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Journal of Mathematical Physics

The Feynman maps and the Wiener integral

Aubrey Truman

Published Aug 1, 1978• 53 citations
Publisher

Abstract

By introducing the family of Feynman maps ℱs, we show that our earlier definition of the Feynman path integral ℱ=ℱ1 can be obtained as the analytic continuation of the Wiener integral E =ℱ−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps ℱs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for ℱ1=ℱ.

DOI
10.1063/1.523873

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Journal of Mathematical Physics•May 1, 2000•1 citation
By introducing the family of Feynman maps ℱs, we show that our earlier definition of the Feynman path integral ℱ=ℱ1 can be obtained as the analytic continuation of the Wiener integral E =ℱ−i. This leads to some new results for the Wiener and Feynman integrals. We establish a translation and Cameron–Martin formula for the Feynman maps ℱs, having applications to nonrelativistic quantum mechanics. We also estalish a (weak) dominated convergence theorem for ℱ1=ℱ.
The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F1-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.
In this paper, we define a new concept of analytic Feynman integral on theWiener space, which is called the generalized analytic Feynman integral, to explain various physical circumstances. Furthermore, we evaluate the generalized analytic Feynman integrals for several important classes of functionals.We also establish various properties of these generalized analytic Feynman integrals. We conclude the paper by giving several applications involving the Cameron-Storvick theorem and quantum mechanics.
A Cameron-Martin formula is established for the Feynman map definition of the Feynman integral in non-relativistic quantum mechanics. This formula involves the Morse or Maslov indices and shows the connection with the Feynman-Hibbs definition of the path integral and the Fresnel integrals of Albeverio and Hoegh-Krohn. Applications are given to the Feynman-Kac-Ito formula for anharmonic oscillator potentials On etablit une formule de Cameron-Martin pour l'integrale de Feynman en mecanique quantique non relativiste. Cette formule fait intervenir les indices de Morse et de Maslov et montre la relation avec la definition de Feynman-Hibbs de l'integrale de chemins et les integrales de Fresnel considerees par Albeverio et Hoegh-Krohn. On donne des applications a la formule de Feynman-Kac-Ito pour des potentiels d'oscillateurs anharmoniques