II [det vii IJ O(V A)] exp {;2 J x [fi' d in Visual Studio .NET

Draw PDF417 in Visual Studio .NET II [det vii IJ O(V A)] exp {;2 J x [fi' d

II [det vii IJ O(V A)] exp {;2 J x [fi' d
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2g tr (J. A)J} (12-68)
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Through a modification of the normalization in Eq. (12-68), we may replace det vii by det vii vIIo 1 where
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vIto = -L\Oij0 3(X - y)
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Usingdet vIIvII o 1 = exp [tr In (vllvll o l)J and tr In (1 + A) = L [( llnJ tr An, we may derive the Feynman rules in this gauge. However, they are not covariant either.
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12-2-2 Integration over the Gauge Group
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The gauge transformations used so far were time independent. In order to make the Lorentz covariance manifest, it is more appropriate to consider time-dependent transformations as well. The method to implement them will appear as a byproduct of a different problem. How can we show the equivalence between the axial and Coulomb gauges Let us drop the coupling to the external source J, replace it possibly by sources coupled to gauge-invariant quantities Oi(X), and consider the functional integral (at a given time)
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(A, E)
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IJ o(A 3),u(A, E)
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(12-69)
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where ,u is the gauge-invariant functional
,u(A, E) = exp
{i J x [fi'(x) + Ji(X)Oi(X)J} d
In a gauge transformation (12-70) where g is considered as a product
TIx g(x) and ,u is invariant. So is the measure
QUANTUM FIELD THEORY
2t1(A, E), since the gauge transformations are canonical. Hence X =
f 2t1(A, E) IJ <5 VA 3)fl(A, E)
As the conditions V' A = 0 or A 3 = 0 are two equivalent ways to define uniquely (in a perturbative sense) a point in each equivalence class, there must exist a go(A) such that (12-71) 9oA 3 = 0 = V' A = 0 Let us compute the jacobian of this transformation. We set
~ -l(A) =
2t1(g)
IJ <5(V' 9A)
(12-72)
where 2t1(g) stands for the infinite product of invariant measures on compact groups, isomorphic to the group G at every space point x, 2t1(g) = Dg(x). The in variance of the measure Dg' = D(gg') entails
~(9A) = ~(A)
(12-73)
We then multiply Eq. (12-69) by 1=
~(A) f 2t1(g) IJ <5(V' 9A)
(12-74)
interchange the order of integrations, and use Eq. (12-73) to write X
f 2t1(A, E)fl(A, E) IJ <5[A 3(x)] f 2t1(g) IJ <5[V' 9A(y)]~(A)
= 2t1(A, E)fl(A, E)~(A)
IJ <5[V'
A(y)]
2t1(g)
<5[rlA3(x)]
(12-75)
Owing to the in variance of the measure 2t1(g), we may change in the last integral g-l into ggo, where go is such that 9oA 3 = O. For simplicity, we denote B = 90A and write
f 2t1(g) IJ <5[rlA3(x)] f 2t1(g) IJ <5[9B 3(x)]
Since B3 = 0, it suffices (at least within perturbation theory) to consider only infinitesimal transformations g and to perform the group integration in the vicinity of the identity
g(x)
= e + a(x)
where a is infinitesimal. Under these circumstances, the measure at each point Dg(x) reduces to the product dai(X) and
9B 3 (x)
8a(x) 8x 3
NONABELIAN GAUGE FIELDS
Hence the integral
is independent of A and may be absorbed in the normalization. Therefore with an overall ,u-independent normalization factor N, we have X
(A, E)L1(A),u(A, E)
I] b[V . A(x)]
(12-76)
To recover (12-68), we have to show that L1(A) is proportional to the determinant of the operator At defined by (12-67). Since L1(A) is multiplied by b(V' A) in (12-76), it is sufficient to compute it for transverse fields A. Then, only infinitesimal g contribute to the integral defining L1 :
L1- 1(A)
(g)
I] b[V
A(x)]
f (o:) I] b(V' {- Vo:(x) + [o:(x),A(x)]})
det- 1 At (12-77)
with, as in (12-67),
We have just shown the equivalence of the axial and Coulomb gauges, as far as the computation of gauge-invariant quantities',u(A, E) is concerned. What happens to the source term, and hence to the Green functions, has not been examined. However, only local and canonical changes of field variables have been performed in the previous derivation. By virtue of the equivalence theorem mentioned in Chap. 9, we do not expect these transformations to modify the physical content of the theory. The previous method is interesting, as it allows us to deal with time-dependent gauge transformations and to impose covariant auxiliary conditions. Let
~(A) =
(12-78)
be such a condition. ~ is a local functional of A, that is, a function of A(x) and of its derivatives, it takes its values in the Lie algebra, and it may depend on AO. Using the same method as above we find X=
f (A) I] b[~(A)] det At ,u(A)
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