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QUANTUM FIELD THEORY in .NET
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dM 2 n(M2) (569) Next we evaluate the equaltime commutator of the potential with its conjugate field. Using Eq. (565), a simple calculation yields <oj [Ap(x), nV(y)] jO)lxo=yO = ib 3(X  y){g~ fo
dM 2 n(M2) (570) We would like this commutator to be canonical, which means equal to ig~b3(X  y). As far as space components are concerned this is achieved by setting fooo
dM 2 n(M2) = 1 (571) In order that this result extend to the time components we would have to satisfy
Z3 Z =
too dM 2 n(M2) This is, however, incompatible in general with Eq. (569). We are going to see that n(M2) is a positive measure. Therefore the validity of both relations leads to a measure of support M2 = /1 2 and can only hold for a free field. The best that can be achieved is to allow for different normalizations of the space and time canonical commutation rules, i.e., to assume that ELEMENTARY PROCESSES
(572) Up to now, no reference has been made to the asymptotic condition. Since dynamics does not affect the various parts of the field in a similar fashion, it would be clearly unwise to assume that (even weakly) Ap(x) tends for XO +  00 to a free field up to normalization. It is more realistic to define a transverse component of this incoming field as AJ in(x) = A~(x) + ~2 opo Ain(x) (573a) and to assume
(573b) This condition entails in particular that
For the three transversely polarized states only the first term contributes, while only the second matters if we deal with a scalar state. The oneparticle contribution to the commutator can then be worked out as If M5 stands for the continuum threshold we obtain
(576) QUANTUM FIELD THEORY
(577) (578) Accordingly, the deviation from canonical behavior is due only to the continuum contribution and vanishes for a free field since (572) reads dM 2 n(M2) /1  (579) The corresponding decomposition for the covariant propagator is
i<OI T[A (x)Av(Y)] 10 ) = p
d k eik.(Xy)[z/gpV  kpkv/~2 (2n)4 \: P  /1 2 + u; 2 kpkv//1 .) 2  m + 18 (580) + (00 dM 2 n(M2) gpv  kpkv/M2] + is
(see the discussion of the problem of covariance at the end of this subsection). Even though the longitudinal component is dynamically decoupled it turns out that the covariant expressions written above imply a nontrivial normalization factor Z 3Z for this term; Eqs. (577) and (578) do not allow us to set Z = 1 or = 1. As soon as we have shown that the measure n(M2) is positive, in spite of the indefinite metric state, the physical interpretation of Z3 will become clear, as it will playa role analogous to Z in the scalar field case. To do so, we extract from Eqs. (564) and (576) the vacuum expectation value of the current commutator: 0 <01 [jp(x),jv(Y)] 1 ) =  f:~ dM 2 2k~(:~)3 (eik.(x y)  y eik.(x ) x n(M2)(/1 2  M2f(gpv  (581) This formula exhibits (1) the disappearance of oneparticle contributions: <OU 11 photon) = 0 and (2) the current conservation. In a space with definite metric we would easily derive from (581) that the density n(M2) is positive. However, we have here b(p;  M2)( 1t"<0IjAx) 10:) <o:ljv(Y) 10) n~~3
3 d k
eik.(xy)n(M2)(1I2 _ M2)2(g
_ kpkv) (582) where net. denotes the number of scalar photons in the intermediate state 10:) and <Oljv(O) 10:)* = <o:Uv(O) 10). In Chap. 4 we have encountered examples of cancella

