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Connected Parts in VS .NET
515 Connected Parts Decode PDF417 In Visual Studio .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. PDF417 2d Barcode Printer In VS .NET Using Barcode encoder for VS .NET Control to generate, create PDF417 image in .NET applications. Some disconnected parts in the transition amplitudes (but not all) were not shown in Eq. (528) but can now be thoroughly studied with the help of Eq. (538). In simple terms some subsets of particles may interact independently from each other in a collision process. It is therefore natural to look for a definition of connectedness in such a way that an Smatrix element is the sum in all possible ways of products of connected ones involving subsets of interacting particles: (540) Decoding PDF417 2d Barcode In .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications. Barcode Generator In VS .NET Using Barcode generator for VS .NET Control to generate, create barcode image in VS .NET applications. We use a shorthand notation for a collection of indices, 1PI) = lEI at(pI) 10), CX(PI) = lEI CX(PI). The connected matrix elements are defined recursively by the righthand side of the equation where the J sum runs over all partitions of the set of initial and final particles. If 1 1 denotes the number of elements of J, we define SC(cx, Ii) Recognizing Barcode In VS .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Paint PDF417 2d Barcode In C#.NET Using Barcode printer for VS .NET Control to generate, create PDF417 image in .NET framework applications. L I_1_1I fn d 3PI n d 3qj cx (p I) <pMc IqJ)Ii(qJ) III,IJI J ! J! I I j J
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s(cx, Ii) e'0(a,a) (543) If we use the definition (539) for the generating functional Z(j) we may write
s(cx, Ii) = exp {fd 3k 2kO(211lcx(k)li(k) + fd 4y a(y)(Oy + m 2 ) ~}Z(j) ~(y) (544) Comparing (543) and (544) it is natural to define the connected Green functions through Z(j) = e Gow Since exp d4y a(y)(Oy arid (545) the relation (545) + m 2 ) D1bj(y)] is a translation operator, we find by combining (543), (544), (546) Let us make this formula more explicit. In the same way as we writet
t For the 2point function this differs by a factor i from the standard choice in the freefield case [compare for instance with Eq, (390)] but it is convenient in the general discussion. We hope that the reader will not find it difficult to convert from one convention to the other according to the context. QUANTUM FIELD THEORY
Z(j) with let us define
<01 T exp i fd 4 X <p(x)j(x) 10> =
I: ~ fd XI .. 'd xn j(XI)" 'j(Xn)G(Xh"" Xn) (547) G(Xh ... ,Xn) = <OIT<p(xd'''<p(xn)IO>
(548) Then the meaning of Eq. (546) is that
SC(()(, a) = fd 3 k 2kO(211l()((k)a(k) I: !" Jd YI ... d Yn a(YI)'" a(Yn) r
(549) If we insert the definition of a and the expansion (541) we see that for n = I
I I + IJ I> 2 we obtain
(550) the basic reduction formulas
expressing connected Smatrix elements in terms of connected Green functions. When compared to Eq. (528) we see that obviously disconnected terms have disappeared. Extra disconnected ones have also been suppressed as Gc has teen substituted for G. Recall that the correct normalization of <p requires dividing the righthand side by zn/2. The case n = 2 has been set aside. According to (540) we have (551) Moreover, we have to assume that G(x) = 0, meaning that the vacuum expectation value of the field vanishes. Using the KallenLehmann representation (517), the twopoint Green function may be written Gc(X,y) = G(x,y) = T<p(x)<p(y)IO>
roo dJl2 (T(Jl2) f d k
(2n)4 k 2  Jl2
+ i8
eik'(XY) implying that Gc(x, y) depends only on x  y and has a single pole in the Fourier transform variable at k 2 = m 2 This means that f d4 YI
d 4 Y2
ei(p. YlQ' Y2) (OYI
+ m 2)(OY2 + m 2)Gc(Yh Yz) = Hence, only the first term on the righthand side of Eq. (549) contributes to <pi sc Iq >and we can readily verify that relation (551) follows. Even if G(x) "" 0, translation invariance requires that it be x independent so that Gc(x, y) differs from G(x, y) by at most a constant and the preceding reasoning still applies. The reader will have no difficulty in proving that (552) where I = (1, ... , n). Thus a natural definition of connected Smatrix elements has led us to a definition of connected Green functions independent of any diagrammatic expansion. A more elaborate discussion will be given in Chap. 6. We note that sometimes the convention that Z(j) = eiG,(j) is used. Let us conclude by noting that in an elastic twobody scattering process the separation into connected parts just corresponds to setting S = 1 + iT and retaining only T in the transition amplitude used to construct the cross section.

