Computer simulations of quantum random dynamics

Abstract

Analytic methods for the Feynman integral are the main topic of this book. A rigorous version of the Feynman integral allows one to control errors in various approximations to the exact quantum dynamics. For quantitative results even the approximate analytic calculations have to be supplemented by numerical methods. In principle, the Feynman integral can be reduced to an approximate finite-dimensional integral by means of the time discretization. However, there are some subtleties in the discretization. This can already be seen from the difference between the Ito and Stratonovitch integrals discussed in section 3.1 (different discretizations lead to inequivalent results in the continuum). The formulation in this book allows us to treat the discretization problem properly. We have reduced the Feynman integral to the Wiener integral. Hence we can apply methods of discretization developed for the Wiener integral [222]. After the discretization of the conventional Feynman integral another difficulty emerges namely, oscillating integrals. By an analytic continuation we have expressed in Chapter 5 the oscillating integral by the Gaussian integral of a complex function whose oscillations can be much weaker than in the original integral. However, such an analytic continuation is possible only for a restricted class of (analytic or meromorphic) functionals. For Gaussian integrals of slowly varying functions (no oscillations) the Monte Carlo methods have been applied with a success for a long time.