Gauge invariance in quantum electrodynamics

ANNALS OF PHYSICS:

52, 122-175 (1969)

Gauge

Invariance

in Quantum

Electrodynamics*

RICHARD A. BRANDT Center for Theoretical Physics, Department of Physics and Astronomy, University of Maryland, College Park, Maryland 20740 A formulation of quantum electrodynamics based on finite local field equations is employed in order to prove and discuss the gauge invariance of the theory in a meaningful and rigorous way. The Dirac and Maxwell equations have the usual forms except that the current operators f(x) and iM(x) are explicitly expressed as finite local limits of sums of nonlocal field products and suitable subtraction terms. The electric current, for example, involves the terms A(x) and : A+):. The field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions JY,II@“, Au, and X”@ (the four-photon vertex function), etc.Application of the boundary conditions E(b = m) = Z’(fi = m) = II(O) = n’(O) = U”(O) = LI($ = m, 0) = X(0,0,0,0) = 0 is shown to completely specify the current operators. It is shown that the theory is gauge invariant in the sense that the divergence conditions k,W”(k) = koX**(k,...) = 0, etc. and the generalized Ward identities k,,N(p, k) = eZ(p - k) - e-J?(p), etc. are all satisfied in each order or perturbation theory. This is shown to be equivalent to the invariance of the field equations under gauge transformations of the second kind. I. INTRODUCTION

The purpose of this paper1 is to use a formulation of quantum electrodynamics based on finite local field equations2 in order to rigorously prove the gauge invariance of the theory and to show that this gauge invariance is equivalent to the invariance of the field equations under local gauge transformations. The field equations will have the usual forms8

l$P”(x)

= j”(x),

* Supported in part by the U. S. Air Force Office of Scientific Research under Grant AFOSR 68-1453. 1 This paper is taken from Ref. (I). Other parts of (I) will be contained in another paper (2). We shall refer to Ref. (2) as B throughout this work. 2 Such equations were derived in perturbation theory in (21). An independent derivation was known to W. Zimmermann at an earlier date. For recent investigations and applications, see references (3)-(6), where earlier references can be found. We shall refer to (3) as A in this paper. s We use the notations and conventions of Bjorken and Drell (7); e.g., c = ti = 1, k - x = kpx,, = kOxO- k . x. Wewritei = y ‘p = yfipp, 0 = a* = a02 - v*, (a/ax,) F(x) = Lw(x) = F’fi(x). Three-vectors will be denoted by bold-face letters.

122

GAUGE

INVARIANCE

123

except that the current operators f(x) and j”(x) will be finite limits of nonlocal expressions. Thus (1.3) where the operator q”(x; 0 has singularities at f = 0 which compensate those of the local product q(x) yU$(x), and similarly for p(x; 5). Gauge invariance in classical electrodynamics means an invariance under the transformations #(x) + e-iea(z)#(x), i&x) -+ eiea(r)#(x), (1.4) AL(x) + Au(x) + 4L44 for arbitrary smooth functions a(x). Here A, is the classical vector potential and + a Schrijdinger wavefunction (matter field). In the usual formulation of quantum electrodynamics, a canonical proof of gauge invariance in this sense has only formal significance because the field equation for the quantized field A, involves the meaningless product $(x) rub(x) of quantum fields at, the same point. In practice the requirement of gauge invariance has been replaced by the requirement that various generalized divergence conditions are satisfied. The simplest of these are the identity kd7,,,(k) = 0 (1.5) and the generalized Ward identity ku/l,(p,

k) = eZ((p - k) - eZ(p).

(1.6)

By the gauge invariance of electrodynamics we shall mean that all these conditions are satisfied. In this paper, on the other hand, one has well-defined field equations for A, and I,L and hence can return to the original form of gauge invariance and discuss its consequences for renormalized electrodynamics. In particular, the requirement that the field equations (1.1) and (1.2) be invariant under (1.4) implies nontrivial restrictions on the subtractions qu(x; .$) and p(x; t) used in definingj@(x) andf(x), especially as the product #(x) y,$(x + .$) is not gauge invariant for nonzero f. We shall show, in fact, that the invariance of (1.1) and (1.2) under (1.4) is equivalent to the requirement that all of the generalized divergence conditions be satisfied. The analysis of the requirement of gauge invariance on the subtraction functions q”(x; 5) andp(x; 8) is given in Section VI, Subsections E to H. A full understanding of this problem requires a detailed knowledge of the general, behavior of qU and p.

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BRANDT

To determine this, in previous sections the renormalized field equations are related to integral equations for Green’s functions and the subtraction functions are determined from conventional renormalization conditions. A preliminary study of Subsections VI.E-H will, however, show what is involved. In order to see what the problem is, let us consider the usual formulation of quantum electrodynamics based on the “unrenormalized” field equations

Calculations with these equations lead to numerous divergences which can, however, by careful use of covariance properties, be isolated into infinite but unobservable mass, charge, and field operator renormalization. For example, for the second order polarization tensor one obtains

where M is a large cutoff mass and U$“(k) is finite for M + co. Now (1.5) requires that n@“(O) = 0 and other requirements of renormalization theory demand that the first two derivatives of the renormalized nuy(k) vanish at k = 0. Thus one must renormalize (1.9) by subtraction at k = 0. The resultant expression is finite and satisfies n““(O) = 0 but is not guaranteed to satisfy (1.5). In this paper we shall in fact prove that the divergence conditions are satisfied in every order of renormalized perturbation theory. Previous proofs are all of a formal nature and do not guarantee that the appropriate identities will hold in perturbation theory. The local field equations (l.l), (1.2) form an ideal setting for a meaningful discussion of gauge invariance. All calculations proceed directly and mathematically from (1.1) and (1.2) without renormalization or regularization. Thus if (1.1) and (1.2) are gauge invariant, the divergence conditions should be satisfied. Note that the expression lim,,, X,&(X)yU#(x + 0 is not invariant under (1.6). It behaves like [-3 for f - 0 and so, after (1.4) is applied, terms like 511Q(x) contribute. They must be cancclled by the q” term in (1.3). In contrast, in the usual formalism based on the divergent equations (1.7), (1.8) and the ad hoc renormalization rules, gauge invariance cannot even be discussed in a meaningful way. Equations (1.7) and (1.8) are not really gauge invariant and, in any case, the need for renormalizations means that they do not involve the correct current operators for the theory. In this approach, gauge invariance is usually maintained by simply discarding non-gauge-invariant quantities or by using the special Pauli-Villars regularization. These procedures cannot, however,

GAUGE

125

INVARIANCE

be formulated in a local way (or even in a simple way) and so they make gauge invariance appear to be rather artificial. With gauge-invariant finite local field equations, on the other hand, gauge invariance is implemented from the beginning in a natural and mathematically precise manner. In Section II we describe how to calculate renormalized Green’s functions in quantum electrodynamics. The Feynman rules and elegant Bogoliubov-ParasiukHepp renormalization prescriptions are outlined. We show how the subtraction points are specified by the requirement that the parameters have their physical values and indicate the restrictions imposed by gauge invariance. Our field equations and the general forms of our current operators are exhibited in Section III. In Section IV we define numerous Green’s functions and proper part functions and then use the field equations and current definitions to derive integral equations relating them. (Most of the details are omitted. They are either identical to those in A or are given in B.) By imposing the conventional boundary conditions on these equations, we completely specify the current operators. (We show in B that our integral equations are all finite and self-consistent by deriving them from renormalization theory. It follows that the iterative solutions of our equations are equivalent to conventional perturbation theory and hence that quantum electrodynamics can be based on finite field equations.) In Section V we show that our theory, and hence conventional theory, satisfies the requirements of gauge invariance. In particular, we derive many generalized Ward identities and other divergence conditions. The arguments (given in the Appendix) are quite involved and indicate the subtleties of gauge invariance. We find in Section VI, moreover, that, given the general forms for the current operators, the gauge invariance of the theory is equivalent to the gauge invariance of the field equations-including the gauge invariance of the electric current operator. II. GENERAL

PROPERTIES

OF QUANTUM

ELECTRODYNAMICS

Our purpose in this section is to summarize the means by which calculations in quantum electrodynamics are performed. This theory describes the Lorentzinvariant interaction of electrons (spinor particles of mass m and charge -e), positrons (spinor particles of mass m and charge +e), and photons (massless and chargeless vector particles) according to the (formal) interaction Hamiltonian density 4(x> = e&4 r?W 4dx). Gw * When necessary to avoid infrared difficulties, we can assume that the photon has a small but finite mass.

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BRANDT

We shall first describe the general calculational procedure and then describe the particular procedure which must be invoked in order that the results describe the particles encountered in the laboratory. This particular procedure is determined by the imposition of the requirements dictated by a specification of the operational definitions of the parameters M, e, and by gauge invariance and charge-conjugation invariance. A. RENORMALIZATIONTHEORY We shall be concerned exclusively with the Green’s functions5

The S-matrix elements can be obtained from these by means of the usual reduction formula (8). A suitable formal starting point for our discussion is the Gell-Mann-Low (9) perturbation expansion, the various terms in which can be represented by Feynman diagrams in the usual way. An arbitrary &h-order Feynman diagram G(V, ,..., V, , 2) connecting the vertices V, ,..., V, with the lines 2 = {fI . .. 8S> corresponds in momentum space to an unrenormalized function6 (2.3)

Here pc is the momentum corresponding to the line 8 and a, represents its discrete spin or polarization indices. Thus if /is an electron line we have

and if /is a photon line we have (2.5) 5 Throughout

this work we denote vacuum expectation values of time-ordered products simply

as . 6 We denote 4-dimensional

momentum integrations

I 1I P = -cw”

dp =-

by 1

s

d’P.

GAUGE INVARIANCE

127

y(u) represents the product of all vertex factors (-ie$) and also contains the overall sign of G. The integration IPint is over all independent internal momenta not fixed by momentum conservation at the vertices. pext represents the external momenta, say, p1 ,..., pzr+s-l .

The expression (2.3), however, is meaningless since the indicated integrand is not integrable. The transformation of the integrand into an integrable function is called “renormalization.” This transformation, which is here treated as part of the definition of the theory, was first successfully employed by Feynman, Schwinger, and Tomonaga (10). The first complete renormalization prescription, however, given by Dyson (10) and Salam (II), has not been rigorously proved to yield convergent integrals in all orders. Bogoliubov and Parasiuk (12), meanwhile, proposed an alternate renormalization prescription which leads to well-defined renormalized Green’s functions, as Hepp (13) has proved. This prescription will be used throughout this paper. One begins by defining a regularized unrenormalized function @ corresponding to G( I’, ,..., V, ,9) by replacing the propagators i&( p, 0) in (2.3) with regularized propagators id;*‘( p, a), r > 0, and by adding more external momenta so that each vertex Vi corresponds to a unique external momentum k, , i = l,..., n. The renormalization transformation psf -+ RF*’ subtracts from P*C certain terms so that the remainder RF*’ has a limit lim r10lim,&,, l@,6 whose Fourier transform defines a tempered distribution on the Schwartz space Y(R4”). These subtraction terms correspond to diagrams obtained from the original diagram G by combining subsets of the original set {V, **. I’,) of vertices into generalized vertices in all possible ways. A given subdiagram G’ rv ( V{ *a. Vk} requires a subtraction only if its superficial divergence v( Vi *a*VA) is nonnegative. Here

where Cconn extends over all /E 5? which connect two vertices in {Vi .a* V&} and lines. In k-space the subtraction term is a Taylor series expansion around arbitrary points up to order v.

r, = 1 for electron lines, r, = 0 for photon

B. SUBTRACTION

POINTS

The well known classes ‘of one-particle irreducible (IPI) diagrams with nonnegative superficial divergences are given in Fig. 1. These correspond to the so-called primitive divergences. It is these diagrams for which subtraction points must be specified for each. theory.

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BRANDT

IIPV{k)

X(P)

l/=2

T4’

U=l

(4 ,kf

I+fp,k)

U-I

X=‘%&~&J

u=o

u=o

FIG. 1. The five classes of IPI diagrams in quantum electrodynamics with nonnegative superficial divergences Y. (k, + k, + ks + k, = 0).

The “proper self-energy parts” DU(k) terms of the Green’s functions

and Z(p) can be analytically

iD”“(x, y) = (AU(X) A’(y))

defined in

= s, e~~~(r-y)iDMv(k),

iG(x, y) = ($(x) J(y)) = 1, e-iy)-(“-Y)iG(p),

(2.6) (2.7)

by the relations D”“(k) = [&Y(k)

+ D”,“(k) 17,,(k) DA’(k),

G(P) = ~&F(P) + WP)

Z(P) G(P)-

(2.8)

(2.9

Here l2 and c3 are arbitrary parameters which have been introduced as an explicit indication of the arbitrariness in the subtraction points for .Z and 17. Thus

[k”gA“+ 17,“Wl D”“(k) = -5&“, G(p) = Similarly,

(2.10)

” ji - fn - Z(p)

(2.11)

the “vertex part” P( p, k) is defined in terms of the Green’s function

F”(x, Y, 4 = (Z/~(X)a&y) A”(z)) = j- j- e-i~.(~-Y)-in.(a-y)~‘(p, k) 2, k

WP,

k)

=

-G(P

+

4

F(P,

4

Gfp)

R“(k)-

(2.12) (2.13)

The “proper vertex part” A’@ k) is then defined by

r”(p, k) = Si%$ + WP, WI,

(2.14)

GAUGE

129

INVARIANCE

where & is another arbitrary parameter. Finally, the proper parts Y and X can be defined in terms of the Green’s functions (AAA) and (AAAA). Further (non-trivial) parameters cannot be introduced here because of the absence of direct three- and four-photon couplings. The procedure described above leads to a class of theories in each of which the scattering amplitudes are well-defined and finite. A particular theory in this class is determined by specifying the numerical values of the parameters m, e, & , & , c3 and of the “subtraction points” associated with each diagram with non-negative superficial divergence. This latter specification then provides operational definitions of the parameters. As was extensively discussed in A (Sections V and Vl),‘I each such theory is invariant under certain simultaneous changes in the parameters and subtraction points. It was shown, in fact, that any change in the subtraction points is equivalent to finite renormalizations of the parameters. Thus each particular theory can employ any values of the parameters. The values of the parameters which we shall use are the conventional ones: & = & = I& = 1, m = physical electron mass, e = physical electron charge. Furthermore, the physical photon mass and four-photon coupling constant are assumed to vanish. We will now determine the subtraction points which specify the particular theory that describes the known electrons and photons. We first note that the general connection between Green’s functions and S-matrix elements requires G(p) to have a pole at j = m with residue 1. Thus, in virtue of the form (2.11) for G(p), we require that (2.15)

Similarly, the vanishing of the physical photon mass requires that IP(k) a pole at k2 = 0 with residue -g I(” so that, by (2.10) we must have II““(O)

= Q17”Y(k)~k=,, = 3, 3J7““(k)Ik=,,

= 0.

have

(2.16)

The conditions (2.15) and (2.16) then guarantee the vanishing of radiative connections to external electron and photon lines. Since v, = 1 and v, = 2, the subtraction points for Z and 17 are now determined. We take the requirement e = physical charge to mean that6 the zero-energy Compton scattering cross section of unpolarized radiation by an electron is given ’ In A only the special case & = la was explicitly considered. The extension to the general case 1, # I& is straightforward. B Other possible operational definitions of the electric charge are considered by G. Klllkn (14) and shown to lead to the same value. 595/52/I-9

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BRANDT

by the classical Thomson (2.14), this requires that

cross section. With (2.15), (2.16), and the relation (2.17)

club, QLvn = 0,

which determines the subtraction point for r since vr = 0. A further requirement which must be imposed on the theory is that of charge conjugation invariance. This invariance requires Y to vanish and so its subtraction points need not be specified. Finally, the vanishing of the four photon coupling constant is taken to require that x+(0, 0, 0,O) = 0. (2.18) Since vx = 0, this condition list of such points. C. GAUGE

determines the subtraction point and completes our

INVARIANCE

The theory is now completely determined. There remains, however, a further condition which it is required to satisfy. The theory must be gauge invariant. Gauge invariance (of the second kind), which is now being considered as a consistency condition imposed on the theory, requires that all observable quantities be invariant under the c-number gauge transformations 4(x) + e-iea(z)#(x),

(2.19)

q(x) + eieaWJ(x),

(2.20)

4Xx) -

(2.21)

4bw + 3Pw

Invariance under (2.19)-(2.21) can formally be shown to place restrictions on many functions occurring in the theory. In particular, the primitive divergences are required to satisfy kd7u,(k) = 0, (2.22) k,“X,,,,,(k,

, k, , k, , kp) = k, . X = k, * X = k4 * X = 0, kuA,(p, k) = eL’(p - k) - eZ(p).

(2.23) (2.24)

These relations imply that 17,,(O) = 0, &m(O, k, 3k, , ka) = ... = Xae& L(P, 0) = --e&-Q).

, k, , k, 70) = 0,

(2.25) (2.26) (2.27)

GAUGE

131

INVARIANCE

Equation (2.27) is called the Ward identity (15) and Eq. (2.24) the generalized Ward identity.s Equation (2.22), together with Lorentz invariance, further implies that 17uy(k) has the form IP(k)

= (k”kv - gupk2) P(k2),

(2.28)

where, by (2.16), P(0) = 0.

(2.29)

Now the conditions (2.25)-(2.27), and hence (2.22)-(2.24) are consistent with the subtraction conditions (2.15)-(2.18). Gauge invariance, in fact, can be used to determine many of the subtraction points. Thus (2.22) implieslO n(O) = (k . a) 17(O) = 0 and (2.23) implies (2.18). Using m = physical mass to obtain Z((h = m) = 0, all of (2.15) and (2.16) then follow by requiring that there be no radiative corrections to external lines. Finally (2.27) and (2.15) together imply (2.17). The parameter e, furthermore, is now required to be the physical electron charge as defined above .I1 Thus the complete set (2.15)-(2.18) of subtraction conditions are derivable simply from the requirements m = physical mass and gauge invariance. The requirements (2.22)-(2.24) are, nevertheless, not guaranteed to be satisfied and still must be considered as consistency conditions. Often these requirements are met by simply discarding any pieces of the functions which do not satisfy them. A more theoretical way of proceeding is to use the Pauli-Villars (10) regularization. Then (2.22)-(2.24) can, in fact, be shown to hold (20). Thus, in this case, only (2.15) and (2.17) need be explicitly imposed. In particular, by virtue of (2.26), no overall subtraction for X will be required. As we shall see in the following, however, gauge invariance can be discussed in a much more natural way within the context of local field equations. III. FIELD

EQUATIONS

AND CURRENT

DEFINITIONS

In this section we shall exhibit and discuss the field equations upon which our discussion of quantum electrodynamics will be based.12 We shall work within the usual Lorentz-gauge Gupta-Bleuler (22) framework, although we shall not have occasion to explicitly use the indefinite metric. We emphasize that this is done s This identity was first used by T. D. Lee (16) and H. S. Green (17). It was first proved, using some equal time commutation relations, by Y. Takahashi (18). lo n’(0) = 0 also requires Lorentz covariance. See Eq. (2.28). I1 This equivalence between the present and former ways of defining the electric charge was first established by Kiallen (19) on the basis of the Ward identity. la Parts of the following discussion are based on material contained in reference (21).

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for convenience only and that we could equally well work without specifying any particular gauge. Thus our (renormalized) field equations are the Dirac equation (ii - m) ij(x) = f(x)

(3.1)

0 Au(x) = p(x).

(3.2)

and the Maxwell equation

Here the field operators are the renormalized ones and m is the renormalized mass. It will not be assumed that the operators satisfy any equal-time commutation relations. The form we shall use for the current operators f and j is that suggested by Wilson (23). This form is a generalization of free field Wick products and of previous proposals by Valatin (24) and Zimmermann (25). In addition to 1,4(x), 4(x), and Au(x), we shall employ several additional local field operators-the “generalized Wick products” : t,& : (x), : A,A,A, : (x), and : &I,& : (x). These are defined and related by the equations (3.3)

2p(X) h(x)

Es : A’A”A” : (x) = ,ljmun 2P(x; = e& : A,&

: (x) = liifa(x;

t, C),

(3.4)

t),

(3.5)

where

Y(t, 4 = 4Qb) y”#(x + 5) - GW - GW A(x) - G%9 &Ax) - Cr%) A”.&) - GT5) %KA(-4- CL (0 :~iw~&):l~ (3.6) 2P(x;

g, 4’) ZE TA’(x + g) A”(x) AA(x + p) - By”(g, F) - Bg(g, - ggg,

5’) A”*“(x)

- JxG(5,5’)

- l?;$(g,

: $j$i : (4,

5’) Aaa6yx) - B$/(g,

5’) A”(x) 5’) 2Pyx) (3.7)

f(x; 5) = eP4x + El #W - 4(5) tW - 4,(63 a944 - 44t3fWl. (3.8)

Here e is the renormalized charge. The limits in (3,3)-(3.5) are assumed to exist in a weak sense which need not be

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specified here.13 The Bi , Ci , DC , which will collectively be referred to as {&}, are covariant functions of .$ with singularities at 5 = 0. These singularities are determined as follows. The fields are assigned dimensions in inverse mass units: dim Z = 0,

dim k(x)

= 1,

dim a,x(x)

dim S/I(X) = dim $(x) = $ ,

= dim x(x) + 1.

Furthermore, dim : $$ : (x) = 3,

dim : AAA : (x) = 3,

dim:A#:(x)=$.

Thus the Ei have definite dimensions di = dim Ei ; for example dim C, = 3, dimC,=2,dimC3=1,dimC,=dimC,=dimC,=O.Thenfor~~Oone has Ei([) - I 8 I+ t o within a small fractional power of / f I. The operators which occur on the right-hand sides of (3.6)-(3.8) as coefficients of the Ei are precisely the ones with dimensions less than or equal to that of the left-hand sides and with appropriate transformation properties. For simplicity, we have not written the allowed terms : AA : and a : AA : in (3.6) since they correspond to subtractions from diagrams with three external photon lines and such diagrams would cancel in pairs. This, of course, is a consequence of charge conjugation invariance and will be completely justified in B. The singular functions Z$ have not yet been specified. They are to be solved for as the field equations are solved. Each choice of the Ei such that the limits (3.3)-(3.5) exist defines a theory. We shall exhibit the (essentially unique) Ei’s such that (3.1)-(3.8) are equivalent to conventional quantum electrodynamics. With these Ets the fields f and ,j become the renormalized current operators of quantum eiectrodynamics. The fields #, $, A, f, j are all relatively local. Let us consider for a moment the electric current operator defined by (3.3) and (3.6). This has the form of Eq. (1.3) and the same interpretation-the singularities of the C,(f) at 5 = 0 compensate those of (matrix elements of) d(x) yU$(x + t). We see in particular that the leading singularity of $(x) 7+&x + 5) (which is realized by its vacuum expectation value) is roughly 1 5 /-3. This is because the leading singularities in perturbation theory are independent of the masses and hence given by simple dimensional arguments. In this case one has exactly I [ /-3 as the leading singularity in zeroth order (i.e., for free fields) and additional logarithmic factors in higher orders. Knowledge of this singular behavior is helpful in deriving integral equations from (3.6) and effectively eliminates the need for equal-time commutation relations. The author (3)13 has previously shown that in the case of neutral pseudoscalar meson theory the appropriate Ei exist in all orders of renormalized perturbation theory and that different such Ei correspond to finite renormalizations of the I3 Thesetopologicalmatterswill be discussed in a future publication.

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BRKNDT

relevant parameters. This was accomplished by showing that the field equation was equivalent to an infinite set of coupled renormalized integral equations which were then derived from perturbation theory. Conversely, the entire perturbation expansion for meson theory could then be derived by iterating these integral equations. Similar arguments (2) show that (3.1)-(3.8) are correct in this sense. Our point of view in this paper will, however, be primarily inductive. Thus we shall use (3.1)-(3.8) to derive integral equations and then, by requiring that the iterative solutions to these integral equations reproduce the renormalized perturbation expansion for quantum electrodynamics described in Section II, we shall determine the functions Ei . We shall now, for definiteness, fix the parameters as done in Section II. m = physical electron mass, e = physical electron charge, & = c2 = c3 = 1. Then equations (3.1)-(3.8) determine a class of theories, one for each set {Ei} leading to finite limits in (3.3)-(3.5). The particular set (to within harmless ambiguities) which defines conventional quantum electrodynamics (i.e., reproduces the renormalized perturbation expansion of Section II) will be determined in the next section. IV. RENORMALIZEiD

INTEGRAL

EQUATIONS*

In this section we shall use the field equations (3.1) and (3.2) and the current equations (3.3)-(3.8) to derive renormalized integral equations for proper part functions. For the most part, the analysis proceeds exactly as in A and so will only be outlined here. However, some new features arise and these will be discussed in detail. Our primary concern is with the Maxwell Eq. (3.2) and the electric current operator (3.3). The other field and current equations will only be used to the extent that they are needed for an understanding ofju(x). We shall derive the integral equations and determine the singular functions in the special gauge in which (k(x)) = 0 [consistent with Eq. (3.3)]. We shall show in Section VI that all our equations are gauge invariant and therefore hold in any gauge obtainable from the above ones by a c-number gauge transformation. A. AUXILIARY FUNCTIONS All consequences of our formulation follow mathematically from (3.1)-(3.8), without regularization, “symmetric” integration, or additional calculational rules. We use the physical parameters but now must choose the functions B1, Ci , Di so that (2.15)-(2.18) are satisfied. We do not use Pauli-Villars regularization and so gauge invariance must be explicitly imposed. This is essential in order that the

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135

i+@'(p,k)

iiaPy"(k)

FIG. 2. The classes of IPI diagrams defining the auxiliary functions II, f’, 2. The blobs contain only e-vertices.

limits (3.3)-(3.5) exist and define finite current operators. Pauli-Villars regularization, on the other hand, only leads to the existence of limits of regularized integrals. (We will see in Section VI that the gauge invariance of the theory follows simply from the gauge invariance of the field equations.) In particular, X will now require an overall subtraction so that generalized vertices with four external photon lines will appear during the process of renormalization. We call these vertices “h-vertices” and call the electron-photon vertices “e-vertices.” Of course, since the physical four-photon coupling constant vanishes, no original Feynman diagram corresponding to 17, Z, r, X, or any other conventional function will possess a X-vertex. However, diagrams corresponding to subtractions for these functions will contain X-vertices. It will turn out that, after the subtractions for 17, r, and X are grouped into integrals of renormalized functions, only one function appears for each which corresponds to Feynman diagrams containing an explicit X-vertex. These diagrams are shown in Fig. 2, where the blobs only contain e-vertices. We call the corresponding functions r^i; p, and x. We emphasize that these functions are auxiliary quantities introduced for convenience only. (Thus, for example, fi is not a part of the photon self-energy but will occur as a subtraction to it.) They are to be constructed from Feynman diagrams and renormalized in the usual way. The coupling constant for a h-vertex is taken to be 1. We did not have to introduce such quantities for our treatment of meson theory in A since there we were concerned with the current : $4 : which corresponds to diagrams with an external fermion line. The electric currentJ”, however, couples to photons. It is the : AAA : term injU which, in fact, gives rise to the new functions. B. EQUATIONS FOR PROPER PARTS In order to exhibit the integral equation for fl, we will need, in addition [Eq. (2.7)] and F” [Eq. (2.12)], the Green’s functions Wx,

Y) = VW

wx,

Y> = (: $44 $: A”(Y)),

WY)),

to G (4.1) (4.2)

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BRANDT

and Dyx,

y) = (: P(X) A’(x) k(x):

The Fourier transforms of these distributions the field Eq. (3.2) we then derive -W”(k)

= P(k)

We further define the new functions Il$ - iv;;(k)

A”(y)).

(4.3)

are defined in the usual way. From D,“(k).

(4.4)

and I?&,, by

= n;;(k)

DA”(k)

(4.5)

f%Ak) = a9,w

Rw.

(4.6)

and

Finally,

using the current definition II““(k)

= ie j

P

(3.4), (3.7), we obtain

[tr y”G(p) P’(p, k) G(p - k) + iCr(p)

- icy”(p)

k,k, + iCyBy

n&,,(k)

+ Cr(p)

k,

+ iC;J;;(k)]

(4.7)

as our renormalized integral equation for 17. We see that the subtractions in (3.3) necessary to define the electric current operatorj+) as the finite part of T&x) y+/(x) h ave gone over into subtractions necessary to make the integral in (4.7) finite. C, corresponds to the usual photon mass renormalization counter term. C, disappears if “symmetric” integration is assumed. C, corresponds to photon wavefunction renormalization and C, to charge renormalization. C, represents the new feature of this equation and corresponds to renormalizations of the four-photon coupling constant A. The functions C&, are usually set equal to zero by “gauge invariance.” We shall see, however, that they are required by gauge invariance, let alone by finiteness. Now we shall exhibit the integral equations for I’ and X: &dq,

k) = ie I

P

[tr fG(p)

&(A

q, k) G(p - k)

+ G&,(P) f?“h, 4 + G‘dp) G&, XOL’%

@I,

(4.8)

, k2 , k,, k4) = ie s [tr y”G(p) Esv8(p, k, , k, , k4) G(p - k,) P + G‘YP)

J%‘~~kW,k,)

+ Gdp)

%“6(k&&&d1. (4.9)

137

GAUGE INVARIANCE

(a)

Cd)

(b)

FIG. 3. The diagrams contributing the form (b), (c), or (d).

to H(q,p,

(a) FIG. 4. The diagrams contributing not of the form (b) or (c).

k)

are those of the form (a) which are not of

(b) to @*(p,

k, , k3, kJ

(cl are those of the form (a) which are

Here we have introduced the new functions H, X, 8, 17,and x. They are defined analogously to the preceding functions (see Figs. 2, 3, and 4). The equations (4.7), (4.8), and (4.9) are similar to those derived in A except for the terms involving fi, r, and x. These functions, which arise from the : AAA : term in,j@(x), are necessaryto ensuregaugeinvariance. They arise from subtractions involving X-contractions. The equations do not involve the Pauli-Villars (or any other) regularization and so explicit subtractions are required in order that the gauge-invariance requirements can be satisfied. Had we used such a regularization, the tilde terms would not be present; but then we could not take the regularization limit outside the integral and would not have a current equation such as (3.3). Our current equation, derived from Wilson’s dimensional rules, automatically contains the : A& : term. The most general integral equation for a proper part function (i.e., a function corresponding to an IPI diagram) which can be derived as above from consideration of vacuum expectation values (je(w) I/J(X) ... G(y) ***A”(z) *e*) has the sameform as Eq. (4.9). These functions are characterized by the existence of at least one external photon line in their Feynman diagrams. The general integral equation relates the functions X,,, corresponding to (r + 1) external photon lines and 2s

138

BRANJIT

external fermion lines to a suitable function S,,, corresponding to r photon lines and (2s + 2) electron lines and to suitably defined functions xV,S and Xi,, : Xr,*;*(k ,...) = ie f [tr y“G(p) &;;(p, + -i&3&)

k ,...) G(p - k)

-xy;*.+k ,...) + C&(p)

x;;,;$(k ,... )]

(4.10)

for (r, s) f (1,O) or (0, 1). We have not given precise algebraic definitions of XT,, or ET”,.,’atthough it is clear these can be supplied. The infinite set of integral equations consisting of (4.7), (4.8), and (4.10) is, of course, not closed. It does not contain equations for G or for I?, F, x?,, . Equations for these quantities can, however, be derived by using the Dirac equation (3.1) and the current definitions (3.4), (3.5), (3.7), (3.8). Equations for fi, F, XT,, can, in fact, be derived simply from the definition (3.4), (3.7) of 9I. These equations are really kinematical relations and, because the corresponding function need not satisfy any a priori conditions, are to a certain extent arbitrary. Let us first turn to the Dirac equation and the current operatorf(x). The definition (3.5), (3.8) of f(x) is almost identical with that of h(x) = : #(x) 4(x): in meson theory, given by Eq. (3.11) of A. Exactly as in A we obtain an integral equation for the electron (= nucleon in A) proper self energy part Z, defined by Eq. (2.9), and a new integral equation for the proper vertex part /1:

(4.11)

-
+ W WP, q, 4 W4

- e-‘r”(p, d &WI.

with A, we have replaced the D#) R,(k)

(4.12)

by &(k) according to

= -iD,(k),

&W Yu = --Dz,w, R,(k) = -D,(k).

(4.13)

QVVis defined algebraically by Eq. (3.30) of A, with the appropriate addition of tensor indices and with Euv the Fourier transform of (+,&PAV). Diagrammatically it is given by the sum of all Feynman diagrams with two external photon lines, two external electron lines, no self-energy parts on external lines, and not of the form shown in Fig. 20(a) of A (with the meson lines replaced by photon lines). Finally, we shall exhibit the most general integral equation arising from the Green’s functions (f(w) 4(x) *** $(;. The relevant proper part

139

GAUGE INVARIANCE

functions correspond to diagrams with at least one external fermion line. The equation relating the proper part function r,,, corresponding to r external photon lines and 2s external fermion lines to the appropriate function d,,, corresponding to (r + 1) photon lines and 2s fermion lines is r;;;(p ,...) = ie 1 [y”G(p + k) AF;;.(k, p ,...) D,,(k) - e?;:;(p k

,...) R,(k)],

(4.14)

for (r, s) f (0, 1) or (1, 1). The Ri will be determined below so that Z:, I’“, and r;:b are the usual renormalized functions in quantum electrodynamics. Then the complete set of integral equations given by (4.7), (4.Q (4.10), (4.11), (4.12), and (4.14), together with the equations for n, p, x, and XT,, , and the relations (2.8), (2.9), (2.13), etc. will be completely equivalent to conventional renormalized quantum electrodynamics. Together they determine all Green’s functions and, in particular, all S-matrix elements to any order of perturbation theory. We must now choose the functions Bi, Ci, Di so that the integral equations derived above and the ones to be derived below define functions which satisfy the boundary conditions described in Section II. We shall show elsewhere (2) that the same equations are derivable from the BPH theory so that they have as iterative solutions the conventional renormalized perturbation expansion of quantum electrodynamics also described in Section II. C. CONDITIONS ON AUXILIARY

FUNCTIONS

We shall begin by determining the functions Bi which specify the current operator P[VKh(x).We must first derive integral equations for n, p, and x. Although we are not particularly interested in these latter functions, we need to know something about them in order to determine the Ci from Eqs. (4.7), (4.8), and (4.9). Since these functions are not directly connected with observables, however, there are no a priori conditions which they must satisfy. It is clear from Eq. (4.7), for example, that the values of ii(k) and its first two derivatives at k = 0 can be chosen arbitrarily, provided we suitably choose C, , C, , and C, . This is, of course, a manifestation of renormalization invariance-now further complicated by the interconnections between the current equations (3.6) and (3.7). Likewise, we can see from Eq. (4.8) that F(q, O)lpZn can be given any value provided C, is appropriately chosen. This is in virtue of the condition

Gd4, w&z = &L

>

(4.15)

which follows from (2.14) and (2.17). Thus any constant added to f can be absorbed into C, since only the sum C,f + CJ’ occurs in (4.8). Of course,

140

BRANDT

since C, also occurs in n and X, etc., questions of consistency arise. These will be discussed below. Finally, consideration of Eq. (4.9) suggests that ~@OOO) can be given any nonvanishing value since any multiplicative constant in it can be absorbed into C, . We shall, therefore, impose definite conditions on fi, p, and a to resolve this arbitrariness. Any other consistent conditions would be equivalent to a finite renormalization of parameters. We thus define the functions fi, p, and a so that they satisfy n(o) = aIT &I,

= aan

= 0,

w&m = 0,

(4.16) (4.17)

0, 090) = x,

(4.18)

and m,

where x;z is the tensor, symmetric in its upper and in its lower indices, given by x$

= gUolgy’gK’ + 5 terms.

(4.19)

The consistency of these conditions with the current Eq. (3.7) and with the requirement that Eqs. (4.7), (4.8), and (4.9) define the conventional 17, II, and X will be established in the course of this section. Their consistency with gauge invariance will follow from the analysis in Section V. We note here that the condition (4.16) is really quite harmless since C, , C, , C, only occur in Eq. (4.7). C, and C, , on the other hand, occur in all the X,,, functions and so the consistency of (4.17) and of (4.18) are not immediate. Condition (4.17) corresponds to the absence of a physical &coupling since it implies that the electron cannot directly couple to the photon by means of such a vertex. Condition (4.18) corresponds to our choice of X = 1 for the four-photon coupling constant occurring in n, etc. There exists one further ambiguity (or, better, invariance property) in Eqs. (3.4), (3.7) which amounts to the arbitrariness of the vacuum expectation value (‘Pi) in our chosen gauge. Below we shall choose (: $a(4 Ym>:> = 0

(4.20)

in order that (y(x))

= &(:

a&(x) #j(x):) = 0.

(4.21)

Thus it is natural to define (2P(X))

= 0

(4.22)

in our gauge. It then follows from Eq. (3.7) that By([)

= 0

(4.23)

141

GAUGE INVARIANCE

since (A”(x) A”(y) AA(z)) = 0

(4.24)

by charge-conjugation invariance. The mutual consistency of (4.20) and (4.22) follows from (3.6) and (3.7) by induction since a given : $& : or : AYAKAA : only involves lower order : $& : and : AYAKAA :. Finally, we shall impose the natural requirement that ‘9PA(x) be totally symmetric in its indices. D. DETERMINATION

OF THE Bi

We are now ready to derive an integral equation for n. To motivate our discussion, let us consider the general Feynman diagram for If in Fig. 2. It can be decomposed, giving rise to the two classesof diagrams shown in Figs. 5(a) and 5(b). In (a), X representsthe (renormalized) proper part described in Fig. 1. The function corresponding to diagram (a) will thus have a nontrivial k-dependence. The (renormalized) function corresponding to (b), on the other hand, will simply be a polynomial of the form (a + bk + ckk) and will therefore be completely cancelled in accordance with the condition j4.16). This means that, effectively, diagrams of type (b) do not contribute to fl, their effect having been relegated into C, , C, , and C, . It is convenient to make a corresponding decomposition of the functions B, , B, , and B, . We write Bi(f, 0 = &(t, 5’) + k(5, 5%

i = 2, 3,4,

(4.25)

and in momentum space (4.26)

Bib P’) = &(P, P’> + &(A $1.

The &‘s will be chosen to remove the diagrams of type (b). We now use the current Eq. (3.4) and the definitions (4.3), (2.7), and (4.2) to derive the relation pVKA(x, y) = ):p4[KuuKA(x + (, x, x + f’, y) - B;:([, - B$([,

4’) iDarA(x- y)

f’) i8DD”A(x- y) - B&Y&,(‘,4’) i~‘~4D”“(x - y)

- B;:;,,(& 8’) @B”A(~- y) - B;;;([,

(a)

5’) V;;(x - y)],

(4.27)

(b)

FIG. 5. The two classes of diagrams corresponding to l7. The diagram corresponding to X is IPI.

x+E x+& x+E x+& P -o-

142

BRANDT

V

X

FIG.

tc x+6'

;

=x<;

Y+x

I

6. Decomposition

\

k

y+

x +

x+6'

of the general diagram corresponding

+x

x+('

x

-o-

Y

x+6'

to WvK”(x + 5, x, x + 5’,y).

where IPyw,

x, y, z) = (A”(w) A”(X) AK(Y) AA(z)).

The diagrams corresponding to Kcan be decomposed as in Fig. 6 so that, calling the connected diagrams T, we have iPKA(X + 5, x, x + 5’, u) = -Dq[) DKA(X - y + 5’) - Dyx - y + 5) Dyq-6’) - DfiK(c$- 5’) D’“(x - y) + TII”~~(x + 5, x, x + c, y).

(4.28)

The first three terms in (4.28) correspond to diagrams of type (b) and are to be exactly cancelled by the & , i = 2, 3,4. Thus, in view of 1I$ 13D”“(f) --f-4

0,

(4.29)

g,‘” + iD”“(e - f’) g,‘,

(4.30)

we must have @F([, 5’) = iD”*(f) g,” + iD*“(-e)

%X5,

f’) = iD”“(S) &‘% + iD’“(-P)

%kiS,

5’) = ; D““(5) g,“5;t; + f W-5’)

gE”& , gol%tv .

(4.31) (4.32)

Conversely, (4.30)-(4.32) are consistent with the dimensionality and invariance requirements. We now transform Eq. (4.27) into momentum space, use the definitions (4.5), (4.6), and (4.28) [with T = DDXDD], and take the 5 --f 0 limit inside the integrals to obtain n”sv8(k) = -i i

I [D,“(p) Df(k - p - p’) D,Y(q) X”‘““(p, k - p - p’, p’, -k) P !D’

+ eBy6(p, p’) - i&r’(p,

p’) k” - &E8(p, p’) k”k”

- iB;;;K(p, p’) fijpyx8(k) + B:$‘(p, p’) II;:(k)]

(4.33)

GAUGE

143

INVARIANCE

as our integral equation for fi. The gauge invariance condition to simplify the first term in the integrand to *&‘(A

P’, k) = D(p)

D(k - p - p’) D(p’)

A?&“(p,

(2.22) allows us

k - p - p’, p’, -k), (4.34)

where we have written (4.35)

DLL”(k)= g““D(k) + k%“D’(k).

Using (2.16) and (2.26) the condition

(4.16) now gives us (4.36)

ly’“(p,

p’) = 0,

&?(P,

P’) = --i gfi

&B,‘(P,

P’) = $,

p’, k) lbzO,

P”‘*(p,

Y& XaBy6(p, p’, k) IIs ,

These values, together with (4.30)-(4.32), completely determine B2 , B3, and B4 . We can further derive integral equations for f and a from the current equation for ‘$I. For f we find isf!“(q,

4 = --i I

I [~$‘(P, 2, B’

+ %!XP,

P’; q, k) -

K%P,

p’) E;*(q,

VI,

P’) Gdq,

k)

(4.39)

where T;?‘(p,

P’; q, k) = R”(P)

&%

- P - P’) &YP’)

%‘XP,

k - P - P’, P’; q, W.

(4.40)

Here B is the function pictured in Fig. 4. We now require that the function p defined by (4.39) satisfies the condition (4.17) in order to determine B6 . Using (4.15), we find B,“%P, P’) = --@~%(p,

p’; q, Wd-m .

(4.41)

Since %“(p, p’; q, 0) is not a function of simply g, however, this expression is not immediately well-defined. We shall define such expressions as f Ic L&(P> k) + gdp,

to mean that one first integrates &(p,

Wlgml

k) + gij(pl

, k) (assumed integrable) and

144

BRANDT

(a) FIG. 7. Decomposition

(b)

(c)

of the general Feynman diagram corresponding

to 2.

then sets j1 = m, p12 = m2. This meaning will be ascribed to all quantities having the form of Eq. (4.41). Next we consider the function 2. As shown in Fig. 7, there are three distinct classes of diagrams contributing to its general diagram. The first, diagram 7(a), is the lowest order contribution x, defined by Eq. (4.19). The second class, which involves the proper part X, corresponds to diagrams of the form of 7(b). To characterize the third class, diagram 7(c), we have introduced a new function Y$k which corresponds to IPI diagrams with six external photon lines. Y must satisfy the gauge-invariance conditions kl,!q:(kI

)..., k,) = *** = k,“‘Y,“~~(k, ,...) k,) = 0,

(4.42)

and hence Y(0 ,...) k6) = *a- = Y(kI )...) 0) = 0.

(4.43)

We shall denote the function corresponding to diagrams of type 7(b) by k, , k, , k, , k4) [each term of which is proportional to a four-dimensional a-function S(p + ki), i = 2, 3, or 41 and the function corresponding to diagrams of type 7(c) by &(k, , k, , k, , k.J. It then follows from (2.26) that

&(p;

x2(P;

0, 0, 090)

=

(4.44)

0

and from (4.43) that Qo,

0, 0,O) = 0.

(4.45)

With these definitions, which can, of course, be introduced of Feynman diagrams, the current equation for ‘% leads to

= -1

without the use

i C g,“gvBgK”&p + kJ %P’ + k2) ‘JJP 9, 1 perm

- i ,Fm g,“~!(k + D,“(P) D:(k,

- P - P’) R”(P’) X’Xk, - P - P’) &YP’) YXP,

- i%%p, p’)[x~yS=+ %XP;

- P - P’, P’, k3 , k4) &P + k3 kl - P - P’, P’, k, , k, , k4)

kl , kg , k3 , k4) + %%(kl , k2 , k3 , &)I

+ %J’(P, P’) JGuv+c(k~ 3k, , k, >kJ 1

(4.46)

145

GAUGE INVARIANCE

as our integral equation for fif. The meanings of the permutation sums are apparent from consideration of Fig. 7. Now the first term in (4.46) is simply x;k . Hence, using (2.26), (4.43), (4.44), (4.45), and X’(0, 0, 0,O) = 0, which essentially follows from (2.26), the condition

(4.47) (4.18) is seen to imply that

B;:~AP,P’) = 0. J-.i 9 P’ In the present context, this is equivalent to

exP,

P’) = 0.

(4.49)

The point is that the integral SJB, is not divergent [or, in x-space, lii

hz B5(5, 5’) < co]

and so the subtraction term involving B5 is not really needed in Eq. (3.7) to define a finite current operator a(x). This can also be seen from consideration of Fig. 7. The term B, in (4.46) represents the overall subtraction to the general Feynman diagram corresponding to 2. Diagrams 7(a) and 7(c) do not contain overall subtractions whereas one such subtraction is required for 7(b). If this subtraction is made at the point k, = k2 = k, = k, = 0, however, the subtraction term vanishes by virtue of Eq. (2.26). It follows that this subtraction is not really needed and hence is finite in general. The choice (4.18) thus amounts to omitting this subtraction, in accordance with Eq. (4.49). E. EQUATIONS

FOR AUXILIARY

FUNCTIONS

We have thus succeeded in using the conditions (4.16)-(4.18) and (4.22) to determine all of the subtraction functions B, ,..., B, . We have, furthermore, derived renormalized integral equations for n, p, and x. To summarize our results, we rewrite these equations as l?““(k)

= -i 1, i,, [S?‘*(p, p’, k) - i&f’*(p, - &p(

p, p’) k”k’ + B::;( p, p’) 17,:(k)],

p,t$(q, k) = -i 1 1 [~$(P, B 2)’ 595/52/I-I"

p’) k”

P’; q, k) + B%‘(P, P’) Gdq,

Ml,

(4.51)

146

BRANDT

and

~Xk,%k,) =xi::- s .’

-i

ISP

9’

@‘$X P’ ; k,k,k,k,) [‘X:X

P, P’ ; W&k,)

+ B%‘( P, P’)

Gw,W,WQl, (4.52)

where .% is defined by (4.34), %” by (4.40), 7Y by ~$XP’;

U&&J

=

C g,“W, perm

+ k, - P’) D(P’)

J?:(k

+ kz - P’, k, , h),

(4.53) +I by %%P,

P’; W&&,)

E D(P) Wb

- P - P’) D(P’)

Y%P,

k, - P -PI,

P’, k,k,k,),

(4.54)

& by (4.37), 8, by (4.38), and B, by (4.41). The general function satisfies a similar equation. F. DETERMINATION

xr;;(k,...)

OF THE Ci

We shall now impose the properties of conventional quantum electrodynamics discussed in Section II in order to derive expressions for the singular functions c 1 ,..., C, which specify the electric current operator j”(x), defined by Eqs. (3.3) and (3.6).12 We begin by requiring Eq. (4.21), which is a consequence of charge conjugation invariance, to hold. It is convenient, in fact, to require that the stronger condition (4.20) holds. Using our gauge definition ((A) = 0) and our normalization (4.22), the requirement (4.20) leads essentially to Ci‘(8 = <6(x> y“#(x + 0) = i tr y”G(f)

= iJ”(S).

(4.55)

Here we have deGned J%V = tr rW0

= -K&4

(4.56)

y+Kx + 5)).

The choice (4.55) is consistent with the required singularity In momentum space we have

structure of C,(t).

C,“(p) = i tr y”G(p) = U”(p).

Now let us use the integral Eqs. (4.7),’ (4.8), and (4.9), the conditions

(4.57)

(2.16)-

147

GAUGEsINVARIANCE

(2.18), and the normalizations (4.16H4.18) to determine the singular functions c 2 ,..., C, . In view of (4.16), (2.16) requires that

G”(P)= i tr y’G(z4r”(p, 0)G(P),

(4.58)

(Y(p)

= --he tr y”G(p) I’“(p) k) G(p - k)l,=, ,

(4.59)

= -

(4.60)

and c?b)

Similarly,

i a,‘%’

tr y”G(p)

P( p, k) G(p - k) Ikzo ,

using (4.17) and (4.15), we see from (4.8) that C&(p) = ie-’ tr y”G(p) Kdp, q, 0) G(~>la=, .

(4.61)

Finally we consider Eq. (4.9). Since x$z .IS symmetric in its lower indices (see Fig. l), we can assume that Q“” is symmetric in p, V, K. This also follows from consideration of Eq. (3.6) since : A,A,AA .* is symmetric. Thus, using (4.18), (4.19) and (2.18), we have C,““‘“(p) = f tr y”G(p) 3s”s(p, 0, 0, 0) G(p),

(4.62)

These results, together with (4.57), give us the complete list of singular functions Ci. G.

DETERMINATION

OF THE

R,

Having determined 2I(x) and j(x), it only remains to determine the currentf(x) given by (3.5) and (3.8). Imposing the condition (2.15) on the integral Eq. (4.11) gives us R,(k) = y”Gtp + W F(P, k) &(k)lfi=m

(4.63)

3

and

Y‘Rdk) = & y”W + k) ptp, k) D,,(k) lBzm. Likewise, the condition

f&W

(4.64)

(2.17) applied to Eq. (4.12) yields

= y”Gtp + k) Q%J, 0, k) &Wl+n

.

(4.65)

This completes our derivation of the Ri or, by (4.13), the Ds and, in fact, of the entire collection {Bi , Ci , Di} of subtraction functions.

148

BRANDT

H. CONVERGENCE AND CONSISTENCY We are now in possession of the complete set of integral equations. The derived Eqs. (4.7), (4.8)-(4.12), (4.14), (4.50)-(4.52), etc. for 17, A, X, A’,,,, 2, r,,,, fi, F, 8, and XT,, with the specified subtraction functions Bi , Ci , Di together with the defining relations (2.8), (2.9), etc., are capable of completely determining a (perturbation) theory. There remains, however, three important conditions which must be satisfied in order that this theory be quantum electrodynamics. The first is that with the specified subtraction functions all the integrals introduced must be convergent. The second condition is that all the integral equations must be mutually consistent so that a unique perturbation expansion for the defined functions is possible. If these two conditions are met, then the requirements (2.15)-(2.18) will be satisfied by construction. There remains, however, the requirements (2.22), (2.23), (2.24), (4.42), etc., of gauge invariance. The satisfaction of these requirements is the third condition on the theory. It is shown in B that the first two conditions are satisfied in the theory defined by our integral equations. This is accomplished by outlining a derivation of these same equations, with the same functions Bi , Ci , Di [given by (4.23), (4.25), (4.36)-(4.38), (4.41), (4.49), (4.57)-(4.65), and (4.13)] from renormalization theory. Thanks to Bogoliubov, Parasiuk, and Hepp, this theory is known to be completely consistent and to yield finite and unique expressions for all the Green’s functions. The finiteness and consistency of our theory will then follow. However, the renormalization theory certainly does not guarantee that the requirements of gauge invariance will be satisfied. If some special regularizations, e.g., Pauli-Villars, is used to ensure gauge invariance, then appropriate limits cannot be taken inside integrals so that regularization-free integral equations cannot, in fact, be derived. We must, therefore, work with a completely general regularization, in which case involved arguments are required to rigorously establish the gauge invariance of the theory. These arguments will be presented in the next section. In Section VI we shall show that the complete gauge invariance of the theory is equivalent simply to the gauge invariance of our field equations. This fact is one of the primary advantages of a formulation of quantum electrodynamics based directly on the field equations.

V. GAUGE

INVARIANCE

OF THE, THEORY

Our purpose now is to show that the formulation of quantum electrodynamics presented in Section III satisfies the requirements of gauge invariance. We shall show, in particular, that Z and A, as given by (4.11) and (4.12) obey Ward’s identity (2.24) and that all the proper parts corresponding to. diagrams with two

149

GAUGE INVARIANCE

external electron and I external photon lines satisfy similar identities. Then we shall show that 17 and X, as given by (4.7) and (4.9), satisfy the conditions (2.22) and (2.23). This will be seen to be a consequence of the Ward identities. Many of the arguments (given in the Appendix) will be rather involved, indicating the very profound role played by gauge invariance. We shall begin by establishing the Ward identities.12 Previous proofs have all been of a formal nature and do not guarantee that the identities will hold in each order of renormalized perturbation theory. The original derivation of Ward (25), as well as its generalization by Kazes (26), were based on the (formal) unrenormalized perturbation expansion. Since renormalization was not explicitly taken into account, integration variables had to be translated in quadratically divergent integrals. This procedure gives rise to surface terms which were not explicitly considered. Other derivations, including the original derivation by Takahashi (28) of the generalized identity (2.24) and the generalizations by Nishijima (27), Chang and Mani (28) Rivers (29), and Kroll (30) employ numerous equal-time commutation relations. But none of these commutation relations have been established in all orders of perturbation theory14 and, furthermore, many involve the infinite (in perturbation theory) renormalization constants. The Bogoliubov-Shirkov (20) derivation, while more rigorous in nature, requires the special Pauli-Villars regularization. We begin our treatment by considering the Eqs. (4.11) and (4.12) for Z and II. The presence of 52 in the II equation prohibits a simple induction proof based on the two equations alone. Likewise the equation for Q involves the function 5. We are thus led to consider the infinite set of coupled integral equations for the functions @Tl”‘a*(pql -a. qT), Y >, 1, corresponding to IPI diagrams with two external electron lines and r external photon lines [see Fig. S(a)], and the functions !q;:;’ “r+1( P41 **. q,+l), r > 0 corresponding to diagrams with two external electron lines and I + 1 external photon lines. These latter diagrams are not required to be IPI but are required to yield IPI diagrams corresponding to Qr’s if the last photon line (qPfl , ++A is attached to the electron line (p) as shown in Fig. 8(b).

qs; ;& r+l

(0)

(b) FIG.

I’ See,however,reference(4).

8. Illustrationof Eq. (5.6).

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150

It is convenient to define further functions. O>“*a*(pql 1-wq7), r >, 1, corresponding to diagrams with the same external structure as those corresponding to Gr but which can be weakly connected by a fermion line although they still cannot have self energy parts on external lines. In order to have equations valid for all r 3 0, we further define @o(p) = -G-l(p)

= WJ) - $ + m

(5.1)

and. (5.2)

@oh9 = G-W. We thus consider the following infimte set of coupled integral equations:

@dp)= iej Ql[%(p,43- &(q1)- <$- 4 &(q1) (5.3) (5.4) (5.5)

Equations (5.3) and (5.4) are identical with (4.11) and (4.12) by virtue of Eq. (5.1) and the relations @p”l = p 1

1

= p

Jp”1 = Q”‘“”

,

(5.8)

2

The remaining equations all have the form (5.6), as can be seen from consideration of Fig. 8. They are all special cases of Eqs. (4.14). Following Kazes (26), we write the generalized Ward identities as k,@f~;::“‘“‘(pkq,

-a* qr) = -e~“‘Llv(,~ql,:** + eG-‘(p)

q,) G(p’) G-‘(P’

G(p - k) 8”:“‘“‘(~

- k)

- k, ql *** q,)-

(5.9)

151

GAUGE INVARIANCE

For convenience we have taken k to correspond to the first photon momentum variable, although a similar equation would hold if we took it to be any of the other r momenta. For r = 0, Eq. (5 9) reads k,P(p,

(5.10)

k) = eG-l(p) - eG-l(p - k).

Using (5.1) and (2.26), this can be written k/W,

(5.11)

4 = eJ?p - 4 - Gp),

which is Eq. (2.24). The ordinary Ward-Kazes identities are obtained from (5.9) by differentiating with respect to k and setting k = 0. From (5.10) we obtain P(p,

(5.12)

0) = e@G-l(p),

or (5.13)

A”(p, 0) = -ePZ(p). We note that since

aqp1 =

-G-l

.auG.

(5.14)

G-l,

(5.12) is equivalent to -eaWp)

= G(P) WP,

(5.15)

0) W-4.

Using (5.12), we find from (5.9) the general relation @$~;-l;“a’(pOq, .*a q,) = @F”“+((pql ... q,) G(p’) I”($,

+ F8(p, 0) G(p) q”‘LYr(pql

0)

.-a q,.) - ei3D8~“‘ar(pql

~1.4,). (5.16)

(a)

(b) FIG. 9. Illustration

of Eq. (5.16).

(c 1

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BRANDT

This equation is illustrated that qT;““‘(pOql

in Fig. 9. It easily follows by graphical considerations

*.a q$ = r’(p,

0) G(p) P;“‘fpql...

q$ - eaP8Q(1~;.*a~(pql..a q,) (5.17)

and @$yypOq,

**a qr) = -ea,B@y”‘(pql

. . . q$.

(5.18)

Let us observe here that, conversely, Eq. (5.16) can be derived from Eq. (5.18). Indeed, any contribution 0: to 0, can be written as @:(P,...) = @:‘(P,...)

GAP - a) @:)(P - 41 ,...I

x GAP - q!sJ*** GAP - qs--1)@:‘(P - 49-1 ,..A

(5.19)

for suitable @t’ with Zri = r. Therefore, in view of -eZOD~‘(p

- qfA1 ,...) = @f$(p

- qiB1 , 0 ,...)

and --~“G&

- qd = GAP - q&?‘)

GAP - qd,

it follows that -eP@~(p,...) contributes to O,.+,(p,...) a sum of terms corresponding to diagrams of the type shown in Fig. 9(c), and it is clear that all such contributions to a,+,(~,...) arise in this way. Since the only other contributions to Or+l correspond to diagrams of the type 9(a) or 9(b), (5.16) follows. Similarly, from the Ward-Takahashi-Kazes identity (5.9) one can deduce the relations

- eG-‘(p)

G(p - k) LC?~““~(~ - k, q1 **a q,),

(5.20)

and kB@fTi-:“‘ar(pkq, ... q,) = e@F”‘“‘(pll;

a** q,) - e@2”‘OLl(p - k, ql ..+ qr).

(5.21)

The latter equations are the Chang-Mani (28) identities. Equation (5.9) is, conversely, an easy consequence of Eq. (5.21). In the Appendix we first establish that the ordinary identities (5.16)-(5.18) and the generalized identities (2.24), (5.9), (5.20), and (5.21) hold in each order N of

GAUGE

153

INVARIANCE

perturbation theory. The same method can be used to derive similar identities involving functions corresponding to diagrams with greater numbers of external electron lines.15 We then use the generalized identities to show that kUflUu(k) = 0 and kl.X01~~*(k&2k3k4) = kzaX”e~S(***) = k,Xa@y8(***) = k48XaB?‘s(***)= 0. The method can be used to show that the general function XF;,j’“..fk,..., q,...) corresponding to diagrams with (r + I) external photon lines satisfies the gauge condition qJ$‘“-(k

,..., q ,...) = 0.

(5.22)

This, together with the previous proof that the functions Z, A, X;;;;., s >, 1, satisfy appropriate generalized Ward identities, shows that the theory defined by our integral equations with the specified values of the subtraction functions Bi , Ci , Di satisfies all of the usual requirements of gauge invariance. It also follows that the corresponding renormalized theory defined in terms of Feynman diagrams is gauge-invariant. In the next section we shall see that this gauge invariance is equivalent to the gauge invariance of our field equations and thus understand why these gauge conditions are satisfied.

VI. GAUGE

1NVARIANCE

OF THE FIELD

EQUATIONS

In this section we return to the configuration-space formulation of our theory given in Section III. We want to investigate the extent to which the gauge invariance properties of quantum electrodynamics are reflected in the currents of the theory. The result we obtain is the best one could hope for. We will show that the invariance of the field equations (3.1) and (3.2) and the induced behavior of the currents (3.3) and (3.5) under the c-number gauge transformations (2.19)(2.21) is equivalent to the gauge invariance of the theory.16 We must immediately emphasize that these statements, as well as similar ones below, are to be taken within the context of the formalism introduced in Section IV. Thus the above equivalence holds under the assumption that the subtraction functions Bi , Ci , Di exist such that the currents and integral equations are finite and consistent. It is also to be assumed that these subtraction functions, as well as all Green’s functions, have singularities (or, in momentum space, high-energy behaviors) determined (within logarithms) by the dimensional arguments described in Section V. These assumptions have been verified in previous sections. I6 These identities are given in Eq. (5.1) of reference (29). I6 In this section the Lorentz gauge condition q a: = 0 shall not be imposed since we shall be concerned with operator identities.

154

BRANDT

A. BEHAVIOR OF C,(t) We note in particular that C,(f) - 5-l and C,(e) - 1 for t - 0. (The precise behavior will be worse by factors like log f”.) In momentum space this means that C3(p) Y I p F3 and C,(p) - I p l-4 for I p I -+ co. Let us consider the decamposition (A.43) of Cr(p) into its symmetric and antisymmetric parts in Y and LY. The symmetric part is indeed seen to behave like 1p 1-3 since J”(p) N I p I-1, As we shall now show, however, the antisymmetric part Cr(p) of C,“‘“(p), given by Eq. (A.41), actually behaves like I p l--4 due to a cancelation between the two terms in (A.41). We note that each term behaves as I p /-3 since G(p) N I p I-1 and G~rY(p, k)L,, -

I p 1-l.

We shall make use of the general form for A(p, k) allowed under the restriction, of the generalized Ward identity (5.11) and of charge conjugation covariance as determined in the appendix of the paper by Kazes (26). We shall refer to equation (A.n) of this appendix by (KA.n). Herep’ = p + k so that P = p’ + p = 2p + I$ and Q = p’ - p = k. We shall denote [(p, k = 0) by /and &&(p, k)l,=,, by 8 k From the general form (KA.l) for A(p, k) we find

where use has been made of the fact that, as a consequence of charge conjugation covariance, & is (even/odd) under interchange of p and p’. Since the leading high energy term in G(p) is proportional to $ and since tr y$u$ = 0, we see that only the first two terms in (6.1) will contribute to the leading term in C3 ., Next we shall make use of the restrictions (KA.3)-(KA.6) imposed on the Ps by the generalized Ward identity (5.11). Writing G-~(P) = $A(p2) + mB(p2),

Eq. (KA.4) implies that &(A = p.A’(p2) and Eq. (KA.5) implies that 2p,/.‘. = p,A’(p2). Hence Y.a&i + 2P”YLA = (Y”P, + YLYP”)A’(P’%

which is symmetric in v, cy and therefore does not contribute to (6.1). Thus the leading term in C, is seen to vanish so that c&p) - I p 1-4; that is, C3([) - 1, Since the theory is finite with such a C3, it is clear that it would not be finite if C,(@ - t-l. Thus this property of C3 is required for consistency and will be included in the underlying assumptions relevant to this section. We note, incidentally, that by Eq. (KA.8) we have 8. = 0 and by Eq. (KA.~) we have a,J2; = -2p& = 0 so that we obtain simply e-WA(p,

0) - adup, o)i = 2~&(~,

0).

(6.21

GAUGE

Furthermore,

INVARIANCE

155

since tr p(uj

+ b) u%z$ + b) = ab tr(y$Y + Y+Y$) = 2ab tr p%~‘~ = 0,

we actually have

He shall not, however, have to make use of this fact in the present section. We shall see in B that (6.3) is a consequence of charge conjugation covariance alone. j3. COMPARISON

WITH CONVENTIONAL

APPROACH

Having made the meaning of the equivalence mentioned above clear, let us ask why we might expect it to exist. Previously one worked with the unrenormalized field equations (1.7) and (1.8) rather than (3.1) and (3.2). These equations involve the bare parameters and undefined products of bare fields. Formally, however; the equations are invariant under the gauge transformations (2.19)-(2.21). But when one uses the equations to construct perturbation expansions of the Green’s functions, one finds that these functions do not obey the requirements of gauge invariance unless they are explicitly made to do so during the course of renormalization. This ad hoc procedure was justified on the grounds that, in a “singular” theory such as quantum electrodynamics, invariance properties of the equations need not be maintained in their solution. We know, however, that the trouble lies in the fact that the Eqs. (1.7) and (1 .S), to whatever extent they, can be given a meaning, are not really gauge invariant. This is because the need for renormalization effectively means that the operators on the right,sides of.Cl.7) and (1.8) are not the true current operators for the theory. Thus the renormalization procedure was required to make up for the fact that (1.7) and (1.8) are not, in fact, gauge invariant. Our Eqs. (3.1)-(3.8), on the other hand, while not obviously gauge invariant, are meaningful and immediately lead to a finite theory without the need for additional assumptions. Thus one would expect that if these equations are chosen to be gauge invariant, the resulting perturbation theory will also be gauge invariant. C. COMPLETE EXPRESSION FOR ELECTRIC CURRENT Before proceeding, we must fill a gap existing in the previous sections. We have omitted the : A2 : terms in j since, as will be shown in references (I) and (2), they do not contribute to our previous equations., We shall :need them now,

156

BRANDT

however, and will therefore briefly discuss how they are handled. The complete electric current operator is the limit of

- Cr : A,A, : - Cp%$ : A,A, : - C~‘%lVKI\ - tr C,@ : I,$$ :].

(6.4)

Here Ci means C,(E) and all the field operators are evaluated at the point x except that #+ means I@ + 5). The two new : A2 : terms serve to renormalize the primitive divergence YaSy(q,k) corresponding to diagrams with three external photon lines (see Fig. 1). The renormalized Y then vanishes by charge-conjugation invariance and therefore has been consistently omitted from the beginning. In a rigorous treatment, however, only this renormalized Y vanishes and thus the values of C, and C, needed to accomplish this must be specified. We find C?(p)

= +

aua”J”(p)

Cyyp)

= $ av”a”J”(p),

(6.5)

and

where use has been made of Eq. (A.51). Use of the appropriate identities satisfied by 0 then shows that qpryq,

ii) = k,Yfly(q, k) = 0.

generalized Ward (6.7)

On the other hand, since Ylr6y is finite, the usual argumentsl’ can be applied to show that it vanishes. We shall give a more elegant proof of this in B. D. GAUGE INVARIANCE OF THEORY

We can now give a unified account of some of our previous results. We have used the conditions (2.15)-(2.18) and (6.7) to determine the values of the functions C, , C, , C, , C, , C, , C, , R, , and R, . These are given in Eqs. (4.58)-(4.60), (4.62), (6.5), (6.6), (4.59), and (4.60). It then followed that our integral equations (4.7), (4.9), (4.1 l), (4.12), etc., defined functions 17, Y, X, II, L’, etc., which satisfied the conditions (2.22)-(2.24), (6.7), etc., of gauge invariance. We saw that as a consequence of the values of the Ri , we could write the relation (A.lO) and that as a consequence of the values of the Ci , we could write the relations (A.42)(A.44), (A.31), (A.32), (A.53), (6.5) and (6.6). The Fourier transforms of these relations are W,(O = W3 yp w3) I’ This is a special case of Furry’s theorem (31). For a modem account see, for example (20).

GAUGE INVARIANCE

157

and (6.9) (6.10) (6.11) (6.12)

c?(g) =

- $

Cs’K”(S) = - 7

f’5”J”(.f),

(6.13)

f”Fg”J“(g),

(6.14)

where (6.15) x&xJyA(g)

= 0.

(6.16)

In (6.8) we have used Eqs. (4.13) to relate the Ri and Di . It was precisely the relations (6.8)-(6.16) which were responsible for the gauge invariance of the theory. That is, gauge invariance holds provided (6.8)-(6.16) are valid, independently of the values of D1, D, , D, , C, , Ca, Ca, or C, , assuming they are such that the integrals are all finite. Looking over the calculations of Section VII, however, we easily observe that (6.8)-(6.16) are necessary, as well as sufficient, in order that our functions satisfy the requirements of gauge invariance. In particular, (6.8) is equivalent to the generalized Ward identity k - A(p, k) = eZ(p - k) - eZ(p) and hence to all of the generalized Ward identities, and, assuming the Ward identities, (6.9)-(6.13) are equivalent to k * I7 = 0, (6.12) to k * X = 0 (or, more generally, to k * X,,, = 0), and (6.5)-(6.6) to k * Y = 0. E. GAUGE INVARIANCE OF FIELD EQUATIONS

Now let us consider the renormalized field Eqs. (3.1) and (3.2).12 The necessary and sufficient conditions which the currents j“(x) and f(x) must satisfy in order that the field equations be invariant under the gauge transformations (2.19)-(2.21) are easily seen to be (6.17) jW -jW and f(x) --f e+a(z)f(x) + e(+,a(x)) e-ie+b(x). (6.18)

I68

BRANDT

We shall refer to (6.17) as the gauge invariance 0f.j“ and to (6.18) as the gauge covariance off. Our task is to determine the necessary and sufficient conditions which the functions Ci, Di must satisfy in (3.6) and (3.5) in order that (6.17) and (6.18) hold. We will then compare these conditions with (6.8~(6.16). F: CONDITIONS ON THE Di We begin with the simpler case of the current Eq. (3.5), (3.8) forf(x). We want to find the relations the Di’s must satisfy in order that f(x) transforms as (6.18) under the gauge transformations (2.19)-(2.21). In the usual inductive sense, we must have e{[A(x + 5) + 4(x + 0-J eeiearfz)$(x)- Dlemieotj- D,u8’e-ieaa,h - D,emi”“f _ L),&e-“e”~} z

where we wrote & = rQ,ol(x). {ieD,&)

e-ieoff+ e$e-iea#,

(6.19)

Using again (3.5) and (3.8), (6.19) reduces to

c?(x) erniea(

- D&) d(x) e-i’a(E’#(x)) z

0.

In order for this to hold for all a(x), we must have je4,(0

= &@

‘Ye.

(6.20)

Conversely, if (6.20) holds, then .f(x) will obey the transformation, law (6.18): But (6.20) is exactly the condition (6.8) equivalent to the generalized Ward identities. It follows that these identities are equivalent to the gauge invariance of the Dirac equation or, equivalently, to the gauge covariance of the current operator f(x). G. CONDITIONS ON THE Ci Next we turn to the definition (6.4), (3.6) of the electric current operator j“(x). We must determine the transformation properties of each term in (6.4) under (2.19)-(2.21). The first term I+?(X)y@$(x + 5) will acquire the factor

where we have written &Y’ = gM,,or(x) etc. and have not written terms of order g* or higher since I,(” I 4(x> 9(x + 6) z 0.

GAUGE

159

INVARIANCE

As a consequence of the definition

: A,(x) Am: = l&wtx

+ 5) A,(x) - am,

(6.22)

the operator : A2 : is seen to transform under (2.21) as : A”(X) A,(x): -

: A,(X) A,(x):

+ A,(x) 44

+ 44

A(x)

+ &>

4~).

(6.23)

Here we have written a,(x) = &X(X), etc. We note that this is the same transformation law the ordinary product A(x) A(x) would have if it were well defined. This is due to the absence of A and : A2 : counter terms in (6.22). We also note that the vacuum expectation value of : A2 : transforms according to (6 : A”(X) A,(x):)

= a!“(X) a,(x).

(6.24)

Next we consider the term : AAA :. To determine its transformation property we must use its defining Eq. (3.7), which we have seen can be written as ‘W”(x;

(, I’) = A; A”A:,

- Bi$A”*B - B&$,,Aa”.EY- tr BFA : t,h,6:,

- &:‘A’

(6.25)

where B, = B3 + 8, and B, = B4 + 8,. Here B, , B3 , and B4 are given by (4.30)-(4.32) and, according to (A.27) and (A.28), 8, and 8, satisfy

S$(&[‘) xv = 0

(6.26)

and (6.27)

These facts are all that we will need to determine a, = a,~, assuming S : +$ : = 0, we find ‘X(x; f, f’) -

2I(x; E, 5’) + Nx + 8) 44

6%.Under

A, -+ A, f 01, ,

Wx + E’>

+ A(x) A(x + 5’) h(x

+ 5) + A(x + 5) A@ + 5’) Wx)

+ A(x + 5) h(x)

h(x

+ 5’) + A(x) a&x + 89 a+

+ 5’)

+ A(x + 5’) hx(x) h+ - ill(() &(x) - q-f’)

+ 5) + a& + 5) a+) a+ h(x) - D(E - 55’) a&,

+ 5’)

-

iD(-.$‘)(t

iD(f)(f’

- ; D(&$’

. 8) h(X)

-

. q2 &x(x) - f D(-E’)((

* 8) aa

. a)2 aa(

(6.28)

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BRANDT

Now, by (4.29), we can replace --iI@) -iD(!g[aa(x

&Y(X) in (6.28) by

+ [‘) - (C’ * a) six(x) - $(tJ’ * q2 &k!(x)].

(6.29)

The sum of the second term in (6.28) and the first term in (6.29) have, according to (6.22), the finite limit : A(x) A(x) : &L(X) for 5 + 0, 5’ -+ 0. The remaining two terms in (6.29), moreover, cancel the twelfth and fourteenth terms in (6.28). Proceeding similarly with the 0(-c’) &X(X) and D(5 - 5’) &.X(X) terms in (6.28), we find that when contracted with a completely symmetric function, ‘+t transforms as a(x)+

'8(x) + 3 : A(x

h(x) + ~A(x)[&(x)]~

+ [&(x)]~.

(6.30)

Again we see that : A3(x): has the same transformation law as does the ordinary product A3(x). This is in part a consequence of our choosing B5 = 0 [cf. Eq. (4.49)]. We finally note that the vacuum-expectation value transforms according to (6 : k(X) A”(x) /P(x):)

= a”(x) P(X) a”(x).

(6.31)

We are now ready to investigate the restrictions on the C;s in (6.4) imposed by the requirement that j‘(x) be gauge invariant. We shall first determine the forms the Cis must have in order that simply (6j”(x)) = 0 under (2.19)-(2.21). As the basis for the usual induction argument, we can assume that (6 : t&(x) &(x):)

= 0.

(6.32)

This is because the current definition (6.4) forJ@ is just the trace with ys the of current definition for : $& : so that invariance of : I&$$ : implies invariance ofjw and, conversely, invariance of jw implies invariance of : $& : since all of our arguments remain valid if the overall left-hand tr y” were absent. One need only replace all occurrences of J‘(t) below by G(5). Transforming (6.4) and using (6.21) (6.24), (6.31), (6.32), we find

- C(E) %- cm %K- cw %A - Cyq) a,a,- Cyq[) a*a,a,- Cpyf) %%~A I3

(6.33)

where we used the fact that e4G([) +t+O 0. Since (6.33) holds for arbitrary 01(x), by requiring (6ju(x)) = 0 we obtain the conditions that the coefficients of each

GAUGE

161

INVARIANCE

of%9%K , %A3%%2ah~,~, , and

O~,CL,OI~ must contribute zero to (6.33). Taking into account the total symmetry of CL,,and IX,,,, , the first three conditions imply that (6.34) G%> = eS”J”(E), + or%>,

(6.35)

= 0

(6.37)

xJx,x,ck”“A(~) = 0.

(6.38)

C,"y"(5) = f 5"PJ%) and

for some functions C, and C, satisfying XJJyy~) and

Since the symmetry properties of CL,OI,, a,,ol,a, , and OI,OI,OI~, on the other hand, are, respectively, the same as the coefficients of C, , C, , and C, in (6.4), the last three conditions require that essentially Cyy”(c$)=

(6.39) (6.40)

and (6.41) Any other allowed parts of C, , C, , or C, would not contribute to (6.4) which is the only place the C’s occur. Thus (Sj) = 0 implies (6.34)-(6.41). Conversely, if the C’s have the forms (6.34)(6.41), then (Sj) = 0. Since the forms (6.34~(6.41) are identical with the forms (6.9)-(6.16), it follows that simply (Sj) = 0 (with the Ward identities) is equivalent to k * L7 = k . Y = k * X = 0, etc. Together with our previous result, this shows that the gauge covariance off and the gauge invariance of the vacuum expectation value of j are equivalent to the gauge invariance of the theory. As we shall now show, however, (Sj) = 0 is equivalent to Sj = 0 so that complete gauge invariance is really involved. This equivalence requires use of the fact that (j> = 0 so that c,qcg = iP([). In other words, it is “(j)

(6.42)

= 0 in all gauges” which is equivalent to “8j = 0.”

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BRANDT

H. GAUGE INVARIANCE OF ELECTRIC CURRENT Performing gauge transformations on the fields occurring in (6.4), using (6.21), (6.23), (6.30), the values (6.34)-(6.41) of C-C, , C, , C, , and the induction assumption 6 : $4 : = 0, we find

+ F

&$J(2A’ol’

8 + T @[J(3

+ 2Ad

+ 201’01”)

: AA : a’ + 3Aa’d

+ dcia’).

(6.43)

Here we have used the fact that $#+ 15 I4 -fe+,, 0 to exclude terms which vanish as 5 - 0. NOW we return to Eq. (6.4) and use the known ‘behavior of the C,(t) for 5 - 0, including the behavior C,(e) - 1 derived at the beginning of this section, to obtain

for 5 - 0. Substituting (6.44)-(6.46) into (6.43) and taking the limit .$ -+ 0, we find 6j“(x) = 0, thus establishing the gauge invariance ofjp(X). Let us note exactly what was needed for the above proof that Sj = 0. In addition to the forms (6.34)-(6.42) for C,-C, , C, , and C, , we used the facts Gtn

t + 09

~4

5-

07

c&3 f -

0,

J(f) 5” - 0,

(6.47)

for g -+ 0, in order to write (6.44~(6.46). The last three of these relations follow from the general dimensional considerations of Section IV while the first relation was established, using the Ward identity, at the beginning of this section. General considerations only require that C&) 4” -+ 0. We can easily see, conversely, that the gauge invariance ofj+), given by the limit of (6.4), requires the forms (6.34)(6.42) and, furthermore, implies that each of (6.47) must hold. Since (6.47) are required for finiteness, we can still say that, assuming finiteness, (6.34)-(6.43) (i.e., (j) = 0 in all gauges) are equivalent to Sj = 0. As we have just seen, however,

163

GAUGE INVARIANCE

we can even make the stronger statement that, assumingnothing but the general form (6.4), Sj = 0 is equivalent to (6.34)-(6.43) and (6.47). We thus see that there exists an intimate connection between the gauge invariance of quantum electrodynamics and its finiteness. Namely, gauge invariance, in addition to uniquely specifying the parts of the subtraction functions Ci which lead to nongauge-invariant subtractions, also requires that the parts which lead to gauge invariant subtractions be only logarithmically divergent for e + 0. In summary, we have shown in this section that, assuming the finiteness of quantum electrodynamics, its gauge invariance is equivalent to the gauge invariance of the field equations. For the electric current, all that was required for this equivalence was (Sj) = 0. Assuming finiteness and (j) = 0, this was shown to imply Sj = 0. The requirement Sj = 0 was further seento have independent implications concerning finiteness. We finally note that, as a consequenceof the established invariance, the values of the Bi , Ci , Di which were previously determined in the gauge for which (A) = 0 are now assuredto be correct in all gaugesobtainable from this gauge by c-number gauge transformations. The specified values of the Bi , Ci , Di always refer, of course, to the gauge for which (A) = 0. Thus the current (6.4) with the specified Ci is correct in any gauge (A,) = a,a provided C&t) = iJ”(f) = i tr y”G(c), etc., are evaluated in the gauge for which (A) = 0.18 ACKNOWLEDGMENT

I wish to express my gratitude to Professor Paul Federbush for suggesting this investigation and for valuable suggestions, discussions, and criticism.

APPENDIX: A. ORDINARY

FR~~F

OF

GAUGE

INVARIANCE

WARD IDENTITIES

We shall first establish that the ordinary identities (5.16)-(5.18) hold in each order N of perturbation theory. Our proof will be by induction on N and so we begin by showing that if the functions Q and @ occurring in the integrands of I8 This statement is, of course, only true for c-number gauge transformations from our original Lorentz gauge [characterized by Eqs. (3.2) and (2.91 with (A> = 0. Other classes of gauges, such as those obtained from the above one by “q-number gauge transformations” and/or changes in the zeroth order photon propagator (2.5), would require changes in the C,‘s. The forms of the C’s would, however, remain the same. Thus, in one of these new gauges, and in any gauge obtainable from it by a c-number gauge transformation, we would have C,p(e) = i tr ypG(g) with G(5) evaluated in the particular new gauge for which = 0. Requiring invariance under these more general gauge transformations does not lead to additional restrictions on the theory but simply establishes connections between different formulations of it. See Zumino (32).

164

BRANDT

Eqs. (5.3)-(5.6) satisfy (5.17) and (5.18), then the functions Q, defined by the resulting integrals also satisfy (5.18). We start with the distinguished Eqs. (5.3) and (5.4):

A%4 4) = ie J, PWP, 4, W - e-‘Wp, 4) &WI,

64.2)

where %(A W = y”G(p + 4 r”(p + k, k> R,(k), 6”(p,q,

k) = y”G(p + 4 Q?P

+ k ~4

(A.3)

&W,

(A-4)

and R,(k) = %(A k&n

9

(A. 5)

y”&(k)

= %“%(P, k&n

1

w-9

y“&(k)

= <“(A

9

(A.7)

0, WI+

Eqs. (5.17) and (5.18), assumed satisfied by the integrands, become L@‘(p, 0, k) = -e%,“P(p,

k) + Wp

+ k, 0) G(p + k) Wp,

k)

64.8)

and (5.12). It follows from (A.3), (A.4), (A.8) and (5.15) that (au = a,*) -e@%(p,

k) = y”G(p + k) WP + k, 0) G(P + 4 r”(p, k) &c(k) - ey”G(p + 4 a”~(p, k) Uk) = y”G(p + WW

+ k 0) ‘3~ + W WP, W

- ea”p(p, WI &,W = y”G(p + k) Q‘%, 0, k) &c(k) = e”(p, 0, k). Evaluation

(A-9)

of (A.9) at $ = m (in the usual sense) gives -ep&(k)

= y”&(k).

(A. 10)

GAUGE

165

INVARIANCE

Now, using (5.13), (A.9), and (A.lO), we get from (A.l) -e@Z(p)

= ie jk [Flu(p) 0, k) + eyPR,(k) - e-lA”(p, = ie

I

Ic[q(p,

= ie R[@(p, I

0, k) - e-ley”R,(k)

0) R,(k)]

- e-lfl“(p,

0) R,(k)]

0. k) - e-lI’@(p, 0) R,(k)]

(A.ll)

= A”(P, o>,

thus establishing (5.13) and hence (5.12). Now we consider the general Eq. (5.6) whose integrand we assume satisfies (5.17) and (5.18). Using (5.7) (5.17), and (5.15) we get (q = q7+3 -e#2F~“‘*Qq1

. . . qrq)

= Y”G(P + 4) rob + 470) G(P + d Q:‘;‘“% + YWJ + d(-e@

= F$yyp,

+ q, q1 **. q,q) DAY(q)

Q:;;.%J + 4, q1+-.q,q) DAY(q)

(A.12)

0, q1 ... q,q).

Using (5.6), (5.18), and (A.12) gives -e8%:“‘ar(pql

... q,) = ie k [~~;““‘(pOq, s - e-lR(k)

. . . q,k)

0:“:; ’ ‘“‘( pOq, . . . q,)]

which again establishes (5.18). We are now ready to prove that all of the identities (5.16) hold in each order of perturbation theory. We use induction on the order N. For N = 1, the identities are simply -eSG, = GFeyflGF, 0 = 0 ,...) 0 = 0 ,...; which are known to hold. We suppose the identities hold in orders l,..., N and consider Eqs. (5.3)-(5.6) with the left sides of order N + 1. Since all the Q’s

166

BRANDT

and Q’s which occur in the right sides are then of orders
WARD IDENTITIES

We have shown in particular that II and Z satisfy the Ward identity (2.27). The generalized Ward identity (2.24) and its generalizations (5.9), (5.20), (5.21) can be similarly established. For the distinguished Eq. (A.2) we get

qJ%, 4)= ie1 W(P + kW(p + k W - e&p + k) G(P + q + k) T(P + k + q, Ml D(k) - e-W-Yp + q) - dWp)l MW

- ie2I {yG(p + q + k) W-J + q + k, k) W) - 46 - (B + B - 4 R,(k) - e-l% + d &(W = e%9 - ez(p + 41, where we used (A.lO), and for the general Eq. (5.6) we get

q+@r+l(~,-) = iejk W(P + WWP + k-1 - eG-Yp + k) G(P + k - q) %(P + k - q,...)l WI - e-lk@T(p,...>- eQ%(p- q,...)l &WI where we have assumed the integrands satisfy (5.20) and (5.21). An induction argument thus establishes (5.9), (5.20), and (5.21) to all orders. Our proof guarantees that all the proper part functions @,. defined by the iterative solutions of the integral equations (5.6) will satisfy the generalized Ward-Takahashi identities (5.21). The same method can be used to derive similar identities involving functions corresponding to IPI diagrams with greater numbers of external electron lines.15

GAUGE

C. k . U(k)

167

INVARIANCE

= 0

We shall now use the generalized Ward identities to show that k - I7 = 0 and and hence can assume that relevant functions occurring in integrands satisfy these conditions. To establish that k .I7 = 0, we shall need the condition k . X = 0. We shall again use induction

f&+“(k)

Using the induction that p”“(k)

k, = 0.

assumptions, we immediately

ks = -i j

/

9 9,

[-i&s,‘“(p,

(A.14) see from Eqs. (4.48) and (4.34)

p’) k”k8 -

fi&:‘(p,

p’) k”k”k,].

(A. 15)

To proceed further, we note that any function X”(k) which satisfies the condition k,X8(k)

will also satisfy the conditions

= 0,

(a,, = a/ako) X,(k)

aJAW

(A.16)

+ a,&(k)

+ k&J,,J?(k)

+ k,a,W”(k)

(A.17) (A.18)

= 0,

= 0,

and UW,(k)

+ a,a,X,,(k) + a,4Jv(k)

as can be seen by repeated differentiation at k = 0 gives us

+ ksa,V,X*(k)

of (A.16). Evaluation

= 0,

(A.19)

of these relations

X8(0) = 0,

(A.20) (A.21)

and ZZ

0.

(A.22)

Hence k”k,a,X8(0)

= 0

(A.23)

and k~kVk&J,Xs(0)

= 0.

(A.24)

Now we return to (A.15) and use the expressions (A.2) and (A.3) to see that the terms in the integrand have essentially the forms k”k&,

= -iDDDa,X”(O)

k”k8 - iDa,D . DXs(0)

k”ks

(A.25)

168

BRANDT

and k”k”k.&,,

= DDDi3&,X”(O) + Da,a,D

k”kvks

+ 2Da,D

. X’(O) kX’k,

where X6(k) satisfies (A.16), according and (A.24), we see that

. D&X’(O)

k*k’k,

(A.26)

,

to Eq. (2.18). Thus, by (A.20), (A.23),

&B;/“(p, p’) k”k, = 0

(A.27)

i?$“(p, p’) k”kvks = 0.

(A.28)

and

This establishes (A. 14). We shall also need the following information: -Wr

~‘G(P)

P(P,

k> G(P

-

kLo = - 2”a,aa,v[tr fG(p)]

- $ a,V,%r fG(p) P(p, k) G(p - k)],=, = $ a,“aDBa,“[trfG(p)l

+

cy(p),

(A.29)

+ Cy@(p), (A.30)

where k&c,“‘“(p)

= 0

(A.31)

kJcak,cyB(p)

= 0.

(A.32)

and

To derive these relations. we first note that tr y@G(p) P(p, 0) G(p) P(p, 0) G(p) = tr GrYGr”Gyu

= tr y”GPGPG. (A.33)

The first equality is a consequence of the form ‘3~)

=

a# +

(A.34)

d

of G(p), the form UP,

0) =

ay,

+

bpd

+

CP, ,

(A.35)

which follows from the general form of A,(p, k) given by Kazes,ls and the relation

la Reference (26), Eqs. (A.l)-(A.6). Restrictions imposed by the generalized Ward identity are already included here; but this is legitimate since our argument is an inductive one.

GAUGE

169

INVARIANCE

The second equality in (A.33) is a consequence of the trace property tr(AB) Next we differentiate k,jYku&“P(p, Evaluation ak”P(p,

= tr(BA).

(A.37)

(5.10) with respect to k twice to get

k) + t31cvP(p, k) + &“P(p,

k) = e&VdG-l(p

-t- k).

(A.38)

at k = 0 and use of (5.13) gives 0) + iYkAP(p, 0) = e8pALYguZ(p) = --~Y~~flY(p, 0) = -i3apAP(p, 0). (A.39)

Now we use (5.15), (A.33), and (A.39) to get ; aPaa,“[tr y”G(p)]

= - ‘2 L$,Otr y”G(p)

rY(p, 0) G(p)

= 4 tr ~“G(PN.~~“~“(P, 0) + %‘TP, 011G(p) + 4 tr y‘%‘“(p) QP, 0) G(p) + Gtr y‘%(p) I”‘(P, 0) G**(p) = tr y”G(p)

. ~koTy(p, 0) . G(p)

+ tr y”G(p)

r”(p,

0) G,“G(p - 0) + v(p)

= Gab-y“Wd F(P, k) G(P - WL,, + MY,

(A.40)

~?YP) = -4 tr ~WPX&“J~A 0) - %“~‘YP,(31G(P)

(A.41)

where

clearly satisfies (A-31). Equations (A.30) and (A.32) are established in the same way starting from three derivatives of (5.10) instead of two derivatives as in (A.38). We are now ready to consider the integral equation for II. We recall the definition (4.57) of J”(p). By (4.58) and (5.15) we have Cr(p)

(A.42)

= -ieaT”(

By (4.59) and (A.29) we have q(p)

= - ;aw(p)

+ qyp),

(A.43)

where C, satisfies (A.31). By (4.60) and (A.30) we have (A.44)

170

BRANDT

where C, satisfies (A.32). Thus (4.7) becomes P”(k)

= ie j

P

tr y“G(p) rY(p, k) G(p - k) + ea’Y(p) - g k,a”a”J’(p)

+ ~kmkBaaa6aT(p) + k&e(p) +

iCLjo”‘(

P)

n&(k)

+ K&C

- ik,kscyB(p)

P> KG(k)

(A.45)

1.

It then follows from (5.10), (A.31), (A.32), (A.14), and k,,IP(k) for aur induction proof) that kJP”(k)

= ie 1 I-eJ”(p)

= 0 (as the basis

+ eJ”(p - k) + ek,XP(p)

- ik,k.aa’Jy(p) + 5 k&.ksaaaBatqp)~ = ie

11B

- -$ k,k,k,gk,a”aBavaYJ”(p) + ...I.

(A.46)

Since G(p) - l/p, (A.46) vanishes so that kJ7uY(k) = 0. Since flu“ must be symmetric, we also have k,li’~Y(k) = 0 so that UC” satisfies the gauge-invariance condition (2.22). D. k * X(k)

= 0

Our final task is to demonstrate that k . X = 0. Again we shall make essential use of various generalized Ward identities. We begin by observing that the function E occurring in (4,9) is a O-type function and not a Q-type function since any O-function will yield a IPI X in (4.9). Thus the appropriate generalized Ward identity satisfied by E is (5.9), which we write as k 2,P6(~,

k, , k, , k.4

=

eWp)

G(P

- eW(p,

-

k2)

WP

-

k2 , k3 , k4)

k3, k4) G(p - k, - k4) G-l(p

- k2 - k, - k4).

(A.47) Here we have defined W(p, k, , kg) to be the O-function Og8 corresponding to diagrams with two external photon lines. These are illustrated in Fig. 10(a). Thus the first term in the integrand of (4.9) satisfies G(P) kwu DBvs(~, =

eW-4

k, @YP,

, k, , k4> WP h,

k4)

- eG(p - k,) @“(p

G(P

-

k, -

k3 -

k,

-

kd

kJ

- k, , kg, kJ G(p - k2 - k3 - k4).

(A.48)

171

GAUGE INVARIANCE

p-k,-k,

(a)

(cl

( b)

FIG. 10. Decomposition

k,

of the general diagram corresponding

to 8.

We write the identity of the form (5.16) satisfied by E for k, = k, = 0 simply as 38vS = @?‘6GrBf rSG@d _ ea8@‘6,

(A.49)

suppressing the variable p. The corresponding identity satisfied by oVs(p, 0,O) is 0~6 = pGp

+ pig-

- eo”Yp.

(A.50)

It then follows from (5.15) and (A.50) that GWG = e2LP@G. Next we use (5.15) and (A.49)-(A.51) Gpv”G

(A.51)

to get = - e3a8a~a~G.

(A.52)

Substituting this into the expression (4.62) for C, gives (A.53) Now the general diagram of Fig. 10(a) corresponding to 0 receives contributions from two distinct classesof diagrams; those which have the form of Fig. 10(b) and those which do not have this form. We denote by 0 the uniquely defined functions corresponding to these latter diagrams. Thus we have tW(pk3k4) = bB( pk,k,) + @,‘6(pk3k4),

(A.54)

where a’“(~,

k, , kJ = i j” @“YP, q, k, + k, - q) Wq) D(q - k, - kg) X%q, e

k, , kd,

(A.55)

as illustrated in Fig. 10. We seethat oqp,

0,O) = 0.

(A.56)

172

BRANDT

FIG. 11. Structure of the general diagram corresponding

to 8.

We know that each diagram contributing to O(p, k, , k4) will behave as p-l (within logs) for large p. The @ and @ contributions to 0 are, however, further distinct in that the 0 part of & alone guarantees the p-l behavior of @ whereas no such part of @ has this property. 2o Indeed, for any decomposition of 8 as shown in Fig. 11 the blob on the left will generate a p-” behavior with n > 1 and it is only the entire diagram which behaves as p-l. This means that &

3

@(P, k, 2k4)

and

$

4

@(P, k, 7kJ

will behave at least as well as p-2 for large p, whereas &

3

@(P, k, 7k4)

and

&

4

@(P, k, 3k4)

will continue to behave as p-l. We symbolize these facts by writing $

@(P, k, , k4) N P-~,

(A.57)

&

@(P, k, , k4) N p-l.

(A.58)

Equation (A.58) also follows from consideration of an expansion of @ in powers of k and use of Eq. (A.56). Next we consider the integral equation (4.50) for x, which we write somewhat more explicitly as zzf(klk2k3k4)

= ~2: - 2i

s 0 D(q) D(q - k, - kddX:(&kd

+ gv’gC,,W3k4)+ gKBK%k3k4)l- 2i j DWk2k3) - 2i j DDX(k,k,)

- i jj

DDDY(k,k,k,).

(A.59)

Taking kza%, the last three terms in (A.59) yield zero since k, * X(k, ,...) = k, * Y(k, ,...) = 0 so that we get

*OThis is the sense of Weinberg’s theorem. Reference (33), EIq. (20) and related discussion.

GAUGE

173

INVARIANCE

Since x$f only occurs contracted with the completely symmetric function C;“*’ [see Eq. (A.53)], we need not explicitly symmetrize (A.60) and so we can write (A.61) where kd,B,y,“,(k&d

= 6i

j

P

D(q)

Nq

-

k,

-

ka)

(A.62)

LX%WJ.

We are finally ready to consider the integral equation (4.9) for X. We multiply by k, and consider the result obtained by substituting (A.48) into the first term, (A.61) into the second term, and k,$F’~(k,k,k,k,) = 0 (as the basis for an induction argument) into the third term. We further use (A.54) in the first term and obtain k,X”“Y”(k,k,k,k,)

= ie 1 [2Pav8(p, k, , k4) - Fvs(p 2,

- k, , k, , k4)

+ .+“(p,

k, , k4) - ~@‘~(p - k, , k, , k,)

+ G’V4

bx~~ + G‘Yp) k3J&%k3 , Ul,

(A.63)

where SaYG(pk,k,)

= e tr y*G(p)

@(pk3k4)

G(p - k, - k4)

(A.@)

&Y6(pk3k4)

3 e tr y*G(p) &+(pk3k4)

G(p - k, - kJ.

(A.65)

and

We next write WP,

k, 3 ka) - X(P

- k, , k, , kd = -&z

* 8,) .WP,

k, , kd + em*, (A.66)

where, since Z(p, k, , k4) ~p-~, the omitted terms integrate to zero in (5.84). However, in virtue of Eq. (A.57), we have WP,

k, , kJ -

X(P

- k, , k, , ka) = -(kz

* 3,) WP,

0, 0) + *me, (A.67)

where the remaining terms still integrate to zero. Thus, using (A.67), (A.51), (A.54), (A.56), and (A.53), we obtain ie

i PPf?p,

k3 , k4) -

*‘=“?P

= ie 9 [-(k2 * a,) 2P”(p, s

- k, , k3 , k4) + iC”Yp)

0,O) + 6ik2&iB”(p)]

by~~l = 0.

(A.68)

174

BRANDT

We now use (A.55) and (A.62) to write the remaining ie * [~@“(p, J9 = -e

k, , k4) - e”(p

tetms in (A.63) as

- k, , k, , k4) + KY(p)

k,BX$,?K(k,,

k4)]

W’“(P, k, , ka) - *VJ - k, , k, , k) + 6WT’Wl Q x D(q) D(q - ks - k4)X%, ks >k) = 0, (A.69) J-J9

the last equality following give the desired result

from (A.68). Equations k,pXu’j”‘s(k,k,k,k,)

= 0.

(A.68) and (A.69) together (A.70)

We can similarly show that k, * X = k, * X = k, * X = 0, thus establishing gauge invariance of X. RECEIVED:

the

June 6, 1967

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. IS. 16. 17. 18. 19. 20.

R. A. BRANDT, University of Maryland Technical Report No. 673, (1967, unpublished). R. A. BRANDT (to be published). R A. BRANDT, Ann. Phys. (N. Y.) 44, 221 (1967). R. A. BRANDT, Phys. Rev. 166, 1795 (1968) W. ZIMMERMANN, Commun. Mar/z. Phys. 6, 161 (1967). W. ZIMMERMANN, Commun. Math. Phys. 8, 66 (1968). J. D. BJORKEN AND S. D. DRELL, “Relativistic Quantum Fields,” McGraw-Hill, New York, 1965. H. LEHMANN, K. SYMANSIK, AND W. ZIMMERMANN, Nuovo Cimento 1, 205 (1955). M. GELL-MANN AND F. Low, Phys. Rev. 84, 350 (1951). J. SCH\KINGER (Ed.), “Quantum Electrodynamics,” Dover, New York, 1958. A. SALAM, Phys. Rev. 82, 217; ibid. 84, 426 (1951). N. N. B~GOLIUBOV AND 0. S. PARASIUK, Acta Math. 97, 227 (1957). K. HEPP, Commun. Math. Phys. 2, 301 (1966). G. KALLBN, Nuovo Cimento 12, 217 (1954). J. C. WARD, Phys. Rev. 77, 293 (1950); ibid. 78, 182 (1950); Proc. Phys. Sot. (London) 64, 54 (1951). T. D. LEE, Phys. Rev. 95, 1329 (1954). H. S. GREEN, Proc. Phys. Sot. (London) 66, 873 (1953). Y. TAKAHASHI, Nuovo Cimento 6, 371 (1957). G. K~~LLEN, Helv. Phys. Acta 26, 755 (1953). N. N. B~~OLIUBOV ANLI D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Interscience, New York, 1959.

GAUGE

INVARIANCE

175

21. R. A. BRANDT, Unpublished doctoral dissertation, Massachusetts Institute of Technology, 1966. 22. S. N. GUPTA, Proc. Phys. Sot. (London) 63, 681 (1950). K. BLEULER, Helv. Phys. Acfu 2% 567 (1950). 23. K. WILSON, Cornell report (unpublished). 24. J. VALATIN, Proc. Roy. Sot. (London) A222,93,228; ibid. 225, 535; ibid. 226,254 (1954). 25. W. ZIMMERMANN, Nuovo Cimento 10, 597 (1958). 26. E. KAZES, Nuovo Cimento 13, 1226 (1959). 27. K. NISHIJIMA, Phys. Rev. 119, 485 (1960); ibid. 122, 298 (1961); M. MURASKIN AND K. NISHIJIMA, ibid. 122, 331 (1961). 28. N. P. CHAWS AND H. S. MANI, Phys. Rev. 134, B896 (1964). 29. R. J. RIVERS, J. Math. Phys. 7, 385 (1966). 30. N. M. KROLL, Nuovo Cimento 45A, 65 (1966). 31. W. FURRY, Phys. Rev. 51, 125 (1937). 32. B. ZUMINO, J. Math. Phys. 1, 1 (1960). 33. S. WEINBERG, Phys. Rev. 118, 838 (1960).