The Semiclassical Expansion in Configuration Space

Abstract

In (I.C), we have seen that perturbation theory in phase space functional integrals is well defined if one uses the notion of discretization. A particular application was the construction of the semiclassical expansion of the propagator I of the Schrodinger and Fokker Planck equations in chapter IX. In the framework of the same application, we shall consider now the perturbation theory in configuration space. The purpose of solving the same problem in configuration space is to show that the notion of discretization also allows us here to give a precise meaning to all the terms in the expansion. Although this was to be expected, the situation is much more complicated due to the occurrence of contractions of the derivatives of q(t), such as ${\nabla _t}qq(t)$ and ${\nabla _t}q{\nabla _t}\cdot q$ which at equal time cannot be determined in a formal way, but require the close examination of the discretized expression. The main difference comes from the fact that after splitting the quadratic part defining the free generating functional Z_o[J], one cannot take the continuous limit separately in Z_o[J], and in the interaction terms, but has to study previously the discrete form of the equal time contractions with derivatives. This is in contrast to the phase space situation where the limit could be taken in the free generating functional Z_o[J,J^★], and all the discretization dependence of the functional integral could be taken into account effectively by a prescription on the action of the interaction Hamiltonian on Z_o[J,J^★]. Another new feature appearing in the configuration space treatment, is the occurrence of undefined terms in δ(0) in the expansion, that appear as terms in 1/ε in the discrete, and which we show to cancel in all orders of η. Similar problems to these have been studied in the literature [C6,G8,L29], mostly with operator techniques, or with formal use of the path integral.