Gauge invariance and renormalization constants

ANNALS

OF

PHYSICS:

Gauge

13, 268-283

(1961)

Invariance

and

Renormalization

Constants

L. EVASS* The Johns G.

Hopkins

University,

FELDMAK~

Ittlperial

Baltimore,

1’. T.

AND

College,

London,

Mar,yEand

&IATTHEWS

England

“We’ve quips and quibbles here in flocks But. none to beat this paradox.” W. S. GILBERT Using only general considerations such as translation invariance, positive definite energy spectrum and gauge invariance, spectral representations have been set up for the vacuum expectation values of two photon and two electron operators in electrodynamics. The gauge dependence of such quantities is thus clearly exhibited, particularly that of equal time commutators and of propagators. Certain constants, related to the renormalization constants, integrals of the spectral functions are defined and shown to be gauge invariant,. The generalized Ward identity is established in any gauge. 1. INTROI>UCTION

Renormalized quantum electrodynamics as a power series expansion is in very det,ailed agreemeut with experiment. In spite of this, it. is still not clear whether it is a well-defined self-consistent theory outside the formally r&her limited context of the power series expansion. The work of Kallen (1) on renormalization constants suggests t#hat it is not,, and Johnson (2) has examined Kallen’s argument in the light of t,he gauge invariance of the theory. More recent.ly Schwinger (5) has point,ed out that a positrive definke energy spectrum, current, conservation and canonical commutation relations are not consist,ent. This inconsist,ency is related t,o the nonvanishing of the photon “self-mass.” Recently the behavior of the propagatiou functions under gauge transformntionx and t.he gzluge invariance of t,he renormalization constant,s have been discussed by several authors WI. In this paper we begin the investigation of some of these problems. In particular, we examine the consequence of gauge transformations. To do this we * Supported in part by the National consin, Madison, Wisconsin. t On leave from The Johns Hopkins Science Foundation.

Science

Foundation.

University. 268

Now

Supported

at t,he Universit.y in part

by

the

of WisNational

GAUGE

269

IhTARIAh’CE

make use of the general framework proposed by Kallen (1) and Lehmann (7) by examining vacuum expectation values of field operators, the basic assumption being the positive definite energy spectrum. III Section 2 we define gauge transformations and discuss the behavior of photon propagators under such transformations. To do t.his it is useful to int.roduce t,he ‘true’ gauges. These are the gauges in which the equal time commutator of two electromagnetic fields vanishes. This is always assumed in the Lagrangian formulation, and wherever possible we relate our results to the convemional ones. In Section 3 we examine the behavior of fermion propagators. In particular we discuss the properties of the 2 factors and self masses under gauge transformations. In Section 3 we prove the generalized Ward ident,ity in any gauge in terms of t,he renormalized quantities. 2. THE

PHOTON

FIELD

A. GAUGE TRAKSWRMATIONS To discuss the phot,on propagat,or we st,art from the general considerations used by Lehmann and introduce a vector field A,, , which is associatedwith p&icles of zero massand int,eract,s with a conserved current. These two general considerat’ions are automatically incorporated if we define t’he current to be j,(r)

= -(grYd2 - a,ay)ii,(zj Ez - K,,;l y .

(2.1)

We require the theory t,o be invariant. under gauge transformations, by which we mean that when il,(.~‘l ---) A,(ri

+ ~,,A(T),

(2.2 j

(A, is replaced by A, + a,A) observable quant,ities do not change. In particular, expressionsinvolving ody j, are unchanged by this transformation. In principle a wide variety of A’s are allowed. They may be c-numbers or operat,ors and need not be covariant expressions. We restrict ourselves to A’s which are translat,ion invariant,. This excludes c-number transformat,ions. Vurt.her, let. us restrict ourselves to A’s that are invariant mlder space rotations. A = A(.c*, n .x) where R,, is a unit time-like vect)or. Thus expressionsinvolving 11, may depend 011a unit time-like vector 7~~, but, do not have any furt,her dependence on space-like direct#ions. I’ollowing Lehmann (7) we may now write the vacuum expectat,ion value of a product of field operat,ors in the form

270

EVAA-S,

FELDMAN,

.4XD

MATTHEWS

which follows from translation invariance and the assumed positJive definiteness of the energy spectrum. The tensor function pPVmust he a linear combination of terms involving’ gPY, k,k, , k,n, + kin, and nin, , with scalar coefficients which are functions of 1,’ and (n.1~)“. A change of gauge in (2.3) produces on the left-hand side new t,erms of the form

,

and

which (due to translational invariance) will alter the coefficients only of those t,erms in prV, which depend explicitly on k, . Using (2.1) , an expression for the vacuum expectation value of the product of t)wo currents in t’erms of psycan he obtained from (2.3). The requirement that’ this observable quanGt,y be covariant,, (t,hat, is, independent, of ,n,,), excludes the possibilit,y of a term depending on wb 111 P,, and requires further t>hat’t)he coefficient of gpv is covariant. We are thus led t,o t,he most general form’

p,,(k) = g&A(P) - %.“I p&t?, (nvk)“)

(2.4)

+ n,k+,l;,

(n.k)/33(k’,

(n.k)“).

The function PI is unchanged by gauge transformat,ions, but, as seen above, /3z and /I3 are gauge dependent, and it is often convenient t)o specify different classes of gauges by placing conditions on /3zand /33. An example of noncovariant gauges are the radiation (or Coulomb) gauges in which3 [a, - n,(n-d)]Ap 1 gev is the tensor

metric

with

diagonal

elements

= 0. (1, -1,

(2.6) -1,

-1)

and the

scalar

product

A.B = AoBo - A.B. 2 Considering the most general gauge transformations subject only to translation invariance there are ten independent elements to the symmetric tensor p,,. constructed from q,,“, k, , 12, and two independent space-like vectors I, cl), l,,@) (giving in all four independent vectors). The further restriction on the current commutator reduces this number to five, those given above, and two more, similar to the coeficient & with n replaced by Z(‘+“. In this more general case all the p’s except PI would be functions of four variables-the components of k, . 3 In the frame in which n, = (l,O, 0,O) one can transform to a radiation gauge by choosing A(t, r) = -L 4* It is then easy to check duced by any h satisfying

s

that transformations V2A = 0.

1 -, r _ r,, &-LO, from

r’)

&‘.

one radiat,ion

gauge

to another

are in-

GAUGE

Applying

this condition

t.o (2.3)) we have t)he restriction IX, - n,(n~k)lp,.

which,

when combined

271

INVARIASCE

with

(2.7)

= 0,

(2.-I ), leads to the relations

(2.8) Or

kk- (n IIk)(71-k) d4k. -“““e(/;) gpy - YU.-!L!--ti.m-k? -k+ (n.k)? 1fl(,+‘)



(2.9)

If we restrict, ourselves to covariant This implies that

gauges, (2. -1) must be independent

(2.10)

Be = 0, and also t)hat’ 0, is independent,

of rl, .

of (,n.k)“, p2 = /32(k2).

(,2.11)

Thus (2.3) becomes td,(a)~~,(O)>

= /- e?%‘(k)

[

gp&(k2)

1

- ‘+/3&i2) I

d4k.

(2.12)

From now on we rest’rict ourselves to covariant gauge t,ransformations. In subsequent’ considerations a very important part, is played by the Landau gauge (sometimes called transverse Lorentz gauge), in which t,he fields A,’ are defined to satisfy a,il,’

= 0.

(2.13)

This implies <,4,‘(2)A,‘(O)>

= / e-“k”B(k)

[(h

- “$)

pdk’j

+ bk,,kJ(k’)

] d*k,

(2.1-I)

where 6 is a const,ant t,o be determined below (see (2.25)). Making a gauge transformat8ion to ,4,(x) where A,(x)

= A;(s)

+ a,,~\(.~),

(2.15)

we have

= 1

+ / e-“;“0(k)

k$’

pZ(k2) d4j,-. (2.16)

272

EVANS,

FELDMAN,

AND MATTHEWS

The extra contribution in (2.16) comes from terms such as & <3,A(r)&h(O)>. Since 9, sat.isfies{2.13), one can choose gauges?

= 0.

and

(2.17)

(This is in agreement wit,h the gauge ksnsformations considered by Bogoliubov and Shirkov.) Thus
= / e-i”“O(k) ‘$

d’k.

(2.18)

We define Lorentz gaugesto be those covariant gaugesin which (2.1) can be replaced by j, = -azAG .

(2.19)

a’aPA4,= 0.

(2.20)

Siuce j, is conserved, this implies

This condition applied to (2.16) implies lc2pp(k2)= 0.

(2.21)

The solutions to (2.21)) p*“(I? j are given by p2L(k2) -= a 8q k” B. COMMUTATION

+ b’6(k2).

(2.22)

RELATIONS

Prom (2.16) and t.he positive definiteness of the energy speckurn, it follows that &i,(z),

A,(O)]> = lm dtc2f nl’k e-‘kre(6) Pl(K2) + k,k, ( b6(d) + ~EK) K2 )] 6(k2 - K2). (2.23)

Hence the equal time commutator is ([A,,(X), ,4,(o)]> = (2?r)“i(~,,,a, + ~,0a,)~3(x)

1 “Lj”

dKZ- b] . (2.24)

In order to be able to compare our expressions with t.he corresponding ones of the conventional formalism and particularly those of Fallen (l), we shall re4 We may anticipate that the implication of this equation in electrodynamics is that the states created by the operation of A on the vacuum are not contained in the Hilbert space defined by A,7 and the corresponding electron field operator $‘.

G.iUGE

273

INVARIANCE

strict ourselves to “true” gauges, which, by definition, are those in which the equal time commutator of two fields vanishes. Thus in the t,rue Landau gauge5

b = ( $ dtc2.

(2.25)

Having chosen b in this way, all other true gauges are restrictled by the condition (2.26) Constructing

the commutator

from (2.181, (2.26) <[A(x),

i(O)]>

=

implies

&at

(2.27)

0.

Xow one can also obtain <[A,(x),

1

d”(O)l> = (%r)“is”(X)

p1dK2- &i”b s (Pl - P2)dK2,

4&(x), .-i”(O)[>= (2a)m,“a” + s,,a,)s”(x) j (p1- p2)dK2. These are precisely the commut’at,ion relations used by Fallen (1) interpret p1dK4= z,‘,

s

(2.28)

(2.29)

if n-e

(2.30)

(t,his is discussedfurther below\, and provided we work in a gauge for which s

p2dtc” = 1.

(2.31)

This implies, by (2.22) a true Lorentz gauge for which a = 1.

(2.32)

and b’ is chosen to satisfy (2.26). From (2.1) and (2.23) c&i e--““‘e(/c)(~2g,,

- k,J&c2pl(i2)6(/i’

- K’)

(2.33)

5 If one is originally in a Landau gauge for which b = 0, one can get into the true Landau gauge by means of the transformation 3, = A, + a,,~ where @X = 0. Note also that by direct calculation of the equal time commutator from (2.9) it is easy to check that all radiation gauges are automatically “true” gauges.

274 which, (j,(s)

EVANS,

FELDMAN,

of course, is independent = 0), is given by

AND

MATTHEWS

of the gauge. The

hy(r’)

noninteracting

case.

= 0,

01 &KY)

= 8(K2).

(2.34)

From (2%)

~[,jp(~),jv(o)l~ = (27d3i(6,0a, + ho a,PW

/ & d2.

(2.35)

It has been shown by Evans (8) that am is positive definite in the radiat,ion gauge. Since p1 is independent of the gauge, this stat)ement is t’rue in any gauge and the equal time commut’at80r of two currents does not vanish. This is in apparent ront,radiction with t!he usual canonical commutation relations for an electromagnetic field in interact’ion w&h spin M particles (3). This feature of t.he theory is somet.imes referred to as the nonvanishing of the photon self-mass, since by analogy with spinless bosons, one can compute and find that it is also proportional to J ~~~~dK2. C. THE

PROPAGATOR

.4iv~

THE

K4ss

OmRaToR

We now consider t,he chronological product’ which is defined as ?‘(A,(r),

A”(O))

= f$ti(x)[A,(z),

AJO)] + >Q.4,(r),

A,(O)}.

(2.X)

From the representations (2.16) and (2.2s) and using i,.?”

one obtains6 d4xelkr (2.37 1 = (h

) d&k*) + $$‘d#c*),

- g

where dt(k2)

=

/

k2

“‘;2;

ie

dK2,

6 Note that, we are assuming a true gauge and have dropped a term proportional

(2.38) to

GAUGE

Similarly,

ISVARIAKCE

275

from (2.18) x(x)

=
A(O))>

(2.40)

e--lkr~.- Pdd__ K’b’ -

lising

t,he true gauge condition

K2 +

d”kdtc’.

(2.41)

it)

(2.26))

(2.42)

Define a( B’) by the relation pl(K2)

=

8(K2)

+

(T(K2).

(2.44)

In perturbat’ion theory [see Ballen (I)], ~(12’) behaves like l/k2 for large k”, so that (2.38) is a convergent int’egral and (2.37) is the renormalized propagator in terms of the renormalized fields. In a Lorentz

The

free

propagator

gauge,

using

(2.22),

in t,he interact,ion

one finds

that

representation

is given

by dl and P,o so that

Some of the propagators which have been used in calculations arise from different values of a corresponding to different Lorentz gauges. Thus a=0 Landau C&=1 Feynman-Dyson. We remark, in passing, that electromagnet,ic self masses of charged bosons to second order in e arise from two types of graph; those in which the photon is emitted and absorbed (a) at different points, and (b) at the same point. Since (a) can be made finite by giving a finite size to the hoson charge distribution, whereas (b) is, so to speak, intrinsically infinit,e, it would seem best to make boson self mass calculations in the FEM (finite electromagnetic mass) gauge in which a = -3 and graphs (b) vanish identically. This appears t,o us to be the best justification of the relation between classical and quantum mechanical self masses given by Mat,theas P.nd Cret.sky (9).

276 Using representat’ions

EVANS,

FELDMAK,

AXD

of t’he form (2.33),

MATTHEW’S

we calculate Pi, where

Since the currents j, are in terms of the renormalized field yuantit’ies, PiV is related to the renormalized self-energy but still cont,ains its true divergences (10). See Ey. (2.53) below. The final term in (2.15) is another manifestation of t,he fact that t,he equal time commutator of two currents does not vanish. The improper renormalized self-energy II,,” , can be obtained by removing the “t,ails” from t,he difference bekeen the true propagat’or and t,he free propagator. (2.46)

- <~(Ax”(dA,“om>}, where the final term is obt’ained by replacing functions

p1 by pi’. In terms of the spectral

(2.47) Note that (in contrast

to P:“y ) k&(k)

(2.48)

= 0,

and also that

or D,, = 0;; + D;&,D;y” where 0;: is the free propagator

(2.50)

+ D:, ,

in the Landau gauge.

Also,

writing

and

II,,(k) = (&” - F) rII(a,

(2.52)

GAUGE

from (2.45)

and (2.47)

277

INVARIANCE

we see that

)

(2.53)

k2=0

which establishes D.

the relationship

'C;NRENORMALIZED

between

I’” and the renormalized

self energy.’

FIELDS

So far WC have considered only the renormalized fields and currents, which are t,he natural variables in this formulation. To make contact with Dyson’s theory, it, is of interest to int’roduce the unrenormalized fields. The relation bct,ween these and t,hc renormalized fields is clearly complicat’ed in an arbitrary gauge since only the transverse part> (in a covariant sense) of the field is affected by radiative correct’ions. However, in the true Landau gauge, we have, from (2.21) and (2.28), <[-&L(x), il”(O)l>

(2.54)

= 0,

tLlld <[A,(x),

ii”(O,]>

= (a7r)“is3(x)(gp.

-

s,,a,,,)Z,‘.

Thus in this gauge we may define the unrenormalized

(2.55)

fields as

A,lL = z:‘2A, .

(2.56)

Then <[d,“(x), <[A,“(x),

Ay’((0)]>

n”“(o,]>

Using (2.50), the unrenormalized

=

= 0

(2.57)

(27r)“iS3(X)(&

-

S,&,).

(2.58)

propagat,or

= 0,: + 0;;: rI$ I~;:: .

(2.60)

In (2.60) l& where

&,,[u]

which is the term the renormalized

= (2, - l)(Pgh,

is the renormalized

-

self energy

subtracted in perturhat’ion field variables.

theory.

/i&J

+ IIXJU],

in terms

Recall

that

(2.61)

of the unrenormalized

P is expressed

in terms

of

278

EVANS,

FELDMAN,

fields. We recognize that, in Dyson’s improper self-energy part. 3. THE

AND

MATTHEWS

terminology

ELECTROK

II0 is just, the unrenormalized

PROPAGATt

)R

We now consider the electron field #(a) and t,he corresponding is defined by the equation -(-td

+ m)!b(.x)

current,

= f(x).

which (S.1)

In t’he true Landau gauge the electron field is denot,ed by I+G~,and in an arbit,rary gauge in which the electromagnetic field is given by (2.15)) the electron field is 4(x)

(3.2)

= eifA(sw4.

On t#he basis of (2.17) we make the plausible assumption operate in a different Hilbert space from the #r(~). Lehmanrl (7) has defined speckal functions such that <$(x)$(O)> and shown

= 1 O(k)e-“““[(irk

t,hat the equal t,ime anti-commutat’or 1 (2?r)3

Z2

- &3)pF1(li2)

j”

<{\L(x),#+(O)l>

d”x

will be equal to the ronventionally

=

+

t)hat operat,ors

,oFP(/P)]d4k,

A(X)

(3.3)

is I

p&t)

dtt = z,‘.

defined constant Zz in the gauge in which



m - up )

= J

(3.5)

tP

where 1 1> is the one-electron stat,e and ZL~the positive energy spinor normalized such that tiP~lP = 1. The rorresponding expression for the time-ordered product iS S(x)

=


=

;~t(x)
From the latter expression (11)) &

lim [S(t,x) t-o+

(3.6)

q(o))> 4(O)}>

+

;,L;<[$(“r),

lJ(O)]>

it is easy to est,ablish (see also Gell-Mann

- S(-t,x)]~~

= j pF1(~‘) di2 6”(x)

= ZiL’S”(x),

and Low

(3.8)

G;IUGE

Similarly

279

Ii\‘VARItl?JCE

the electron self mass can be defined by

(3.10)

= I [(m - K)PdKS)+ Pm(K2)1 dKZ, where G(.ij

=
q(o) j>.

(3.11)

Equivalenfly (3.12) The equality of the two expressions may be established as above, by evaluating t,he anticommutat,or in terms of the density functions. We now consider the gauge dependence of these quantities. To do this one must know how X(z) and G(x) transform under gauge transformations. Let the chronological product in the true Landau gauge be x’(z) Since the operators

A(x)

=
q(o))>.

(3.13)

commute with +7(x), it follows i~c.~cr~--h(o))>S’(.r). S(x) =
that in a general gauge (3.14)

For small values of c (IS - = 2E(X(X) de Since the gauge transformations g

- X(O))S’.

(3.15)

form a Lie group =%(X(x)

(3.16)

- X(O))S.

Hence’ S(s)

r*lX(d--X(0))ST(x), = e

(3.15)

= er2~X~‘~-h(o)1.

(3.18)

or
4(O))> been

(14).

obtained

by

Landau

(Id),

Bogoliubov

(IS),

Johnson

280

EVANS,

FELDMAK,

AND

MATTHEWS

Similarly
g(o))> = [cr(f’(.t+),

= (C(X)

-

p(o))>

-

~‘(~ax(.~jjSr(~))~t2~h(3)--h(“)) ~‘(~ax(.~)), G(o) j>]ea2(h(r)-X(u)) (3.19)

We also require certain propert,ies of X(X), which follow from (2.42). x(0+,x>

= X(0--,

xj,

(3.20)

A(o*, xj = 0,

(3.21)

also KwOf,

x)Ll

(3.22)

= 0.

Thus -1

zs

=-

1 lim [S(t, x) - X( --t, x)174 d3rC (27r)3 s t-n+

= h3 1 ,‘il exp [E2(X(x)- k(O))I(fi’(4 x)

,S’( - t, x) )?I dx

(3.23)

= (z,‘)-‘, where zzT is defined in the true Landau gauge. Similarly, by (3.9) and (3.19) and using (3.20) t,o (3.22) 6nz -=zz

1 lim [G’(t, x) --Lt2yaX(t, x) S’(t, (“a)3 I 14+ - G”( -t, x) + c”yax( - t, x)S’(

Xl

- t, X)]yph(r’--X(“”

dYLT

(3.24)

A similar argument shows that in boson electrodynamics the renormalizat,ion constant ZB and self mass 6~’ are also independent of the true gauges. The expression for t,he self mass, analogous to (3.12), which is suggested by the Lagranginn formulat.ion, is somewhat more complicat~ed owing to t.hc explicit dependence of t#heBose current on the elect’romagnetic field. The results obta,ined above on the invariance of the renormalizat’ion constants are in apparent contradiction with the results of Bogoliubov and Shirkov (23) and of Johnson and Zumino (6 j . However, these aut,hors have defined the con&ant, Zq such that <0 1+ I 1> is normalized to dm/e, u, in every gauge. In our notation, this implies t,hey are considering, not gauge transformat,ions of t,he type (3.2) but rather the following transformation: l/l(r)

=

t9.(“) ~2irA(z)~7(x), @

(3.23)

GAGGE

281

Ii’iVARIASCE

The additional factor efZACo”’ just accounts been the two formulations. 4. THE

VERTEX

for t,he apparent

discrepancy

he-

PART

For completeness we shall discuss the behavior of t,he vertex part in order b ohtnin the generalized Ward identit,y. (:I) Thr grnwalixed Define

Ward

identity

in amg gauge”

IT, =
y@(z)

)>.

(A.1 1

.4@and # are related t’o the true Landau fields by (2.13) and (3.2). Proceeding as in Se&on 3, hy considering dV,/dt for small Ewe find how P, is related to I’,,‘, where 1;,’ = .

(1.2)

We ohtain T’,

i’,“,’

=

+

it

[

6

(A($’

-

2:)

-

k(jJ

_

z))$‘(.e

_

P

i)

1

p-=)--X(O)).

(4.8)

The vertex part Fp is defined as follows, Vp(? Y, 2) =E

s S(:tz - d)DJy

- y’)ry(x’,

y’, z’)S(z’

- 2) d4hz4y’d42’,

(1.J)

where D,, is given by (2.49). Xot’e that \ve have built into the definition (4.4) the fact, t>hat

-ah a V)&‘(z, y,z)=0. This implies we are assuming, at equal times, [B(x),

A;(o)]

= 0.

(4.6)

From the definit’ion of D,, we have

a,D,,(.v) = a,a'ii(.c),

(3.7)

where j\(n) is given by (2.42). Thus from (1.4) $ and

P

vu = c 1 SCx

0 Okubo (3.5).

- T')au2 D", x(y - ~wr(.d, y',i)s(d il

(14) has t.reated most of the aspects of this We follow t,hrough to oht,ain Eq. (4.11).

problem

- 2) dx'dy'dz'. having

derived

Eqs.

(4.8) (4.3)

282

EVANS,

FELDMAK,

.%ICD

MATTHEWS

Since X(1/ - g’) is a function of the difference y we have

7~‘~ hy an integration by parts

Also from (4.3) and using (4.5) and (3.14) we obtain - vfi = idJx(y d; /J

- xj - A(!/ - z)]S(z

- 2).

(4.10)

Eyuatling the right-halld sides of (4.9) and (4.10) we have in moment.um spwe

(p - Y’eQJ - y) jS(pj(p - yM?,(p, y)S(y! - i[S(p) - S(q)ll = 0

(4.11)

which is tnle for all A, and is a statement of the generalized Ward identit,y ill any gauge. 5. CONCLUSIONS

By expressing the phot,on and electron propagators A la Lehmann (7), as integrals over speckal functions, it, has been possihle to exhihit very explicitly the dependence of these quantities on the gauge. Operator gauge t,ransformations are considered since these are required if the fields are to he translation invariant in any gauge. The convent.ionaI assumption that all clect.romagnetie field components commut,e at equal t,imesis equivalent t,o restricting the allowed gauges to a rertain class which we call “true.” We have Mahlished t’hat all the properties of the photon propagator and self energy are as given in the conventional renormalized theory. In particular the photon propagator separat(esinto two parts, one of which is gauge independent and includes all the “radint,ive correct,ions,” and another, which depends on the particular choke of gauge. The Landau gauge is that in which the gauge dependent part of the propagat’or vanishes identically. All &her cwvariant gauges can he ohtained from the true Landau gauge by gauge transformations. It appears t’hat only in this gauge do the fields d,’ and #’ commute with the transformation field A(X). Also, it, is only in the Landau gauge t.hat. t,he unrenormalized fields may he int,roduced simply as scale transforms of the renormalized fields. In evaluating in t#ermsof the spectral functions a positive definit,e gauge invariant const,ant appears, which in a Lagrangian formulation might, be interpret,ed aa a photon self mass,hut it is not clear within the limit’ed scope of t,he present paper whether the appearance of such a t)erm is paradoxical or inconsistent. The same constant also appears in <[j,(x), A,(O)]>. The renormalization constant’ of the electron field as defined by (3.4) was shown to he gauge invariant,, where the gaugetransformat,ion is given hy (3.2). These transformations, however, change t)he normalization of the one part,icle

expectation values, i.e., . The gauge transformation (3.25), which is the one used by Johnson nlld Zumino (6), keeps the cluantity invariant, hut, makes the renormnlizat ion constant’ gauge dependent. ACECNO\\.LEI)GYENTS

We wo~tld like to thank Professor hbdus Salam and Dr. T. W. B. Kibble for stimulating discussions. We are also indebted to 11r. K. Johnson and Dr. B. Zumino for stimulating disrussions on the definition of the electron renormalization constant. One of us (G. F.) is indebted to I1.S.I.R. (Department of Scientific and Industrial Research) for financial support, and Professor Salam for the hospitality of Imperial College.

l~~uww):

Sovcmher

6, l!)(X) REFERENCE?

! 2.

3.

4. 5. 6’.

7. 8. 9.

20. 11. 19. 15. 14.

(:. Kar,r~~s, nlso Kg/.

in “Handbuch der Physik,” Vol. 5, Part I. J. Springer Verlag, Berlin, 1958; ljanske T’idenskab. Srlskab, Mat.-fys. Xedd. 27, ?;o. 12 (,1953). K. JOHNWS, Phys. Her. 112, 1367 (1958). J. SCHWISGER, Phys. Hw. Letters 3, 2% (1959). .I SCHWISGER, Phys. Ret,. 115,721 (1959). B. ZIIAIISO, .J. Math. J’hys. 1, 1 (19tiO). K. lJ~~~x~os ASD B. Zcmso, Phys. Kez~. Letters 3, 351 (1959). H. LEII~~SS, Strove cijttenlo [S]. 11, :3*2 (1954). L. Evas~, thesis, The Johns Hopkins ITniversity, (June, 1960). P. 'C. MATTHEWS .~XD J. L. URETSKI-, Piqs. Rep. Letters 3, 29i (1950). A. QAI,.4hI. E'h!/s. Ret,. 82, 217 (1951). M. GEL~.-;LI.~IN ANI) F. E. Low> Ph!/s. Krr. 95, 1300 (1951). L. I). LASD~C AND I. iI1. I
(kUB0.

~\‘1/02'0

C~WW~O

[la],

15,!140

(l!%o).