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II [det vii IJ O(V A)] exp {;2 J x [fi' d in Visual Studio .NET
II [det vii IJ O(V A)] exp {;2 J x [fi' d PDF417 2d Barcode Recognizer In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. PDF 417 Generator In VS .NET Using Barcode creation for VS .NET Control to generate, create PDF417 image in .NET applications. 2g tr (J. A)J} (1268) PDF417 Scanner In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Barcode Generator In .NET Using Barcode maker for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Through a modification of the normalization in Eq. (1268), we may replace det vii by det vii vIIo 1 where Bar Code Scanner In Visual Studio .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. PDF417 2d Barcode Encoder In C# Using Barcode creation for Visual Studio .NET Control to generate, create PDF417 image in VS .NET applications. vIto = L\Oij0 3(X  y) PDF417 Maker In .NET Using Barcode drawer for ASP.NET Control to generate, create PDF 417 image in ASP.NET applications. Create PDF417 In Visual Basic .NET Using Barcode maker for VS .NET Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. 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. Printing GS1  13 In .NET Using Barcode generation for .NET Control to generate, create EAN 13 image in Visual Studio .NET applications. Code 128A Printer In Visual Studio .NET Using Barcode generation for VS .NET Control to generate, create Code128 image in .NET applications. 1222 Integration over the Gauge Group
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,u(A, E) = exp
{i J x [fi'(x) + Ji(X)Oi(X)J} d
In a gauge transformation (1270) 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 (1271) 9oA 3 = 0 = V' A = 0 Let us compute the jacobian of this transformation. We set ~ l(A) = 2t1(g) IJ <5(V' 9A) (1272) 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) (1273) We then multiply Eq. (1269) by 1=
~(A) f 2t1(g) IJ <5(V' 9A) (1274) interchange the order of integrations, and use Eq. (1273) 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)] (1275) Owing to the in variance of the measure 2t1(g), we may change in the last integral gl 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 ,uindependent normalization factor N, we have X (A, E)L1(A),u(A, E) I] b[V . A(x)] (1276) To recover (1268), we have to show that L1(A) is proportional to the determinant of the operator At defined by (1267). Since L1(A) is multiplied by b(V' A) in (1276), 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 (1277) with, as in (1267), We have just shown the equivalence of the axial and Coulomb gauges, as far as the computation of gaugeinvariant 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 timedependent gauge transformations and to impose covariant auxiliary conditions. Let ~(A) = (1278) 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)

