A Framework for Non-Gaussian Functional Integrals with Applications to Quantum Field Theory and Number Theory
J. LaChapelle
TL;DR
This work addresses the lack of a universally rigorous infinite-dimensional measure for functional integrals by proposing a framework that realizes functional integrals as a countable family of Banach-valued Haar integrals on locally compact topological groups. Localization is achieved through a directed family of surjective homomorphisms, enabling integration on measurable subspaces and the transfer of algebraic structure between a Banach algebra \mathfrak{B} and the observable-functional space \mathbf{F}(G). The authors develop a hierarchy of integrators—Gaussian, skew-Gaussian, gamma, Poisson, matrix gamma, and Liouville—that provide both localization mechanisms and links to cohomological, BRST-type, and DH/BV localization phenomena, with applications to quantum field theory and number theory. The framework supports a bridge between operator methods and functional integration, culminating in a path toward non-commutative C*-algebras via functional Mellin transforms, and suggests broad avenues for future exploration in symmetry, renormalization, and gauge theories. Overall, the paper supplies a versatile, rigorous toolkit for constructing and manipulating non-Gaussian functional integrals within a Banach-algebraic and group-theoretic setting, with promising implications for mathematical physics and analytic 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 Kähler 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.