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Xo+ in Visual Studio .NET
Xo+ Recognizing PDF417 In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Drawing PDF417 2d Barcode In .NET Framework Using Barcode generator for VS .NET Control to generate, create PDF417 2d barcode image in Visual Studio .NET applications. + 00 Read PDF417 2d Barcode In .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Drawer In .NET Using Barcode printer for .NET framework Control to generate, create bar code image in VS .NET applications. = A~ut(x) Bar Code Scanner In Visual Studio .NET Using Barcode reader for .NET Control to read, scan read, scan image in .NET framework applications. PDF417 2d Barcode Creation In C# Using Barcode drawer for .NET Control to generate, create PDF 417 image in VS .NET applications. (48) PDF 417 Generation In .NET Framework Using Barcode maker for ASP.NET Control to generate, create PDF417 image in ASP.NET applications. Print PDF 417 In Visual Basic .NET Using Barcode encoder for Visual Studio .NET Control to generate, create PDF417 image in Visual Studio .NET applications. The precise mathematical meaning of these expressions depends on the source j. We may require at least that the matrix elements of some local average of the fields between normalized states satisfy these relations. This weak limit relies crucially on the assumption of an adiabatic vanishing of the source as Itl+ 00. UPC A Creator In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create UPCA image in Visual Studio .NET applications. EAN 13 Creation In .NET Using Barcode maker for VS .NET Control to generate, create UPC  13 image in .NET applications. QUANTUM FIELD THEORY
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(410) S lout>
Suppose that at time t =  00 the system is in a definite state, the vacuum for instance, i.e., contains no (physical) photon. The final state has some computable probability to contain zero, one, two, etc., emitted photons. For example, the probability amplitude to remain in the ground state is <0 outlO in> = <0 inlSIO in> = <0 outlSIO out> For this interpretation to make sense, we should verify that the probability = 1<0 outlO in>12 is less than one. Theprobabilitiespl' P2,"" the final polarizations, and the angular distributions are computable in an analogous way. Therefore, the operator S contains all the information about the final state. We now turn to its determination. Equation (47) implies that A~ut(x) = Afn(x) d4 y [Gret(X  y)  Gadv(x  y)]P'(y) (411) The second term of the righthand side of (411) is, of course, a solution of the homogeneous equation, and is nothing but the classical field Ag1 radiated by the current j (1206). The combination G() == Gret  Gady has already been encountered in Chap. 1 [see Eq. (1173)] : G()(x) = Gret(x)  Gadv(x) = (2~)3 d4p e ip ' x e(po)c5(p2) = 2n e(xO)c5(x 2) =  d(x) It coincides up to a sign with the commutator d of scalar massless free fields (356). This enables us to rewrite (411) as A~ut(x) = Sl Afn(x)S = Afn(x)  i
d4 y [Afn(x), Ain(y) j(y)] (412) INTERACTION WITH AN EXTERNAL FIELD
This is reminiscent of the formula (281): Indeed, only the first two terms of the righthand side do not vanish, since A and B represent free fields, the commutator of which is a c number. Therefore, we may write S = exp [i f
d4 x Ain(X).j(X)] exp [  i
(413) d x Aout(x) j(X)] This form does satisfy all the conditions, including unitarity in the indefinite metric space. Only a cnumber phase, depending possibly on j, is still arbitrary in S; the latter does not affect the physical quantities. It is convenient to rewrite S in normal order. We decompose Afn as a sum of annihilation Af~+) and creation Af~) operators. We observe that aIlAf~+) = aIlA~~i), and therefore S must leave the positive metric physical subspace invariant. The commutator of AIl() and AV(+) is a cnumber function: (414) Therefore we may use the identity eA e B
eA+B+[A, Bl/2 (415) which is valid whenever [A, [A, B]] = [B, [A, B]] = 0, and write the normal form ofS: S = exp [  i
exp [  i x exp
d4 y Ain(y) j(y)] d4 y Al:)(y) j(y)] exp [  i
d4 y
Al: )(y). j(y)] (416) {i ff
d4 x d4 y [Al:)(x) j(x), Al: )(y). j(y)] } We introduce the Fourier transform of the current j(x): JIl(k) = d4 xjll(x) e ik x
(417) The real character of j and its conservation law are expressed by
jll( k) = jll*(k) (418) 168 QUANTUM FIELD THEORY
The exponent of the last term in Eq. (416) is
~ ffd4x d4y [A~:)(x) oj(x), A~:)(y) oj(y)] = ~ f 2k~;;n J*(k) jt'(k) J(k)Iko=lkl
As expected, only the Fourier components of the source corresponding to lightlike arguments do contribute. Moreover, for P = 0, we may decompose kt< It(k) + Jt'r(k) (419) where J1(k) is a number and Jfr(k) is a spacelike vector orthogonal to k. For instance, if k = (kO, k), we introduce the spacelike fourvectors G1 = (0, e1), G2 = (0, e2) with ei = e~ = 1, e1 e2 = e1 k = e2 k = 0. We may then choose Jfr(k) =  Li= 1,2 Ji(k)Gf(k) with Ji(k) = Gi J(k). Using this decomposition, it is easy to see that 0 0 0 0 In other words, only transverse components contribute to the last term of (4 i6):

