A Framework for Non-Gaussian Functional Integrals with Applications

Abstract: Functional integrals can be defined on topological groups 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 a space of functions, and instead provides for a topological realization of localization in the infinite-dimensional domain. This yields measurable subspaces that characterize meaningful functional integrals and a scheme that possesses significant potential for representing non-commutative Banach algebras suitable for mathematical physics applications. 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 are expected to play an important role in generating $C^\ast$-algebras of quantum systems. Several applications and implications are presented.