Integration with respect to closable set functions
The present paper gives yet another approach to the mathematical formulation of Feynman's path integral. The central idea is to take the closure of the associated operator valued set functions with rspect to those set functions which are analytic continuations to imaginary times. These set functions are related to the Wiener measure, so the usual theory of integration with respect to operator valued measures applies. The closability is facilited by analytic continuation in the upper half- plane. It is shown that the dynamical group of a quantum mechanical system can be written as the integral, in the sense above, of a complex multiplicative functional, with respect to closable operator valued set functions for certain classes of potentials. Potentials with zero relative bound to the free Hamiltonian of the system are included. The case of Coulomb interactions is covered. The representation is also valid for the attractive \(1/r^ 2\)-potential, although the associated dynamical semigroup may no longer be unitary. The Feynman-Kac formula is well known to be useful for establishing estimates on quantities associated with quantum mechanical systems. It remains to be seen whether the apparatus of the present paper is useful or not.
mathematical formulation of Feynman's path integral
integration with respect to operator valued measures
scientific article; zbMATH DE number 3946909
scientific article