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Consequently, : e-a'a: In> = <>nO In>, proving (5-33). Now let Q stand for an operator sufficiently regular to give a meaning to the subsequent operations. We expand it in terms of its matrix elements as
In this expression IX and Ci are taken as independent variables while the operators in front of the matrix element are in normal order. Therefore we obtain
Q = : e a' 8/8a+a a;8ii-a'a:
<01 e aa Q e ria' 10> la~ii~o
This is not quite the desired result where we would like to find the matrix elements of Q between normalized coherent states, i.e., states obtained from the vacuum by the action of the unitary operator eria' -aa. However, the relation enables us to rewrite
Q =: e a' a;8a+a8/8ii-a'a: e aii
<01 e aa - iia' Q eiia'-aa 10>la~ri~O
Moreover, we may commute ea. with the derivatives acting to its left in order to be able to use the condition IX = Ci = 0 at the end. Under the normal product sign a and at commute and we note that e a' 8/8a translates IX by an amount at. These remarks lead to the final expression (5-34) It would, of course, be incorrect to conclude superficially from (5-34) that the sole knowledge of diagonal matrix elements of Q in the coherent state basis is sufficient to reconstruct the operator itself. As the formula indicates IX and Ci have to be taken as independent variables in order to carry out the required derivatives.
Returning now to the physical Fock space with infinitely many degrees of freedom we can write the unit operator as
OOlf- -I L, dk1"'dkp k1, ... ,kp><kb ... ,kp1 p.
(5-35)
where the initial term is understood as the vacuum projector generalization ofEq. (5-33), namely,
10><01, given by a
(5-36)
so that the S matrix can be represented as S = :exp x
dk [ain(k)
ba~k) + ain(k) b&~k)l
10>1<x=~=o
<01 exp
d 3 k [ain(k)a(k) - ain(k)&(k)]S
x exp
d k [ain(k)&(k) - ain(k)a(k)]
(5-37)
210 QUANTUM FIELD THEORY
in complete analogy with (5-34). Due to our normalization convention the measure d 3 k and not dk appears in the coherent state exponent. We shall now combine this result with Eq. (5-32) applied to the case where the B are set equal to unity. Introducing a source on the right-hand side of this equation it takes the simple form
e(k ) e(k~)[a~(kl), ... , [ai~(kp), S] ..]
Ii [fd4Yr eik' Y'(D y, + m2 ) bJ(yJ exp i fd4Xj(X)CP(X)/ j=O ~JT r= 1 Since the vacuum is stable, that is, S 10) = 10), it then follows that
<01 exp
d 3 k[ain(k)o:(k) - aln(k)&(k)]S exp
d 3 k [aln(k)&(k) - ain(k)o:(k)] 10 )
{f d y ti(y)(D
+ m 2 ) bj~y)}<OI Texp i f d4 x j(x)cp(x) 0 ) /j=O
We have written ti(y) for the solution of the homogeneous Klein-Gordon equation expressed as
ti(y)
d 3 k [o:(k) eik
y + &(k) e- ik . y]
Inserting this matrix element into Eq. (5-37) we see that S = : exp
{f d y CPin(y)(D
0 + m 2 ) bj~y)}: <01 T exp i f d4 x j(x)cp(x) 1) /j=O (5-38)
We have naturally been led to introduce the free in-field with its Fourier decomposition [reciprocal of (5-30)]
CPin(y) =
dk [ain(k) e- ik y + aln(k) eik Y]
We should remind the reader that a proper normalization of the field cP requires its replacement by Z- 1/2cp in the last T product. The outcome of this lengthy algebra is a compact relationship between the S matrix and a generating functional for the interacting field Green functions Z(j) =
<01 T exp i
0 d4 x cp(x)j(x) 1)
(5-39)
This bears some formal resemblance to the external source problems considered in Chap. 4. The difference is, of course, that we do not have any simple closed expression for Z(j) in the present case.
The functional ZU) lies at the heart of field theory. We might be tempted to think that it contains far more information than is needed to compute the on-shell transition amplitudes. With present-day techniques, however, this is not true for several reasons. The full Green functions are needed to
ELEMENTARY PROCESSES 211
remedy in a consistent way the ultraviolet infinities occurring in the perturbative approach. Offshell equations are also required to discuss bound states. Furthermore, short-distance behavior of interactions is ideally studied in the field theoretic framework. Nevertheless, notable achievements are to be credited to the S-matrix approach which abstracts from a relationship such as Eq. (5-38) only very general properties of scattering amplitudes and proceeds from there. Section 5-3 will give a simple-minded sketch of this line of thought which lies beyond the scope of this book.
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