P H Y S I C A L R E V I E W V O L U M E 1 3 9 , N U M B E R 2B 26 J U L Y 1 9 6 5

Gauge Independence and Path Independence* F . ROHRLICH

Syracuse University, Syracuse, New York

F . STROCCHlf

Istituto di Fisica delV Universitd, Pisa, Italy (Received 21 January 1965)

Quantum electrodynamics is formulated in a gauge-independent as well as path-independent way. The theory is manifestly Lorentz covariant. It can be obtained from the path-dependent electrodynamics of Mandelstam by an averaging process. The commutation relations and propagators in this gauge-independent formulation correspond to those of the Landau gauge in the conventional theory, giving this gauge a funda­mental significance. The method is also applied to the gauge-invariant theory of neutral vector mesons with nonvanishing mass interacting with a conserved current. The resulting theory is gauge-independent and describes a pure spin-1 particle, the spin-0 contributions being identically zero.

I. INTRODUCTION

RECENTLY, considerable attention has been paid to a formulation of quantum electrodynamics in

terms of fields rather than potentials.1"4 That makes it possible to work without any reference to particular gauges and thus avoids all the complications and un-physical features associated with any of them. According to this philosophy the fundamental quantities are the fields and the quantization is carried out directly on them by specifying their commutation relations. The potentials appear as derived, strictly auxiliary quanti­ties and their introduction into the theory is merely a mathematical device for simplifying the field equations, just as in the classical theory.

However, the formulation by Rohrlich3 is not mani­festly covariant. I t corresponds to the Coulomb gauge, in which surface-dependent terms occur and only the scattering matrix is invariant. The intermediate equa­tions involve the normal vector to a space-like plane. On the other hand, the DeWitt-Mandelstam formula­tion is completely covariant, but involves a "path de­pendence" of certain quantities.

In the present paper we propose a formulation of electrodynamics in terms of field strengths which is both manifestly covariant and path-independent. I t thus combines the advantageous features of both pre­vious gauge-independent formulations. The basic quan­tity is the gauge- and path-independent operator CXM

which is defined in terms of the field strengths FM„ in Eq. (9) below.

* Support of this work by the National Science Foundation is gratefully acknowledged.

f Present address: Laboratoire de Physique Theorique et Hautes Energies. Orsay. Seine et Oise, France.

1 S. Mandelstam, Ann. Phys. (N. Y.) 19, 1 (1962). For a more general approach see M. Levy, Nucl. Phys. 57, 152 (1964).

2 B. S. DeWitt, Phys. Rev. 125, 2189 (1962). 3 F. Rohrlich, State University of Iowa Report 62-15, 1962 (un­

published). See also F. Rohrlich, Proceedings of the Midwest Theory Conference, Argonne National Laboratory, June 1963 (unpublished).

4 F. J . Belinfante, Phys. Rev. 128, 2832 (1962).

B

The arbitrariness in the choice of the "gauge," which deprives the conventional potential of a direct physical meaning, no longer affects the theory. Thus, the gauge-independent operator dp may have a direct physical interpretation as photon-field operator. The Lorentz condition is automatically satisfied by dM as a conse­quence of the antisymmetry of F^. No spin-zero field is therefore involved in the theory, as is the case, e.g. in Stueckelberg's formulation.5 The commutation rela­tions for <2M are derived by assigning those for the fields FpV\ they will be proven to be the same as in the Landau gauge. The Lorentz condition is therefore consistent with them as an operator equation, and no problem arises requiring ad hoc measures such as the use of an in­definite metric with the consequent appearance of un-physical states. The success of the Landau propagator (Z1—Z2, finite) may thus be justified by its connection with a gauge-independent formulation of quantum electrodynamics. I t may be interesting to note that the recent attempts at a finite electrodynamics by Johnson et at} make use of the Landau gauge for the photon propagator in first approximation.

These results may be extended to the theory of mas­sive vector bosons interacting with a conserved current, where analogous problems of gauge in variance arise.7,8

Here too, a gauge-independent field operator OfcM may be introduced. The physical meaning of gauge inde­pendence in this theory is that (^ described a pure spin-1 field. The commutation relations for CfcM, derived from those for F^, are

In this formulation, the renormalizability of the theory is therefore explicit.8

5 E . C. G. Stueckelberg, Helv. Phys. Acta 11, 225 (1938). 6 K. Johnson, M. Baker, and R. S. Willey, Phys. Rev. Letters

11, 518 (1963); Phys. Rev. 136, B l l l l (1964). 7 V. I. Ogievetski and I. V. Polubarinov, Zh. Eksperim. i Teor.

Fiz. 41, 247 (1961) [English transl.: Soviet Phys.—JETP 14, 179 (1962)].

8 A. A. Slavnov, Zh. Eksperim. i Teor. Fiz. 44, 1119 (1963) [English transl: Soviet Phys.—JETP 17, 754 (1963)].

476

GAUGE INDEPENDENCE AND PATH INDEPENDENCE B477

II. DEWITT-MANDELSTAM FORMULATION

Maxwell's equations may be simplified by introducing the potential Ay, (/z=0,- • •, 3), through the equation

FyV=dyAv-dvA». (1)

Actually, Eq. (1) does not determine the potential Ay uniquely. The field tensor FyV does not change if Ay undergoes a gauge transformation (of the second kind),

Ay->Ay=Ay+dyA,

where A = A(#) is a scalar function which vanishes at infinity

A(x)—>0 as %*—» cc (/*=(),• • -,3).

If the electromagnetic (e.m.) field is coupled to a matter field the gauge transformation on Ay must be accom­panied by a gauge transformation (of the first kind) of the matter field, in order that the field equations remain unchanged,

yjz —> rf/f = exppeA]^.

The arbitrariness in the choice of Ay may be regarded as the origin of difficulties which arise in connection with the quantization of the e.m. field.

The DeWitt-Mandelstam1,2 formulation of electro­

dynamics tries to overcome these difficulties by intro­ducing gauge-independent field operators. The tensor FyV and a suitably defined gauge-independent operator ty are chosen as the basic fields.

We start by defining a gauge-independent potential. Let dp and \p be the potential and the matter field opera­tor in some arbitrary gauge. A gauge-independent po­tential A^ may be defined by the following gauge transformation:

A^ap+dnA, (2) where

,o QZC

A=- / a„(z)—dt. (20

Here 2M(x,£) are four arbitrary single-valued differential functions of the space variables Xy and a parameter £, which transform like a four-vector and satisfy the boundary conditions

z»(x,0) = x11, (3)

lim z^(x, £) = spatial infinity.9 (4)

The path integral in (2 ) is to follow a space-like path. I t is easy to show that Ay is gauge-independent. One

has in fact

dA Ay dy^ = •

dx"

r° daff dz<> dz* r° d dz* r / d£- / a,(z) i f = - /

7-00 dz? dx» d£ J-oo d£ dx* J.

+

i.e., r° dz* dzv

Ay(x) = J Fv<T(z) d£. dx» d£

(5)

Then, if #M and a,/ describe the same field, but in two different gauges, the operators Ay and A/,

and

ru dz* , A= — / a<r(z)—d£

dz* f" dz* AJ^aJ+dyA', A ' = - / aa

f(z)—d£,

coincide (gauge independence of Ay). Thus Eqs. (2) and (2') enable us to pass from a class of gauge-dependent potentials to a field variable which is gauge-independent.

The gauge-independent matter-field operator is de­fined by

ty ~ exp[ieA~]\{/,

where A is given by Eq. (2'). The gauge independence of ty is easily shown:

>P = exppeA]^ = exppeA']

X e x p [ > ( A - A ' ) > = e x p | > A ' > ' .

0 daff dzp dz*

^ 6ZP dx* d£ 0 da„ dz*

«

dz*\° d%— a<r(z)

d£ dx* dx»

f° dz* dzv

= / Fvv(z)- ~d%—ay, 0 J —00 dx» d£

These operations are possible also in the quantized theory as long as the paths for A and A' are the same. I t is not difficult to verify that & coincides with the field operator introduced by Mandelstam.

As the Eq. (5) uniquely determines the gauge-independent potential once the field tensor FyV is given, electrodynamics may be formulated either in terms of A y or in terms of FyV. The latter alternative has been followed by Mandelstam.

The above definition of Ay needs two remarks. First, on starting from Eq. (5) and using d\FyV-\-dyFv\ + dvF\y=0, the relation (1) may be obtained only if explicit use is made of condition (4). Secondly, according to the definition (5), Ay depends not only on FyV but also on the chosen path 2M(#,£) along which the integra­tion is performed. Thus the theory, while no longer gauge-dependent, now becomes path-dependent, and this may be regarded as just another kind of "gauge."

The above-mentioned path dependence may be re-

9 Spatial infinity here means any limit which is sufficiently re­mote in a space-like direction from that the field vanishes there; at this limit Ay may without loss of generality be set equal to zero. We shall return to this assumption later.

B478 F . R O H R L I C H A N F F . S T R O C C H I

moved by performing a sort of average over all possible paths. We shall see that such a procedure leads to re­sults related to the usual gauges.

III. NONCOVARIANT PATH AVERAGE

Following Belinfante,4 we put

and choose the Lorentz frame in which

e o = o , e 2 = l , £ = - r = - | z - x | .

The lines JSM(X,£) are then confined to straight lines at / = const, converging to the field points x in arbitrary directions given by unit vectors £. Equation (5) re­duces to

Putting

A nix) -f Fi,{z)m (6)

(sum over dummy indices). The potential A^ now de­pends on the direction of e. This dependence is removed by averaging Eq. (6) over all possible directions t. By straightforward manipulation one obtains

r dil r d ( 1 \ A,(x)= \A—= d*zF*(z)—( — )

J 4TT J dzi\kirr/

= -[<Pz£V'Filt(z)l/(4wr). (7)

Here dtt is the infinitesimal solid angle in the direction e. Equation (7) gives explicitly

curlE(z,/) r <kvh{z,t) _ r i°(x) = / dH , A = / dh — . (70

J 4:wr J £wr

Thus the potential Ay. coincides with the potential in the Coulomb gauge.

As it should be expected, the average performed above has led to noncovariant results due to the choice of the plane / = const as a privileged plane in a given Lorentz frame. This breaks the manifest covariance just like the definition of the Coulomb gauge, divA=0.

IV. COVARIANT PATH AVERAGE

In order to obtain a Lorentz-invariant result we must perform a covariant average, i.e., an average over paths covering a Lorentz-invariant region. Now, the Lorentz-invariant subregions of Minkowski space are the space-like region outside the light cone, the time­like regions inside it, the light cone itself, and the whole space. Averaging over straight lines covering any of the first two regions leads to divergent integrals unless special prescriptions are given.1 Therefore, we shall average over all straight lines converging to the point where the potential has to be calculated.

Zv = %v-\-evi;, €vev=zhl (8)

(with the plus sign if the line is space-like and the minus if it is time-like), and using Eq. (5), we have:

A^x)--/

0 gv ^.v

.?. = / F,v(x+y)-d{,

Jo 4f

where the new variables yv~zv—xv and £=(yvyv)2, have

been introduced. We now wish to average over all directions of the unit

vector ev. To this end we note first that in 4-dimensional Euclidean space

d*y=d3ydy4=ld{d2n(dfjLE/(l+fXE2)2),

f=(yMy^)2, HE=yi/\y\,

where 0 < f < <*> and — <*> <ju< + <*>. The transition to Minkowski space is governed by the relation y^=iy° but permits two alternative distortions of the path along the real y^ axis to one along the imaginary 3/4 axis, viz., by a rotation by \-K— e or — (J71—e). Those alternatives give contours in the complex y° plane which avoid the poles according to y0==k(\y\+ie) or 3/°==b(|y| — ie). They correspond to the D1R and D1A

functions, respectively. Time symmetry is preserved if we take the contour for i(DiR—DiA) = i(DR—DA) = Dp as indicated in the Appendix. The choice of any of these contours is completely determined by the asymptotic conditions.

The integrals in Minkowski space are thus evaluated over jjf=y°/\y\ — — i^E with the above contour choices in the ju plane,

d±y=d3ydy0=ld£d2£l(dtji/(l-ix2y).

The surface of the pseudosphere in M4 is

/ dsQ= / d2tt / =2TTH. J J J^OL-M 2 ) 2

The direction average of A^x) now becomes, choosing one of the above contours, CR, CA, Cm, or CIA,

G>n(x) 2wH J

A^x^d^

1 r yv d\x = F^x+y^d^Q-

2THJ 4f (1-M2)2

2TH J F^ix+y)- -d*y

(y«ya)2

=— [Fll,(x+y)—(-)d*y. 4TT2 J dyv\y

2/ (9)

GAUGE INDEPENDENCE AND PATH INDEPENDENCE B 479

Here y2E=yaya and the integral extends over all Minkow- The well-known retarded and advanced functions are

ski space, restricted only by the path Ca. The resultant here denoted by DR and DA- Thus, dp are obviously Lorentz-covariant.

The result (9) could also have been obtained as fol- Q retrx\— j j) rx_z\j f z ) ^ z

lows. One notes that 1/x2 is the Green function for the J inhomogeneous d'Alembert equation, 1 f j^oc+y)

= / d*y. (15) D ( l A 2 ) = -47r2i5(^), (10) ATTH JcA y2

provided that both sides of the equation are regarded as The asymptotic behavior of this function leads to the distributions, the left side being defined by one of the identification paths CR, CA, CIR, or C\A. Consequently, for any of 1 f fFVix{x-\-y)\ these paths <Vn(*) = — : / d>[ Wy4. (16)

F,t,(x)= Fflv(x+y)dada[~ )d*y ^ . tA rN .

4^2 J \y2/ The expression (15) is easily seen to be equivalent to the Sommerfeld representation of the retarded potential.10

~~?' f , v / 1 \ The decomposition (13) can thus be written as = — daFp,(x+y)d" [-Wy, P

4TT2 J \y2/ aM= c t / e t + a / n = aMa d v+ aM

out (130 since the surface integral vanishes. If the homogeneous depending on the choice of the contour, CA or CB, re-Maxwell equations hold, this becomes spectively. Of special interest is the decomposition

i r , u A»(x) = A^(x)+Mx), (17) F^(x) = — J (d^Fva-dPF^a)d

a\^-jd^y. w h i c h i g o b t a m e d a s t h e arithmetic mean of Eqs. (13')

and which is time-symmetric. I t plays an important role Hence, if we define a'^x) as in (9), this equation yields in the classical theory of point charges.11 Here

M v v » M" v J a M <+>(a ; )= | ( a / e t +a Ma d v )=— : / dA

y (18) Equation (9) may be put into the form: _, sir i J cp y

L r»pr^£y+±. fJF^x+y)\^ ®M=mr+®r>) aii(x) = — d»FVfi(x+y)—+— d>[ )d±y 47r27 y* br*J \ y* J I f _ /F^(x+y)\ ^ i_ r JF^(x+y)\

TTHJCP \ y2 / % f J^x+y) i f /F^x+y^ STH J CP \ y = / d*y-\ / dH Wy,

4?r2 J y2 4TT2 J \ y2 / They also satisfy (14). We note parenthetically that a decomposition of (£M

t 1 ^ ' according to the contours CIR and CIA also leads to (13)

where an integration by parts has been performed and ™d <14). a n d t h e a r i t h m e t i c m e a n o f these two leads use has been made of the inhomogeneous Maxwell a s ° ° ^ v _ _ _ . eauations antisymmetry of rM„ assures that the Lorentz

' ^ , D _ _ • condition is satisfied as an identity:

Equation (12) provides a decomposition of a„ into a ^ a u^JL f ^F Jx+y)dv—-dAy "bound" and a free field M 4^2 J ^ 3>2

a,= a^+a^\ (13) x - t because = FfiV(x+y)dW—diy=0. (20)

DCt <« = - / „ and D ^ = 0. (14) ^H J y2

Consider, for example, the bound field produced by the The absence of a spin-zero field is thus proven also for contour CA\ according to the Appendix, the case of interactions. This result combined with (14)

yields charge conservation. i r j*(x+y) f , N , N

—; / d*y= / J,i(x+y)DA(y)d*y [TTH J CA y2 J

kirHJr V2 J 10A ' S o m m e r f e l d > A n n- Physik 33, 649 (1910). See also C. A y Miller, The Theory of Relativity (Oxford University Press, Oxford,

/" . , N T W v 7, f ^ , v . , v 1A U F . Rohrlich, Phys. Rev. Letters 12, 375 (1964). See also

: / 3Az)DA\Z—x)d*z== \ DR{x—z)j»{z)d*z. F. Rohrlich, Classical Charged Particles (Addison-Wesley Publish-J J insr Company, Inc., Reading, Massachusetts, 1965).

B480 F . R O H R L I C H A N D F . S T R O C C H I

The path average performed on the potential can also be applied to the electron field. The path-dependent, though gauge-independent field

ty(xJe) = L(x,e)ip(x), L(x,e) = exp aa{z)€°di

involves the unit vector ev of Eq. (8). The path inde­pendent ^ is now related to the path-dependent one by

ty(x) = 1

2TT2<

*(x,e)dstt = L(x)xP(x)

in complete analogy to (9).

V. GAUGE- AND PATH-INDEPENDENT ELECTRODYNAMICS

In the classical case we can start with the nonlocal modification of the usual Lagrangian density

L= -iF^+j.W+LflJ'-], (21)

where d„{x) is defined by (9). The basic fields are there­fore FVP and ^ . The homogeneous Maxwell equations then imply (11) and the antisymmetry of F^ implies (20). The inhomogeneous equations follow in the usual fashion.

We note that of the various gauge-dependent choices which would follow from (11) and which lead to different

representations of the commutation relations,12-14 Eq. (9) fixes the one which is gauge-independent.

Consider now the quantization of the free fields,

£Fflv(x),FPa(xf)li = idfJLp,v<rD(x—x'),

—giicdvdp—gvpdpdo. (22)

In order to find the corresponding commutation relations for dM, we note first that (10) implies, for any function f(x) with appropriate asymptotic behavior and any one of the indicated contours, that

1 f 1 f{%) = / d"—d^f{x^zy)dAy

4TT 2 ;

4TT 2 ;

The vector operator

wH J \ W ±y)d*y. (23)

(d-i)^d~\^-

can thus be defined by15

&-\f(x) = ± fdl-)f(x±y)d*y. (24) H J \y£.

A simple calculation now yields

[aM(ff),G,,(aO] = 4TT

8P[ -)d'<(— )d*yd±ylF»p(x+y),Fvff(x/+y'n

= t\ — ) I dp ^4TT2 yV \y'2)

d*yd*yf(-g»vdp^dj^+gpad^dv^

~g^dp^dv^+gpvd^df^D{x-xf-\-y-yf)

= —i(gn»—d \dv)D(x—x')=—i[ g> )D(x—x'). d„d,

(25)

The gauge- and path-independent potentials therefore have nonlocal commutation relations which are formally just those employed in the Landau gauge.

However, the essentially nonlocal nature of the po­tentials plays an important role: Depending on the asymptotic conditions, a different contour is to be taken in the definition of d_1

u, Eq. (24), which is implicit in (25). I t follows that the commutation relations of the interacting fields depend on their asymptotic behavior.

I t is obvious that the Lorentz condition (20) holds as an operator equation and that it is consistent with the commutation relations (25). The usual difficulties aris­ing from the inconsistency of (20) with the commuta­tion relations in various gauges therefore do not occur here.

In exactly the same way as in (25) one finds

/ d,dv\ (CtM(^)a,(^/))o=— i\ guv \D+(x—xf)

12 A. Peres, Nuovo Cimento 34, 346 (1964). 13 B. Zumino, J. Math. Phys. 1, 1 (1961). 14 S. Kamefuchi and H. Umezawa, Nuovo Cimento 31, 429

(1964); 32, 448 (1964). XP When / is a generalized function, Eq. (24) is a convolution of

such functions. These are more conveniently discussed in momen­tum space. In a future paper it will be shown that a consistent and unambiguous formulation of the operator d_1 can be given in that space for all the generalized functions which occur in quantum field theory. The present paper is therefore restricted entirely to % space.

G A U G E I N D E P E N D E N C E A N D PATH I N D E P E N D E N C E B481

and the propagator / dpdA

<r(aM(*)Gr(*'))>o= -if g> wc(x-xf). (26)

Again, we have formally the same expression as in the Landau gauge. The success of this gauge in the recent formulation of quantum electrodynamics by Johnson et al.6 may thus find here an independent justification.

VI. MASSIVE NEUTRAL VECTOR-FIELD INTER­ACTING WITH A CONSERVED CURRENT

The theory of a massive neutral vector-field inter­acting with a conserved current may be given a form similar to electrodynamics.7 Gauge invariance plays a similar role in the formulation to the one it plays in electrodynamics—the Lagrangian, the field equations, and the equal-time commutation relations being in­variant under gauge transformations.

The simultaneous equations

( • - w2)4v = - j M = - igh^, (27)

are invariant under the simultaneous transformations

but the condition \p —> exppgA]^,

(28)

(29)

usually imposed to eliminate the spin-0 field, is not gauge-invariant unless A is restricted to solutions of the wave equation. Of course $M itself is also gauge-dependent, as is evident from (28).

A gauge-independent formulation is, however, pos­sible by using the fields 4*^ as basic operators; they are assumed to satisfy

dx<^„+ dM<£„x+ dyfan=0. (30)

A gauge-independent potential <£ is now defined in terms of <£M, by

i r 1 %(x) =— / <i>liV(x+y)dv—d^yy (31)

4TT 2 7 C I y2

where Ci is the contour of any inhomogeneous D-f unction. The first Eq. (27) now reads

(D-m2)^=-j,, (270

where j ^ is conserved on account of the second Eq. (27). Equation (30) and (31) combine to yield

d^-d^^fa,, (32)

d*<SV=0 (33)

exactly as in (11) and (20). Equation (33) is now gauge-independent, in contradistinction to (29). I t requires a conserved current as source in Eqs. (27) and (27').

and

This result was to be expected, because only the spin-0 part of the field #M depends on the gauge.7 A gauge-independent formulation therefore must have the ab­sence of this part as a consequence,

In the free case one quantizes the basic fields $M, according to

\j>^(x),(j)Pa(x/)']==idtlp,p<rA(x—xf) (34)

in analogy to (22). Exactly as before, the commutation relations for the ^ are found to be

C*M(*) / dlldy\

M*)l= -i\gp> Wx-x') (35) n

dudv

and its propagator is / dMdA

(T(%(x)M%')))o= -i{g»* j A c ( x - * ' ) , (36)

where the definition (24) is used again. This result makes the renormalizability of this theory

manifest, no Dyson transformation being required.16,17

ACKNOWLEDGMENT

One of us (I. S.) would like to thank Professor M. Levy for an interesting discussion.

APPENDIX 1

The invariant functions Da(x) will be assumed known and defined as before.18 I t is then easy to verify that for any function f(x) for which the integrals converge

4TT

or, symbolically,

— / —f(x)d±x= \ Da(x)f(x)dAx r2i J ca x2

1 1

ATTH x2 = Da(x).

(Al)

(A10

The contours Ca are depicted in Fig. 1. The above equa­tions hold for DR, DA, DIR, and DiA, as well as for D+)

FIG. 1. The contours in the x° plane.

16 F. Dyson, Phys. Rev. 73, 929 (1948). 17 R. J. Glauber, Progr. Theoret. Phys. (Kyoto) 9, 295 (1953). 18 M. Jauch and F. Rohrlich, The Theory of Photons and

Electrons, (Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1955) (second printing, 1959). The second printing contains corrections of various misprints.

B482 F . R O H R L I C H A N D F . S T R O C C H I

Z>_, and D. Slight modifications are necessary for Dp and Dh

1 1 | = 2DP(x) (A2)

\CP and

4:ir2i x 2

1 1 I

4:7T2i X2 I Ci = %iDi(x). (A3)

I t is to be noted that the above contours Ca(x) for in­tegration in the complex x° plane are different from the contour Ca(p) for the corresponding Fourier trans­form.18 In fact, one has CR(x) = C+(p), C+(x) = —CA(p), etc. Only the contour for D is unchanged, C(x) = C(p).

APPENDIX 2

In this Appendix we sketch the proof that d^x) as de­fined by (9) is independent of the particular contour in the x° plane, as long as this contour corresponds to an invariant function which satisfies the ^homogeneous equation (10). To this end it suffices to show that

/ Fl>r(x+y)d'DH(y)diy^0, (A4)

where DH(y) is an invariant function satisfying the homogeneous d'Alembert equation. Substitution of (11) and use of (20) yields

/ dv(d,avDH- andj)n)d*y= -( j -J p ^ D n - apdoDH)d*<r. (AS)

Since DH depends only y° and | y | and is independent of the direction of y, the integration over the directions can be carried out. Using the Fourier transform we have for any function f(x+y) and with w= | k | ,

/ f(x+y)DH(y)d*y= / f(k)eik-^+^-ik0^+^d*kDH(y) | y12d\ y\d2tt J (2x)2 J

1 r r sinw y = - f(k)eik'*-ik0*0e-ik0y°d*y ^ ( | y | , y ° ) | y | d | y |

7T J J CO

Now all homogeneous functions DH can be written as

DH=oiD1+(3D,

where a and P are constants. Since D\ involves the principal part of 1/y2, the above integral will vanish in the limit y° —* ± <*> by the Riemann-Lebesgue lemma. But D(y)^e(y)8(y2). The |y | integration above then yields a term proportional to

2ie-ik°9° sincoy°= e*(»-*0)»0-er*(«+*,)»0. (A6)

Again, the limits y° —> ± oo would give a vanishing re­sult were it not for the possibility of a factor 8(w—k°) or S(co+&°) in / (£) .

Consider the first such factor. In the limit y° —-> =h °o only the first term in (A6) will survive. But the result is independent of whether the limit y° —> + °° or y° —> — <*> is taken. Thus, the two integrals over the spacelike planes in (A5) will exactly cancel. The same is true for a factor 8(o)+k°). This proves Eq. (A4).

Thus, the definition (9) of the total potential CtM is indeed independent of the choice of the inhomogeneous contour G . The two terms in the decomposition (13) are each of course not independent of these contours. And, in particular, the equal-time commutation rela­tions for the retarded fields <$/et which follow from (25) are^also contour-dependent.