A Framework for Non-Gaussian Functional Integrals with Applications to Quantum Field Theory and Number Theory

Abstract

We define and develop a framework to understand functional integrals as countable families of Banach-valued Haar integrals on locally compact topological groups. The definition forgoes the goal of constructing a genuine measure on an infinite-dimensional 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 that 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 non-Gaussian functional integrals are expected to play an important role in generating $C^\ast$-algebras of quantum systems. To illustrate and test the framework, examples and applications are presented in the contexts of quantum field theory and number theory.