Stationary phase method and Feynman path integrals

Abstract:   In 1948, Feynman expressed the fundamental solution for the Schrödinger equation using apath integrals. Feynman explained his path integral as a limit of finite dimensional integrals, which is now called the time slicing approximation. However, in 1960, R. H. Cameron proved that the measure of path integrals does not exist. 

In this talk, using the time slicing approximation,we prove the existence of path integrals and show some properties of the path integral similar to some properties of the standard integral. More precisely, we give a gerenal set of functionals for which the time slicing approximation converges on compact sets with respect to the starting point and the endpoint of paths.

Especially, to prove the convergence of the time slicing approximation, we explain oscillatory integrals, Fujiwara's stationary phase method in a large dimension, and so on.