Some applications of vector space measures to non-relativistic quantum mechanics

Abstract

We give a new definition of the Feynman path integral in non-relativistic quantum mechanics — the Feynman map f. We show how, in fairly general circumstances, the Cauchy problem for the Schrödinger equation can be solved in terms of a Feynman-Itô formula for this Feynman map f. Exploiting the translational invariance of f, we obtain the so-called quasiclassical representation for the solution of the above Cauchy problem. This leads to a formal power series in ℏ for the solution of the Cauchy problem for the Schrödinger equation. We prove that the lowest order term in this formal power series corresponds precisely to that given by the physically correct classical mechanical flow. This leads eventually to new rigorous results for the Schrödinger equation and for the diffusion (heat) equation, encapsuling the result quantum mechanics → classical mechanics as ℏ→0 → O.