Mellin-type Functional Integrals with Applications to Quantum Field Theory and Number Theory

TL;DR

This paper introduces a novel infinite-dimensional Mellin transform framework for functional integrals, enabling new tools in quantum field theory and number theory, with applications to resolvents, determinants, and L-functions.

Contribution

It develops the functional Mellin transform, extending the concept of Mellin transforms to infinite-dimensional integrals, and applies it to quantum field theory and number theory.

Findings

Constructed Mellin-based QFT generating functionals for bosons and fermions

Connected functional complex powers with scattering amplitudes

Provided a Mellin perspective on renormalization and entropy

Abstract

Conventional functional/path integrals used in physics are most often defined and understood, either explicitly or implicitly, as the infinite-dimensional analog of Fourier transform. In this paper, the infinite-dimensional analog of Mellin transform is defined and developed. The associated functional integrals are useful tools for probing non-commutative function spaces in general and $C^\ast$-algebras in particular. Functional Mellin transforms are used to define the functional analogs of resolvents, complex powers, traces, logarithms, and determinants. Several aspects of these objects are examined and applied to various constructs in mathematical physics. As substantial applications, we construct Mellin-based QFT generating functionals for bosonic and fermionic degrees of freedom, explore connections between functional complex powers and scattering amplitudes, interpret renormalization from a functional Mellin perspective, define a parameter-dependent entropy that formally justifies the replica trick, and explore $L$-functions associated with functional traces and determinants.