Other Approaches

Abstract

For the construction of the semiclassical expansion of the propagator I, we have benefited _from the work of C. Dewitt [D14,D17] on some technical points as the study of the Jacobi equation [App.9.l] and the lattice calculation of the inverse of the propagator ${\bar G^{\mu v}}$ (t_i,t_j) [App 10.1]. In fact, one of the aims of these workswas precisely to calculate the correction terms to the WKB-approximation in curved spaces. The functional integral methods that were used [D12,D13] do not seem to give an interpretation however to the undefined quantities that appear in the expansion, and that were recognized in [D14,D15,]. In D17,D18 a method is followed based on the work of Truman and Elworthy [E2,E3] who use stochastic differential geometry to arrive at the WKB-approximation. At this point, we can agree with the procedure, since we indicated in section 7.6 that the manipulations we make have a stochaic origin, and that, like for the Ito and Stratonovich integrals, the lattice offers the opportunity to control the stochastic nature. In this sense, we do not see any fundamental opposition between the two methods, and cannot agree with the assertion that a “kinetic energy term g_αβ(Q)Q^αQ^β cannot be treated by the lattice approximation and must be handled by methods suitable for stochastic variables on curved spaces”[D¹⁷]. We seem to have shown here the opposite.