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Josai mathematical monographs

A note on Feynman path integrals for Schrodinger equations

希世子 古谷

Published Jan 1, 1999• 0 citations
Publisher

Abstract

In this paper, we shall introduce generalized vector measurcs for thepath integrals for Schr6dinger equations. We prove that [Φ(V)](γ)Ξei∫■is integrable with respect to our generalized measure for any real function V(x)which continuous in x e RN＼{O}

DOI
10.20566/13447777_1_113

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In this paper, we shall introduce generalized vector measurcs for thepath integrals for Schr6dinger equations. We prove that Φ(V)Ξei∫■is integrable with respect to our generalized measure for any real function V(x)which continuous in x e RN＼{O}
Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.
In the late forties Richard Feynman (1948) proposed a path-integral form of quantum mechanics that proved to be equivalent to the conventional Schrödinger-equation form but possessed a much more heuristic force (see Feynman and Hibbs (1965) for a detailed exposition of Feynman’s theory). Our consideration of continuous measurements in the following chapters will be based on Feynman path integrals. A theory of quantum continuous measurements will be formulated directly in terms of path integrals instead of a limiting process of the type used in Chapter 2 .