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Figure 9-2 Crossed ladder diagrams.
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amplitude corresponding to the set of crossed ladder diagrams, with the exchange of scalar bosons of mass Jl and coupling constant 9 (Fig. 9-2). It can be represented by a path integral analogous to (9-32) and (9-33). The resulting amplitude for large s (square of the center of mass energy) and small transfer t = - q2 is given by
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T(s, t):o= -2s fd 2 b e,q'b [exp
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(9-36)
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The reader may determine the behavior of (9-36) as a function of s ---> CIJ and generalize further to electrodynamics where the corresponding amplitudes exhibit bound-state poles with the correct nonrelativistic limit [see Sec. 2-3-2].
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9-1-2 Trajectories in the Bargmann-Fock Space
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Up to now we have encountered integrals over trajectories in configuration space, Eq. (9-12), or phase space, Eq. (9-26). In both cases the boundary conditions involved the q or p at each limit. We want to introduce a new type of trajectories suited to a generalization to field theory. It is strongly inspired by the harmonic oscillator case since a field theory may be considered as an assembly of interacting oscillators. This will provide an easy parallel with the treatment of fermionic systems. We make use of the coherent states of Bargmann and Fock discussed III Chap. 3. This gives a representation of destruction and creation operators a=---
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at = - - -
Q - iP
(9-37)
in a space of analytic functions of a complex variable denoted with a complex conjugation bar fi. or z. The reason for this choice will appear in the sequel. If the original oscillator problem involved a frequency w we could first perform a canonical transformation Q -+ Qw -1/2, P -+ PW 1 / 2 before introducing a and at according to (9-37). The analytic functions under consideration generate a Hilbert space with scalar product
<g If> =
and we have the correspondence
dZ dz ZZ -.- e- g(z)f(z) 2m
(9-38)
(9-39)
QUANTUM FIELD THEORY
leading to a pair of adjoint operators. An orthonormal basis in this space is
f,.(Z)
G yin!
(9-40)
A unitary transformation maps this description on the more conventional set of square integrable functions of the configuration variable q. To the In correspond the well-known Hermite wave functions of the oscillator. Of course, the In are eigenfunctions of the operator H = ata = z%z with eigenvalue n. In the conventional approach we characterize an operator A by its matrix elements <q'l A Iq) in such a way that the action of A on a state vector ljJ yields the wave function
[AljJ](q) =
dq' <qIAlq')ljJ(q')
Let us proceed similarly in the Bargmann-Fock space. If In) stands for the state corresponding to the function In in Eq. (9-40) we write
n,m An,m = <nl A 1m)
(9-41)
Accordingly, for any state f,
(9-42) The kernel naturally associated to A, and denoted A(z,
[AfJ(z) =
fd;~[ e-~~
such that
A(z,
~)I(~
is therefore
_ A -+ A(z,~) =
L Ct An,m ylm: C! n,m yI n:
(9-43)
For sufficiently regular A, the function A(z,~) is an analytic function of the two complex variables ~ and Z. Incidentally, it is the desire to write the operators in this form which justified the choice of the complex conjugation bar over the argument of the analytic functions. The representation (9-43) enjoys the following fundamental superposition property as a consequence of the orthogonality of the In:
AIA2(Z,~)=
d a~ e-i1~Al(Z,11)A2(ij,~) 2m
(9-44)
Returning to Eq. (9-41) we rewrite it as
FUNCTIONAL METHODS
We recall that the projector on the ground state 10><01 may be expressed in terms of a normal product as (9-45) Consequently,
= " L..,
An ,m : a tn e - at a am:
.-~ V <)""",,-
= " A~,m L..,
V n! m!
-----=== ~
atnam
(9-46)
This normal form suggests the definition of a normal kernel to represent the operator A. We denote it AN(Z, z) to distinguish it from our previous definition with z and z considered as independent variables (9-47) To obtain the relation between AN(Z, z) and A(z, z) we may either use Eq. (9-46) or observe that the Hilbert space of entire functions is endowed with a reproducing kernel analogous to the Dirac delta function in the form
f(z) =
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