Remarks on the Feynman representation
type: Item
instance of: scholarly article
title: Remarks on the Feynman representation
zbMATH Open document ID: 0614.47037
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1985-01-01T00:00:00Z
review text: There is a substantial literature on Feynman's path integral in quantum physics, but there is still not a satisfactory mathematical theory of integration with respect to such object - unbounded set functions, in the sense that approaches which yield a rich structure do not admit, or have not been proved to admit, sufficiently many integrable functions to provide a representation of solutions to Schrödinger's equation for natural potentials. The present paper deals with an approach to path integrals developed in an earlier paper of the author [J. Funct. Anal. 66, 381-405 (1986; Zbl 0589.46034)]. Not much is known about the class of integrable functions in this sense, except that the integrand of Feynman's representation of the solution of Schrödinger's equation is integrable for a wide class of potentials (ibid.). Proceeding along similar lines, this paper shows that the class of integrable functions is complete for a slightly stronger integration process, which still enables the treatment of Coulomb potentials, but not attractive inverse square potentials. Although completeness is a most desirable property possessed by a locally convex space, it is no easy task to verify that a net converges in the underlying topology of the space of integrable functions. The proof that Coulomb potentials give rise to integrable functions on the space of paths appeals to an extension of the Trotter product formula to the situation of uniform convergence as a parameter varies of a set.
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 81S40
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 3994365
zbMATH Keywords: closable set function
zbMATH Keywords: Feynman's path integral
zbMATH Keywords: the class of integrable functions is complete
zbMATH Keywords: Coulomb potentials
zbMATH Keywords: Trotter product formula
author: Brian Jefferies
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195178518
OpenAlex ID: W2048988836
cites work: Q5799972
cites work: Integration with respect to closable set functions
cites work: Feynman integrals of non-factorable finite-dimensional functionals
cites work: Q3874765
cites work: Q4076072
cites work: Feynman Integrals and the Schrödinger Equation
cites work: Approximation of semi-groups of operators
cites work: On the Product of Semi-Groups of Operators
DOI: 10.2977/PRIMS/1195178518
description: scientific article; zbMATH DE number 3994365