QUANTUM FIELD THEORY in .NET

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QUANTUM FIELD THEORY
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nl(M2) = n(M2) n2(M2) = n(M2) -
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Z;Z b(M2 -
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<OJ [AAx), Av(Y)] jO) =
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t" f2k~(:kn)3
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eik.(x-
x [n(M ) (gPV -
~V) + Zr ~v b(M2 -
m2)]
(5-68)
which exhibits a transverse part weighted with n(M2) and a longitudinal one concentrated at M2 = m 2 We insist now that the equal-time commutator vanishes. On the right-hand side the corresponding contribution is proportional to a gradient of a b function. Setting its coefficient equal to zero yields the sum rule
Z3 Z =
dM 2 n(M2)
(5-69)
Next we evaluate the equal-time commutator of the potential with its conjugate field. Using Eq. (5-65), a simple calculation yields
<oj [Ap(x), nV(y)] jO)lxo=yO = ib 3(X -
y){g~ fo
dM 2 n(M2)
(5-70)
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
(5-71)
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. (5-69). 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
(5-72)
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)
(5-73a)
and to assume
(5-73b)
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 one-particle contribution to the commutator can then be worked out as
If M5 stands for the continuum threshold we obtain
(5-76)
QUANTUM FIELD THEORY
(5-77) (5-78) Accordingly, the deviation from canonical behavior is due only to the continuum contribution and vanishes for a free field since (5-72) reads
dM 2 n(M2) /1 -
(5-79)
The corresponding decomposition for the covariant propagator is
i<OI T[A (x)Av(Y)] 10 ) = p
d k e-ik.(X-y)[z/gpV - kpkv/~2 (2n)4 \: P - /1 2 + u;
2 kpkv//1 .) 2 - m + 18 (5-80)
+ (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. (5-77) and (5-78) 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. (5-64) and (5-76) the vacuum expectation value of the current commutator: 0 <01 [jp(x),jv(Y)] 1 ) = -
f:~ dM 2 2k~(:~)3 (e-ik.(x- y) -
y eik.(x- )
x n(M2)(/1 2 - M2f(gpv -
(5-81)
This formula exhibits (1) the disappearance of one-particle contributions: <OU 11 photon) = 0 and (2) the current conservation. In a space with definite metric we would easily derive from (5-81) 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
e-ik.(x-y)n(M2)(1I2 _ M2)2(g
_ kpkv)
(5-82)
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-
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