A Framework for Non-Gaussian Functional Integrals with Applications

Abstract

Functional integrals can be defined in terms of families of locally compact topological groups and their associated Banach-valued Haar integrals. The definition forgoes the goal of constructing a genuine measure on the domain of integration, but instead provides for a topological realization of localization in the infinite-dimensional domain. This leads to measurable subspaces that characterize meaningful functional integrals and a scheme that possesses significant potential for constructing and representing non-commutative Banach algebras. The framework includes, within a broader structure, other successful approaches to define functional integrals in restricted cases, and it suggests new and potentially useful functional integrals that go beyond the standard Gaussian case. In particular, functional integrals based on skew-Hermitian and K\"{a}hler quadratic forms are defined and developed. Also defined are gamma-type and Poisson-type functional integrals based on linear forms suggested by the gamma probability distribution. These and their generalizations are expected to play a leading role in generating $C^\ast$-algebras of quantum systems. Several applications and implications are explored.Comment: This is the first of two papers representing an expanded version of arXiv:1308.106