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

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.