechanics of particles was explained very clearlyby Sakurai and we will begin by quoting from his book [21].

"The classical limit of the quantum theory of ra-diation is achieved when the number of photons becomesso large that the occupation number may as well be re-garded as a continuous variable. The space-time deve-lopment of the classical electromagnetic wave approxi-mates the dynamical behavior of trillions of photons.In contrast, the classical limit of Schrodinger's wavemechanics is the mechanics of a single mass point obeyingNewton's equation of motion. Thus it was no coincidencethat in the very beginning only the wave nature of lightand the particle nature of the electron were appearent".

This description of the classical limit for the EMfield is also indirectly contained in Bohr's corres-pondence principle. After all, photons are the excitationsof the EM field and hence the number of photons can beviewed as a quantum number, characterizing the state ofa certain oscillator of the EM field. Classical limit '( .

1I

133

(1

means large quantum numbers which in turn imply highintensity of the field. ,

i''\!,.f

t

We may give a more formal argument confirmingthis view by noticing that Ih multiplies all creationand annihilation operators in the expansion of thevector potential of the EM field into a Fourier series, I

t;~;

' •. -+A(r) 2 -+-+

-+ -+ -+ ik·r£ (k,A)a(k,A)e + h.c.2vw (45)-+L

k ,A

In the classic~l limit the average values of the EM fieldmust approach their classical values and this can happenonly if matrix elements of annihilation and creationoperators go to infinity as l/In so that the averagephoton numbers go to infinity as l/n.

To describe the transition to the classical limitfor the EM field systematically let us consider an ar~bitrary system of charged particles, described by non-relativistic quantum mechanics, interacting with thequantized EM field.

The Hamiltonian operator of the system can besplit into three parts

H (46)HF + HA + HI '

where

HF ~ Jd3r

-+1 oA 2 -+ 2(~(at) + (VXA) )c

. , (47a)

134

L 1 +2 1 Leiej

HA = . 2m. Pi + 2 + + Ia, ~ i;;tjI ri-rj

2e. ei +2 +L ~ + + +H = - - p. ·A(r.) + L --z-2 : A (r.) : ,I . m. c ~ ~ i 2m.c ~~ ~ ~

(47b)

(47c)

where the colons denote normal ordering of creation andannihilation operators.

We will study the time evolution of this systemin the interaction picture, in which the time evolutiongoverned by the Hamiltonian HO = Hp + HA is separatedout

i .-RHt -:!:Hte - e tt 0- V(t)

where. t

V (t)_ T -fi J dT H (T)_ exp 0 I

andi .-H t ~

HI(T)_ n 0 --H t-e H e110I

(48 )

(49)

(50)

The EM field evolves in the interaction picture accordingto the free Maxwell equations:

~++

-+ -+ + -+ + -iwt+ik'rA(r,t) = L 2 E(k,A)a(k,A)e + h.c.k, A vw(51)

,135

i'"

S~~~~.4.1. Classical limit in terms of photon number states ,fi

,_It I

tr- We will first calculate matrix elements of the

evolution operator V(t) between photon number statescontaining a large number of photons. We will followthe method described in our recent papers [22]. Thecalculation will first be performed for a single modeof qlectromagnetic radiation i.e. only one term inthe sum (51) survives the classical limit. Generalizationto many modes is straightforward.

Let us represent photon number states In+m> byharmonic wave functions of the phase ¢

In + m> + eim¢ , ( 52)

where n is a fixed reference point and m varies from -nto 00. An arbitrary state vector can be represented as afunction of the phase ¢

1jJ L ck Ik> + 1jJ (¢ )k

Lm=-n

c eim¢n+m (53)

and the scalar product takes on the form

(<I> I '1') J21l d¢ q,* (¢ ) 'I' (¢)21l

o(54)

Annihilation and creation operators act on these wavefunctions in the following way,

a + e-i¢ (n + ! 1-)1/2i 3¢

( 55a)

o

136

at -+ (n + * _8_)1/21 8~

ei~ . (55b)

We also obtain

1 8ata -+ n + I ~ . (56)

If the reference pOint is large and if we restrict our-selves to values of m much smaller than n, we can appro-ximate formulas (55) by expanding them into powers ofl/In and keeping only two lowest terms

-i~ _1__ 8 )a -+e In (1 + 2ni 8~ (57a)

1 _8)at -+ In (1 + 2ni 8~ ei~ (57b)

In the classical limit, when ~ -+0 and n -+ 00 in such away that nn is kept constant we may keep only the firstterm in the expressions (57). Since the representation(52) and (57) can be used for every mode, we obtain thefollowing formula for the vector potential operator inthe classical limit

)-e- 2 -+-+ -+A (~t.~)= \ nn(k,A)c -+(k A) -i(wt-k'r+~ (k,A)) hCL "'t' -+L 2v £, e + •c. ,k,A w

(58)

where ~ on the l.h.s. stands for the collection of allphases. Owing to its dependence on the phases, ACL isstill an operator, even though it has the same appearanceas the corresponding classical object. Upon introducingthis operator into the formula for V(t) we obtain

I~J

1137

.'<:,{~

Il

VCL(t~~)i t CLT exp {-R ~ dT HI (T)} (59)

where

eiHCL(t)=-l: mcIii

2-+ -+ -+ ei 72 -+Pi (t)'ACL(ri,t~~)+~~ACL(ri,t~~).

12m.c1

t~t(60)

Formulas (59) and (60) for the evolution operator are thesame as in the case of a classical external field, butowing to the operatorial character of ACL' we can stillobtain from them the transition amplitudes between photonnumber states. Assuming again that only one mode of theEM field is important, we obtain in the classical limit

< n+m I V ( t) I n> 2Trd~ -im~V (t~~)J _'t' e CL2Tr (61)o

or inversly

VCL(t~~) l:eim~<n+mlv(t) In> .m

(62)

Thus, by Fourier analysing the evolution opeLator in anexternal field with respect to the phase of the field weobtain photon transition amplitudes in the limit when n-+oo•

\'

Having obtained the classical limit of the transitionamplitudes, we may proceed to calculate the time evolutionof the density operator in this limit, assuming that in-itially the photon field was in the n photon state (onemode only). In the interaction picture we obtain the formula

138

P (t) v (t) i n> P A< n i Vt (t) (63)

where PA is the initial density operator of the atomicsystem. The atomic density matrix PACt) at time t isequal to the trace of P (t) with respect to the photonstates

P A (t) L<n+miV(t)in>PA<nivt (t)in+m>m

(64)

substituting (61) for the matrix elements of V and vtand using the identity

L eim(.p-.p') = 21T <5 (.p-.p')m

we arrive at

21T d tPA (t) = f ~ VCL(t;.p)PA VCL(t;.p)

0

(65)

(66)

Therefore, the density operator of the atomic system inthe limit of large n is related to the density operatorin the presence of an external EM field through the phaseaveraging operation. Such averaging is carried out inquantum optics whenever one uses the so-called semi-classical approach in the description of the EM field,but it is introduced then as an ad hoc time averagingprocedure (for harmonic oscillations time and phaseaveraging are equivalent).

All the results of this section were obtained

~.

"'"

139

under the assumption that the number m of emitted orabsorbed photons is much smaller than the number ofphotons in the beam n, so that the reference pOintpractically did not change during the interaction. Inother words, the depletion of the photon beam by theatomic system is negligible. This is true only when weare dealing with not too many atomic systems at a timewhich in addition do not possess the ability to emitcopiously photons. This is the case for bound systemswith discrete levels, but scattered particles may ra-diate many photons. Such difference in behavior is dueto the presence of the factor 1/1; in the expansion (45)of the vector potential. For bound-bound transitions,w can not be very small due to energy conservation,butfor the transition between scattering states w can besmall and the effective coupling constant between theatomic system and the EM field increases making theemission of photons more likely.

If the number of photons emitted (or absorbed) bythe atomic system is large, it is natural to assume thata single act of emission (or absorption) does not in-fluence significantly the state of the emitters. Thisassumption leads us directly to the concept of coherentstates of radiation.

4.2. Classical limit in terms of coherent states

Coherent states appear in a natural manner in QEDwhen we assume that the quantized EM field is coupledto a classical, external current. These states madetheir appearance in QED for the first time in connectionwith infrared divergencies. In their fundamental paper 4

140

on this subject, Bloch and Nordsieck [23] made thecrucial assumption that in the description of electroninteractions with low energy photons" in first approxi-mation the motion of electron is treated as given". Thisassumption leads to the appearance of coherent states ofthe EM field. The term coherent states for eigenstatesof annihilation operators was introduced quarter of acentury after the work of Bloch and Nordsieck by Glauber[24], who recognized the importance of these states forquantum optics.

The description of the sources of radiation interms of a given external current is justified whenthese sources are so strong that they are practicallynot affected by the emission of photons. This is pre-cisely what happens in the analysis of infrared diver-gencies, as can be seen from formula (51). As we ob-served before, the effective coupling constant increaseswith the decrease of w, so that the current coupled tofield oscillators with low frequencies is strong andthis is the physical reason why the emission of soft(low frequency) photons can be described in terms ofan external current.

There is more to the similarity between the problemsof the classical limit and the infrared divergencies than.only the appearance of classical currents in the limit,when w + O. One may indeed give a physical argument thatwhen w + 0 we must obtain exactly the classical limit.For low frequencies i.e. long wavelengths only themacroscopic features of the radiating sources are rele-vant and they can always be described by classicalphysics. Moreover for low frequencies the number of

141

11~I

IJ

photons must be very large if the energy stored in thefield is to be finite.

The usefulness of coherent states in the analysisof various aspects of the classical limit has been shownin a large number of publications. In a broader contextof continuous representations, coherent states were usedto investigate the relation between Quantum and ClassicalDynamics by Klauder [25], who later formulated the weakcorrespondence principle [26]. According to this principlethe expectation values of the dynamical variables in co-herent states have the same form as in the classical theory.Equations of motion were studied with the use of coherentstates by the present author [27]. It was shown that inthe limit, when n + 0, the expectation values of quantumoperators satisfy the classical equations of motion. Re-cently, Hepp [28] proved this result with full mathe-matical rigor using unitary operators W(q,p) (cf. formula(11)) rather than (unbounded) position and momentum ope-rators. All these works were incomplete in one respect:there was no discussion of the classical limit of thetheory of particles interacting with a field. That iswhy the problem of the classical limit is still unresolved.

I'

We will not discuss the classical limit of quantummechanics in terms of coherent states referring readersto Refs. [25-28],because when used in quantum mechanicsthese states are defective in one important respect. Withthe only exception of the harmonic oscillator, the choiceof coherent states in quantum mechanics is to a largeextent arbitrary [25,27]. It is only in the theory offields, where coherent states appear in a very naturalway.

142

There is a vast textbook literature, where onecan find information about the properties of coherentstates (cf. for example [29,30,31]). We hope that thereader has at least a rudimentary knowledge of thissubject.

Let us begin with a brief discussion of the in-teraction of a single mode of radiation with an externalcurrent. The interaction Hamiltonian and the evolutionoperator for this system in the interaction picture canbe written in the form (cf. Refs. [29,30] or [32])

HI (t)* -iwt t iwta (t) a e + a (t) a e ( 67)

V(t) ei1jJ. ( t *)e~ na +n a e-lnI2/2 inate ( 68)

. *an aeHe

where

1 t .1'1 =-nJdTa(T)e~wT

o(69)

and e11jJis a phase factor which is unobservable. Actingon the vacuum, the evolution operator produces a coherentstate

v (t) 10> ei1jJe-lnI2/2 einatlo> (70)eH lin> .

A coherent state is also obtained when V(t) acts onanother coherent state. Therefore, in the world ofclassical currents it suffices to consider only co-herent states.

-..

143

-,'1

Disregarding the phase factor in the formula (70),we can write the coherent state in the canonical form

1 z> t *eza - z alo> (71)

There are various heuristic arguments which indicatethat coherent states of the radiation field are thequantum analogs of pure classical states (i.e. theclassical states which are characterized by givenvalues of field variables and not by statisticaldistributions). For a single radiation mode, theclassical field is characterized by the intensity andthe phase. It is believed that the coherent state Iz>is the analog of the classical state, whose intensityis 1 zl2 and the phase is the phase of z. This beliefis based on the following facts. First, the averagevalue of the field operator a + at in the coherentstate 1 z> is equal to z + z*. Second, the product ofdispersions of the hermitian and antihermitian partsof the operator a attains its minimal value of coherentstates (gaussian functions saturate the uncertainty re-lation). Third, the values of z(t) for a mode driven byan external current a(t) are identical in the classicalcase and in the quantum case,

z (t)1 tin J dT a (T) eiwT

o(72)

,j

The analogy between coherent states and pure classicalstates can be extended to statistical mixtures of suchstates. j,

Statistical mixtures of coherent states formed

144

according to the rules of quantum mechanics (the den-sity matrix of a mixed state is equal to the super-position of the projectors on the pure states con-tained in the mixture)

fd2Z

P = -TI-- 1 z> p(z)<zl (73)

should correspond to classical statistical mixtures ofpure classical states. The representation (73) of den-sity matrices is known as the P-representation. Theintegration in the formula (73) is extended over thewhole complex plane of the variable z.

The role of the P-representation in the theory ofthe EM field is very similar to the role of WWM trans-forms in quantum mechanics and these two concepts, as amatter of fact, are mathematically closely related.Both concepts are used to formulate the quantum theoryin terms of statistical distributions which are in turnamenable to classical interpretation.

The possibility to interpret P(z) as a classicaldistribution function was stressed by Sudarshan in thecontext of the so-called optical equivalence theorem(cf. Ref. 29 p. 192).

We are now ready to state the following classicalcorrespondence property valid for systems driven by ex-ternal currents (for simplicity, we give it for one mode):

The time evolution of the density matrix, when ex-pressed as the change in time of the weight functionp(z), coincides with the change of the distribution

I! (J

1)i IJ

145

function determined by the classical theory.

"

This statement follows from formula (68) andt *from the fact that the operator exp (zoa + zoa)

displaces the variable z of the coherent state by zo0In the interaction picture we obtain

PI (t)d2 tV(t) J-zi z>p(z)<zlv (t)

TI

2 2=Jd/I z+in (t»P(z)<z+in (t)I=JdTI zlz>P(z-in (t))<zl. (74)

This formula translated to the Schrodinger picture reads

P (t)

i--H te n 0

i-H tPI (t) en 0

. t-~wa ae2

Jd zIZ>P(z-in(t»<zlTI

d2 . . tJ z 1 ~wt. ~w 1--- z>P(ze - ~n(t)e )<zTI

(75)

The change in time described by the formula

P t (z) p(zeiwt _ in(t)eiwt) ( 76)

is the same as the change of the distribution functionpredicted by the classical theory. For example, we wouldhave obtained the same result, equating the functionP(z) with P (q,p,t) and solving Eq. (27) for the harmonicoscillator. All the results obtained in this subsection

,

146

can be generalized to include many modes of the EM fields.In most instances this generalization will just involveinserting sums over all the modes of the radiation fieldand it will not lead to any clarification of the one veryimportant problem. Under what precise conditions we canuse external, classical currents to describe the sourcesand whether this can be done in the classical limit. Fromthe existing literature on the infrared problems startingwith the paper by Bloch and Nordsieck [23] we learn thatthe use of classical currents is justified for very softphotons. Hence, if the classical limit is understood notonly as limiting case of large photon numbers but also asa transition to very low frequencies and long wavelenghts,we can then obtain the equivalence of the quantum and theclassical descriptions.

4.3. Classical limit and the low-frequency limit

The low-frequency limits of QED transition ampli-tudes have been extensively studied after it was shownby Low [33] that such soft-photon amplitudes are uniquelydetermined by the requirements of Lorentz invariance,current conservation and some simple analytic properties.With the help of these low-frequency or low-energy theorems,one can establish also the connection between classical andquantum results in several simple cases. For example, fromthe Klein-Nishina formula describing the Compton scattering

9.Ii.dn

2ro'2

nUl 2--(l-cose)2 21 + cos e {l+mc }

nUl 2 2 nUl .] ,[H (l-cose) --2] (Hcose) [1+ (l-cose)--2mc mc

(77)

I. U

• 1

II II

147

one obtains the classica Thompson formula

2dO' rd~ = 20 (1 + cos2e) (78)

by either taking the limit n ~ ° or the limit Ul ~ 0.This equivalence of the two limits is simply a resultof the energy-momentum conservation at each vertex:p + bk = q . If the electron four-momenta are kept~ ~ ~fixed when we take the limit b ~ 0, then even virtualelectrons must have the same momenta as the initialelectron, as if the photons had the frequencies and thewave vectors equal to zero. What seems to be difficultto achieve is the simultaneous transition with n to zeroand with the intensity of the photon beam to infinity,keeping their product finite. Such a simultaneous transi-tion is essential if we are to account for finite amountsof energy being transfered in electromagnetic interactions.In this respect some results obtained by Brown and Goble[34] are encouraging. They have been able to perform an in-finite summation over soft-photon contributions and evento include, to some extent, a change in the electron four-momentum due to the interaction with the soft photons.However, the main part of the change in four-momentum wasattributed to some unspecified nonelectromagnetic inter-actions and it was not shown that the change attributedto photons is introduced in a self-consistent way.

i:i','

/IlW:ii

Complete calculation still remains an unfulfilledprogram. Even in a very simple case of the Coulombscattering of an electron, the classical formula forthe bremsstrahlung spectrum [35], ~

~4lt

148

I (w)222 2

l~ ~ (~)2 (~)2 ~n (Amv )3 c 2 v Z 2mc e w

(79)

has not been obtained from full QED in the classicallimit.

Our review of the classical limit of QED would notbe complete without mentioning recent papers by Monizand Sharp [36]. They have addressed themselves to theproblems of runaway solutions and preaccelerations whichemerge in classical electrodynamics, but seem not toappear in QED. First they showed classically that pre-acceleration and runaways disappear when one assumes anextended electron whose radius L is larger than theclassical electron radius [37]. Next they summed up acertain class of terms in the nonrelativistic approxi-mation to quantum theory, showing that the Compton wavelength plays the role of the radius L. Thus, in theirapproach the electrostatic self-energy of the electronremains finite even in the limit of the pOint electron.

However, as was shown by Weisskopf [38], the electro-static term is just one part of the self-energy the re-maining two being the part due to the magnetic moment andthe part due to the interaction with zero-point fluctua-tions of the EM field. It is a very important questionwhether the results of Moniz and Sharp also hold in therelativistic theory, but even their nonrelativistic appro-ximation strongly indicates that the inconsistencies ofthe classical theory will disappear if we give the electronits quantum extension n/mc. Complete answer to this ques-tion must also contain an explanation how is the vilationot cau.ality avoided for classical extended electrons.

fj

If

It u

...

149

I wish to end here my review of various topicsconnected with the classical limit of QED, hoping tohave convinced you that the first sentence of the In-troduction is indeed true.

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150

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>.\ I (4'~

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!\ I ~

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'\1

151 1..,'!!..,....•

t"I.

'\;~

.~il:'JI

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