Canonical commutation relations for vector fields

7.A

I

Nuclear Phyics 31 (1962) 464---470; (~) North-Holland Publishing Co., Amsterdam

I

Not to be reproduced by photoprint or microfilm without written permission from the publisher

CANONICAL

COMMUTATION

RELATIONS

FOR VECTOR FIELDS

KENNETH JOHNSON

Institute for Theoretical Physics, University of Copenhagen, Denmark Received 22 August 1961 Abstract: The commutation rule [~b~,/0] = 0 and its relation to the sum rule for the bare mass of a vector field is discussed. Consistency with perturbation theory is demonstrated.

In a recent note 1) a relation between the bare mass and physical mass of a vector mesons coupled to a conserved current was given. In that derivation an assumption was made, which, although reasonable, cannot be directly justified on the basis of the "ground rules" (see below) usually assumed in such derivations. Thus, most unkindly put, one cannot say that the relation was "derived", but only follows from that assumption. The "ground rules" which are imposed are: I) the field equations, II) eovariance, etc., III) canonical commutation rules, where in the case of III) for the theory in question we mean (all at equal time)

[~L 4"] = 0 = EG°L G°'],

[4, k, 6 l°]

= iakla

[¢L 0] = [ 6 %

3)(x _ y),

= 0,

{0, 0"} = •(3)(x-Y) •

(1) (2) (3) (4)

Since all other operators are defined in terms of these, we cannot independently assume the form of their equal time commutation rules. However, the current operatorjU(x) is defined in terms of a limit applied to the operators ~*, ~ in a domain where certain matrix elements have a strong singularity. Hence it is not to be considered legimate to use the above rules "formally" with respect to such singular operators. The commutator in question, which leads to the sum rule for the mass given previously is 1) E~bk(x),jO(y)] = 0. Thus, with respect to the rules stated above it is not possible to "derive" this commutator without detailed use of (I), that is by solving the theory. Indeed, we should realize that (I) is not complete until we specify a way of obtaining ju from ~/,. The formal * Permanent address: M.I.T. Cambridge, Mass. 464

CANONICAL COMMUTATION RELATIONS

recipe is

meaningless, that is the equation jr(x) = ½e0[~(x)v u, $(x)]

465

(5)

must be supplemented by a rule for handling the singularity in certain matrix elements of ~(x + ~)T~$(x) as ~ --* 0. The only guide to fix this rule is that the resulting operator should be conserved (dt,f ( x ) = 0), and "as finite as possible". Nevertheless, one might be tempted to extend the ground rules by making the assumption that, in spite of the singular nature of the product, the formal equation (5) may be used to obtain the commutators [~bk,jo] = 0,

(6a)

[-GOk,jo] = 0,

(6b)

[~bk, f ] = 0,

(7a)

[G°k, j']

(7b)

= 0.

These are grouped in two sets since we shall show that not all of these rules can hold simultaneously (in particular (6a) and (7b)). If we use covariance and the field equations, we obtain the expression 1)

i(O[~(x)j'(y)[O)

=(D~o"v-~"av)f~(-~2+m~o)~. A'+).

(8)

Thus, consistency with (6b) follows trivially, and from (6a) we find the sum rule

1 _ fdx a(x).

m~

;~

(9)

Again, consistency with (7a) follows trivially, but from the field equations we obtain

[~k,/] = [_ Gok_ a~o, j,] = [_6o~+ 1m2 Oko~GOs--12 dkjo,fil, I-

so if we use (7b)

[a,jo, j,].

[6k, j,] =

m~

If we again use (8) and the field equations, we find

f

dx(x 2 - mo~)~ = 0.

(10)

But both (9) and (I0) cannot hold simultaneously (as was emphasized previously i)) since then d r (~2 22

f~

too) ~ = 0,

466

KENI~rETH JOHNSON

that is, a positive definite integral would vanish. Hence, also (6) and (7) cannot hold together. Thus, all derivations of sum rules such as (9) or (10) must fail if they are based on a formal application of the canonical rules to such singular field products a s ~1~.

The remainder of this paper will be concerned with a discussion of why (6) is to be preferred over (7), and how this can be guaranteed even in perturbation theory. It is dear, first, that (6a) must hold if the theory is to become quantum electrodynamics in the limit of m o --* 0. For, in that limit the longitudinal part of the vector field is undertermined by the field equations, thus, it should be possible to freely put it equal to zero. This would not be possible if (6a) were not valid, at least in the limit as m o ~ 0. (It is easy to show t h a t j ° commutes with the transverse part of ~bk). Hence the proof that the physical mass of the photon vanishes if the bare mass vanishes still is valid if we make the assumption that vector meson theory approaches quantum electrodynamics as m o -* 0. It is clearly impossible to maintain a rule such as (7b) which leads to (10), in the limit of mo ~ 0. For then one would obtain at least two contradictions, first, the vanishing of a positive integral, and second, the lack of commutation betweenj ° and an operator left undetermined by the field equations in this limit. Further, we see that the sum rule for the mass obtained from (6a) is much more convergent than any which involve the integral J" x 2 trdx. In fact, (9) written in terms of the renormalized spectral function is 1

_

1

f~dx

m2 +-3. where the expression on the right is finite, even in perturbation theory. Hence mo 2 is characterized by the same convergence property as z 3 , the charge renormalization. This greater convergence should be related to the relative independence of the charged and meson field degrees of freedom at equal times. (3) is an abstract formulation of this independence. [~b",j°] = 0 = [G°k,j ° ] is a more concrete requirement. In a consistently formulated theory one would expect these to be a consequence of (3), since this means the simultaneous measurability of the charge density and fields at a given time. I f (7) held it would mean that the current density and fields are simultaneously measurable, less probable if the coupling between the fields is local, since a measurement of the flux of charge involves at least an infinitesimal displacement of the charge in time. Finally, we have remarked that the current is in effect defined by the procedure by which the matrix elements o f j ~ are calculated from those of ~k, that is, by the method of handling the singular terms in the operator ~(x+e)~"~(x). The only a priori constraint on the treatment of these singularities is that the resulting operator should be conserved. Now, it is clear that the expression for the current can be "modified" by including in this limit as a factor an operator F~(@) which has the property that it is a function of the field operators ~b" only in the neighbourhood of x and F~(~b) ~ 1

CANONICAL.COMMUTATION RELATIONS

467

in the limit ~ -~ 0. A limit applied to ~(x+~)~,~'lp(x)F~ would not alter the " f o r m a l " expression for the o p e r a t o r j ~ nor its locality, nor would any matrix elements where the o p e r a t o r ~(x+~)V~'~(x) was not singular be affected. However, matrix elements where ~O(x+s)~"~b(x) is singular as e ~ 0 would be changed. In fact, i f F w e r e such that the difference between the cases w h e r e the factor were present and left out was only in terms p r o p o r t i o n a l to dp"(x), the conservation law a , j ~ = 0 would be satisfied by b o t h operators, and hence it is clear that the requirement o f conservation is not sufficient to define j"(x) uniquely. Hence, we should impose as a further stipulation the requirement that F should be chosen so that the current operator is as well defined (non-singular) as is possible. In this case, that means we should try to define the current so that the sum rule for the mass is (9), that is, so that

[dpk(x), jO(y)] = 0. We shall n o w present a heuristic agreement to show firstly that this can be done, and secondly that, with such an F , the definition for the current becomes one which has previously been noted is required in q u a n t u m electrodynamics (i.e., in the mass zero limit) in order to maintain local gauge invariance. I f we calculate the c o m m u t a t o r of dpk(x) with ~(x+e)7°O(x) = g/*(x+e)~O(x), "naively", that is by expanding for small s °, and using the field equations, the c o m m u tator

[dpk, q,*(x + ~)q,(x)] is o f order e, but in fact, that formal calculation is justified only for those matrix elements o f the coefficient o f 8 which are finite. It is just those which are not finite that can be changed b y F. Thus let us try to choose F so that the above c o m m u t a t o r is o f higher order in ~ and (hopefully) will strictly vanish in the limit e ~ 0. I f we calculate using the canonical rules we find [~*(x + 8)¢(x)Fe, dpk(y)] = [~b*(X + e), dpk(y)]~b(x)F~ + ~*(X + e)~(x)EFe, dpk(y)] and [~O*(x + e), dpk(y)] = [qt*(X + e), dpk(y + e)] + [~b*(X + e), dpk(y + e-- e) -- dpk(y + e)] = - e°[~O*(x + e), ~k(y + 8)]

--,

[0*(~+.),j°(y+.)]

o, mo

?no

Hence one has

= 8° ~~° ok~3~(~ - y)~*(~ + ~)~,(~)v. + 0 " ( ~ + . ) 0 ( ~ ) [ V . , dpk(y)]. mo

468

KENNETH JOHNSON

Let us define F, so that [F~, ~bU(y)] = - e ° e o okf(a)(x_y)F,,

m2

that is, so the terms formally of order e vanish. I f we suppose such an F~ can be chosen, then (6a) will hold so

[~O(x),ffk(y)] Hence, I F , , q~k(y)] =

=

i okf(a)(x_y).

_ieoeO[q~O(x),tkk(y)]

"F,.

We see that this is satisfied by

Cx+e

-ieoJ

F, = e x p ( - i e o e " ~ u ) = exp

\

and also F, ~ 1 and e ~ O. Thus, if we propose to define j " in terms of

Cx+~ \ -~(x +e),~'ff(x) exp ( - i e o J x d~Uq~u(~))

(I1)

in the limit as e --* 0 (after symmetrizing on both signs of e), we will obtain an operator which will be more convergent and for which the commutation relation [jo(x),

= 0

should hold. Such a factor has previously been noted as necessary in the mass zero case (2, 3) in the presence of an external field to maintain local gauge invariance. Further, in a matrix element where the singularities are the same as in the vacuum, where in perturbation theory ~(x+e)~d/~(x) N l / c a + . . . on symmetrizing the limit on positive and negative e, we would obtain from F an extra term, formally, 1

which is quadratically infinite (but proportional to tk~ and so "conserved"). However, since we know that the theory is more convergent if we include the factor F, this must just cancel a similar, quadratically infinite term in ~(x + e)7"~k(x). In fact that is just what happens as an elementary second order calculation shows (see the appendix). The author would like to thank Professor Niels Bohr for the hospitality extended to him at the Institute.

Appendix We wish to sketch a derivation to show that, in the lowest order approximation to the polarization *), the mass renormalization for the vector mesons is consistent with

CANONICALCOMMUTATIONRELATIONS

460

the sum rule (9). To do this, it is most illuminating to use the formal expression s) exp

( -ieo ~-6 j-~) F(~)exp(½iJ"A.Jv)

(outlin) = which, when expanded in powers of J", generates the meson field Greens functions. In this formula F(~) is the vacuum to vacuum amplitude for the charged particle whose current is coupled to an external potential ~ . Thus, F(~) is a gauge invariant functional of q~z. It has previously been noted how to calculate F and maintain consistency with gauge invariance (2). This requires that the current be defined as in (11), in the external field ~z. In this case we find, in the approximation considered F(q~) = exp

(½i(~"p,~~)

where in momentum space p,~ has the form 6) P~v = (g~Vq2 where P.~ = polarization

qUqV) d2 q 2 + 2 2

(gu~-q.qv/q2). (outlin) = exp

Thus, in the lowest order approximation to the

-½ieg ~fi p"V~

exp

(½i(J~A~aJt~)).

If we separate A~, into its longitudinal and transverse parts

quq~] 1 _ q~q~ A~,~= g.~+ m2 / q2+m2o P~,,A+ q2m~ , A-

1

q2+m2 '

then (outlin) = exp

( ½iJ ( q2~o qq 0) J )

exp

(--½iegb~ (½i(JPAJ)) 6 p -~)exp 6

= exp (½iJ (~-~) J) exp(½iJP(A-l +e2p)j). Hence the effective Greens function in this approximation is A'-' = q2+m~+4q2

['d2

3

p(~)

q2+22"

The physical mass is given for the zero of this function, that is

22_m2}"

470

KENNETH JOHNSON

It is clear t h a t this is consistent with the p r o p e r t y t h a t as m o ~ 0, m ~ 0. It is also clear t h a t this e q u a t i o n is equivalent to the sum rule (9). F o r we see t h a t A ' ( 0 ) - 1 = m o 2 a n d this is j u s t the same as (9). I t is hence o b v i o u s t h a t (9) holds to all orders o f pert u r b a t i o n theory, since the p o l a r i z a t i o n tensor, P~v, has the f o r m given a b o v e to all orders. Hence A'(0) -1 = mo 2 to all orders.

Note added in proof: Recently Schwinger 7) has suggested t h a t when m o = 0 it is possible t h a t the threshold for the vector states is n o t at zero b u t at a finite mass. I n this situation it is clear that q2Sd,~p(2)/(q2+22 ) m u s t r e m a i n finite at q2 = 0. This can h a p p e n only if p ( 2 ) d e v e l o p e d a f - f u n c t i o n at 2 = 0, b u t otherwise h a d a finite mass threshold. This situation could n o t arise in the case mo ~ 0 except in extremely artificial cases. However, it is d e a r l y logically possible t h a t p(2) could develop such a 6-function at zero in the limit m o ~ 0. This is one aspect o f the f u n d a m e n t a l defect in viewing electrodynamics as the limit o f a field with mass. References 1) 2) 3) 4) 5)

K. Johnson, Nuclear Physics 25 (1961) 431 K. Johnson, Nuclear Physics 25 (1961) 435 K. Johnson, Nuovo Cimento 20 (1961) 773 J. Schwinger, Prec. Nat. Acad. of Sci. (USA), 37 (1951) 455 J. Schwinger, unpublished; also see for example B. Zumino, Journal of Math. Physics 1 (1960) 1 6) See for example J. Schwinger, Phys. Rev. 82 (1951) 664 7) J. Schwinger, Phys. Rev. (to be published)