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from the lower bound of integration to operate directly on the state Iqz, ... , in). The time <4rc!ieril'lg of the fields
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T<p(YI)<P(XI)
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<P(YI)<P(XI) { <P ( ) <P ( YI ) Xl
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has just the right property of setting the operators in the convenient order relevant for the two limits. Without affecting the previous expression we can therefore substitute the T product and proceed as in the first step with the result that
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<Ph" .,out 1<P(XI) 1qz, .,in)
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+ iZ- 1/Z
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d4YI eiPl'Yl(DYl
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+ mZ) <Pz, .. . ,out 1 T<p(Y1)<P(X1) 1qz, .,in)
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On the right-hand side the first term will yield disconnected terms. Once two particles have been reduced the answer looks like
<Ph"" out 1qh"" in) = <PI."" in 1S 1qh"" in)
+ (iZ-1/Z)Z
= disconnected terms
d4x1 d4Y1
eiPl'Yl-i~l'Xl(DYl + mZ)(Dxl + mZ)
(5-27)
x <pz, ... , out [ T <P(Y1)<P(X 1) 1qz, ., in)
Disconnected terms obtained so far involve one or two <5 3 functions. The same reasoning can now be carried further until all incoming and outgoing particles have been reduced '~
<P I. ... , Pn, ou t 1q 1, ... , q[, in)
<P I. ... , Pn, in 1S 1q 1, ... , qI, or t )
= disconnected terms
+ (iZ-1/Zt+ 1
d4YI ... d4xlexP(itPkOYk- tqrOXr)
(D y1 + mZ)"'(D x, + mZ)
<01 T<p(YI)"'<P(XI)IO)
(5-28)
The remarkable feature of these expressions obtained by Lehmann, Symanzik, and Zimmermann is the relation they provide between the on-shell transition amplitudes and the general Green functions of the interacting theory. The latter are precisely the vacuum expectation values of time-ordered field products. Relation (5-28) implies that in momentum space the Green functions have poles in the variables PT, where Pi is conjugate to Xi. Up to a normalization constant, the S-matrix element is nothing but the residue of this multiple pole. We also notice a remarkable symmetry between incoming (q1, ...) and outgoing (PI. .) momenta. It is convenient to replace the outgoing set (PI," .) by an incoming one (- PI. ... ) with negative energy component. In this way all momenta are on the same footing.
QUANTUM FIELD lliEORY
5-1-4 Generating Functional
It is possible to obtain a compact operator form for the set of relations (5-28). To simplify notations let us absorb a factor Z- 1/2 in <P in such a way that within this subsection the normalization of <P is taken as lim <P = <Pout. We shall
also use a unique symbol for creation and annihilation operators, in agreement with a previous remark, by setting for an arbitrary four-vector k on the mass shell P = m 2 :
J (k) = 1 al (-k)
(5-29)
in such a way that
at;, (k) = ie(kO)
fd X e 80
<Pin
(5-30)
Let B 10 , Bn be local Bose fields. Equation (5-28) is equivalent in operator form to the relation
[ST[B 1(X1) Bn(x n)], at:,(k)]
-ie(kO)S
d4 y
eik y(Oy
+ m2 )
(5-31)
x T[<p(y)B(X1) B n(x n )]
The proof follows easily by taking in-state matrix elements, using ai:S = Sao~t, and expressing a"" through (5-30) in terms of the field. The desired relation is then obtained as above by replacing boundary terms by an integral over the time derivative. Iteration yields the more general commutator
[ ... [ST[ B 1(xd Bn(x n)], ai~(k1)]' .. . , ai~ (k p)]
= (-
i)Pe(kY)
e(k~)
d4 Y1 ... d4 yp exp (i
kr '
Yr}OYI + m
) ..
(OYp + m2 )
(5-32)
x ST[<P(Y1) <p(Yp)B 1(xd Bn(xn)]
This could be applied to the case of several types of particles and fields.
To understand the use of (5-32) as a mean of generating S itself, let us remind ourselves of the construction of Fock space as a product of Hilbert spaces, each one corresponding to a definite mode (indexed here by the momentum k). For a given oscillator, with fundamental operators such that [a, at] = 1 and a vacuum state 10), the normalized excited states are In) = at "j(n!)'/210) The projector 10) <01 has the following representation:
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