ABSTRACT
In this chapter we apply a Hubbard-Stratonovich transformation to the Grassmann integral representation to obtain a bosonic functional integral representation. The main reason for this is not only that we would like to have integrals with usual commuting variables amenable to standard calculus techniques, but is merely the following one. By writing down the perturbation series for the partition function, evaluating the n’th order coefficients with Wick’s theorem and rewriting the resulting determinant as a Grassmann integral, we arrived at the following representation for the two-point function G(k) = 〈a+k ak〉
G(k) = R
(5.1)
If we would have started with a bosonic model, the representation for G(k) would look exactly the same as (5.1) with the exception that the integral in (5.1) would be a usual one, with commuting variables. Regardless whether the variables in (5.1) are commuting or anticommuting, in both cases we can apply a Hubbard-Stratonovich transformation and in both cases one obtains a result which looks more or less as follows:
with an effective potential
(5.3)
Thus, we have to compute an inverse matrix element and to average it with respect to a normalized measure which is determined by the effective potential (5.3). The statistics, bosonic or fermionic, only shows up in the effective potential through a sign, namely = −1 for fermions and = +1 for (complex) bosons ( = 1/2 for scalar bosons). Besides this unifying feature the representation (5.2) is a natural starting point for a saddle point analysis. Observe that the effective potential basically behaves as |φ|2 + log[Id + φ]. Thus, it should be bounded from below and have one or more global minima.
