9.3.2. Functional Integrals in Quantum Field Theory
The path integral method can be applied to quantum fields by simply replacing the dynamic variables with fields. For example, the scalar field version ofeq(9.31) is
(9.32)
where S is the action. Since the sum over pathsnow becomes a sum over functions, (9.32) is often called a functional integral. In any case, the fields in the integral are functions, not operators. Therefore, the adjoint field operator is represented by the complex conjugate , and the functional integral ofHermitian fields is real.
For Fermions, the antisymmetry of the field operators requires the corresponding fields in the functional integral to be Grassmann variables [see appendix A.7].
Now, functional integrals are extremely difficult to evaluate. The practical value of the method thus depend on our ability to extract resultswithout actually calculating the integral. For illustrative purposes, let us‘evaluate’ the Feynman propagator (9.14) for a free scalar field. To this end, we introduce a generating functional
(9.33)
where 0 is the free field Lagrangian density (7.7). Furthermore, we set by a suitable rescaling of the measure . The propagator (9.14) is then given by
(9.34)
[See appendix A for a description of the functional derivative.] Obviously, higher order green functions can be similarly generated. Note that the so-called sources, J and J * , are mathematical quantities introduced solely for book-keeping purposes. They need not have any physical meanings.
In manipulating integrals, a basic technique is integration by parts. Obviously, this is useful only if the surface term vanishes. For simplicity, we shall work exclusively with boundary conditions that have this property. Note however that there are important cases that violate this condition. Indeed, nonlinear field equations often possessive topologically non-trivial solutions, which exist only under special boundary conditions. Best known examples are solitons, monopoles, instantons, vortices, etc. With this proviso, the exponent in (9.33)
can be integrated by part once to give
The Euler-Lagrange equation for this ‘action’ is
(9.36a)
where we have renamed the field as so that always denote the solution to the homogeneous solution . [Reminder: we are interested in , not .] Now, (9.36a) is a nonhomogenous equation whose solution is given by
(9.36)
where g is the green function defined by
(9.37)
Now, assuming and to be arbitrary functions related by (9.36) but not satisfying the equations of motion. Then gives
[ (9.37) used ](a)
Now,
where we have integration by parts the 2nd term twice and made use of (9.37). Thus, gives
(9.35)
where we have made use of the fact that g is real and symmetric. Putting this into (9.33) gives
(9.38)
where
where the change of ‘variable’ from to does not affect the value of the functional integral is formally over the entire function space. Since , we have succeeded in evaluating the generating function without actually calculating the functional integral itself, provided g is found. Now, with the help of
eq(9.37) is easily solved bythe method of Fourier transformation. This gives
(9.39)
which is singular at . Specifically, there are 2 poles on the -axisat , where . This means (9.39) is not defined as it stands. On the other hand, if we perform the k0 integration by means of a contour integral in the complex k0 plane, we can deform the contour infinitesimally to avoid the poles, thus obtaining finite values for the integral. However, the value thus obtained depends on the manner in which the contour is deformed. This gives rise to various kinds of green functions, each satisfying its own set of initial conditions.
It is left as an exercise (Ex.9.3) to show that the correct choice for the free field propagator (9.14) is the causal green function given by
(9.40)
where . Thus, either the contour is pushed upward (downward) for , or the poles at are pushed down (up). Eq(9.33) thus becomes
(9.41)
where is given by (9.40). In this form, the quantity may be viewed as providing a converging factor to keep well defined.
For spin 1/2 particles, the Feynman propagator is a 44 matrix with components
(9.42)
It satisfies
(9.43)
where , and is given by
(9.44)