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Some disconnected parts in the transition amplitudes (but not all) were not shown in Eq. (5-28) but can now be thoroughly studied with the help of Eq. (5-38). 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 S-matrix element is the sum in all possible ways of products of connected ones involving subsets of interacting particles: (5-40)
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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 right-hand 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)
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L -I_1_1-I 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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(5-41)
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with SC(O, 0)
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O. Similarly, s(cx,li)
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<Olexp fd3pcx(P)a(p)sexp fd3QIi(Q)at(q)10)
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such that s(O, 0)
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1. The recursion relations (5-40) are simply interpreted as
s(cx, Ii)
e'0(a,a)
(5-43)
If we use the definition (5-39) 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)
(5-44)
Comparing (5-43) and (5-44) it is natural to define the connected Green functions through Z(j) = e Gow Since exp d4y a(y)(Oy arid (5-45) the relation (5-45)
+ m 2 ) D1bj(y)]
is a translation operator, we find by combining (5-43), (5-44),
(5-46)
Let us make this formula more explicit. In the same way as we writet
t For the 2-point function this differs by a factor i from the standard choice in the free-field case [compare for instance with Eq, (3-90)] 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)
(5-47)
G(Xh ... ,Xn) = <OIT<p(xd'''<p(xn)IO>
(5-48) Then the meaning of Eq. (5-46) is that
SC(()(,
a) = fd 3 k 2kO(211l()((k)a(k)
I: !" Jd YI ... d Yn a(YI)'" a(Yn) r
(5-49)
If we insert the definition of a and the expansion (5-41) we see that for n = I
I I + IJ I> 2 we obtain
(5-50)
the basic reduction formulas
expressing connected S-matrix elements in terms of connected Green functions. When compared to Eq. (5-28) 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 right-hand side by zn/2. The case n = 2 has been set aside. According to (5-40) we have (5-51) Moreover, we have to assume that G(x) = 0, meaning that the vacuum expectation value of the field vanishes. Using the Kallen-Lehmann representation (5-17), the two-point 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
e-ik'(X-Y)
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. Yl-Q' Y2)
(OYI
+ m 2)(OY2 + m 2)Gc(Yh Yz) =
Hence, only the first term on the right-hand side of Eq. (5-49) contributes to <pi sc Iq >and we can readily verify that relation (5-51) 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 (5-52) where I = (1, ... , n). Thus a natural definition of connected S-matrix 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 two-body 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.
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