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Moreover, <OISIO) = <ala) = 1
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<11 S 11') = <11 1')
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where indices have been omitted from the one-particle states and we have only translated the stability of these states. The unitary S matrix has to commute with Poincare transformations to implement the covariance of the theory
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One can also express deeper locality properties of the S matrix if, following Bogoliubov and Shirkov, one allows for space-time dependence of the coupling constants in the lagrangian. We refer the reader to their textbook for a systematic use of this approach.
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5-1-3 Reduction Formulas
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Here we shall use the asymptotic conditions (5-14a, b) to relate S-matrix elements tli) the general Cheen fnneti0ns of the interacting fields. Consider the transition amplitude <Ph"" out Iq1,'''' in), where for simplicity we have omitted the smearing test functions necessary for normalization. We shall reduce this element by extracting one by one the in- or out-creation operators which allow us to build 'S> 7 '" _ the initial or final state. By definition
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<PI, ... ,out Iqh'''' in)
<PI,"" out I aln(ql) I q2,"" in)
- Id xe t
iq, . x !.OPh ... ,outCPin(X)q2, ... ,Ill ) 8< I I .
The last integral is performed at an arbitrary time t. Let us therefore choose a large negative value to enable us to substitute Z- 1/2 cP for cpin: <Ph ... ,out I q 1, ... , in)
1 Z-1/21d3 x 1m
t-+-oo t
_iq,.xl~< Ph"" -:- Vo
OU tl
. cP ()I Q2,"" III ) X
Now for an arbitrary integral ( lim - lim ). ( d 3x If;(x, t)
t-++oo t-+-oo
t,-++oo,tl-+-oo
(tr dt: (d 3x If;(x, t)
Using the asymptotic condition for arbitrarily large positive times we find <Ph'''' out IQl,"" in) = <PI,"" out'l a~ut(Qd IQ2,"" in)
+ iZ- 1/2 {d4XOO[ e- iq ,. x80<Pl,'''' out Icp(x) IQ2,'''' in)]
(5-25)
QUANTUM FIELD THEORY
where the integral extends over all space-time. The first term on the right-hand side of Eq. (5-25) is a sum of disconnected terms (corresponding to situations where at least one particle is not affected by the collision process):
<PI,"" out Ia~ut (ql) Iq2,"" in)
L 2pf(2n)3J3(Pk I
x <PI, ... ,p;;, ... ,outlq2, ... ,in)
We have assumed here that all particles belong to the same species. The necessary changes when this is not the case are obvious. Observe that disconnected terms disappear when none of the initial and final momenta coincide. Let us now take a closer look at the second term on the right-hand side of Eq. (5-25). Since this is understood as a kernel to be tested with wave packets, partial integrations over space variables are legitimate. This is, of course, not the case for time variables, otherwise all the calculations would collapse. With this in mind, denoting by m the mass of the particles and recalling that qi = m 2 ,
= d 4 x e- iq ,. X(O + m2)<f3, out I<p(x) IIX, in)
The first stage of the reduction formula takes the following form:
<Ph"" Pn, out ql,"" q/, in)
f f f
d4 x oo[e- iq ,.
x80 <f3, out I<p(x) Ia, in)]
d4 x {[( -d + m2) e-iq,.X] <f3,out I<p(x) Ia,in)+ e- iq "x o5 <f3,out I<p(x) Ia, in)}
L 2pf(2n)3J3(Pk I
ql)<Ph"" jik, .. . , Pn, out Iq2,"" qh in)
The same steps can now be repeated again and again. To be specific, we shall do this for a particle in the final state. As far as the disconnected terms are concerned, nothing new emerges and we shall omit to write the corresponding contributions. This is, however, not true for the second term, for which we may write
<PI,"" out I<P(XI) Iq2,"" in)
I lim iZ- I / 2 fd3YI e ~~oo
<P2,"" out aout(pd<P(XI) q2,"" in)
ip ,'
x <P2,"" out I<p(YI)<p(xd Iq2,"" in) We would like to replace the last integral by a four-dimensional one as we did before. Clearly some trick is needed in order for the operator ain(Pl) coming
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