Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

J. LaChapelle

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.

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

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 -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 -functions associated with functional traces and determinants.

Paper Structure

This paper contains 34 sections, 29 theorems, 263 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Outline of paper
  4. Functional integration scheme
  5. Functional Mellin transform
  6. Motivation
  7. The definition
  8. Algebraic properties
  9. Mellin functional tools
  10. Functional resolvent
  11. Functional trace, logarithm, and determinant
  12. More Mellin functional tools
  13. Positive powers
  14. Some loose ends
  15. Functional Mellin and QFT
  16. ...and 19 more sections

Key Result

Theorem 2.4

If $G$ is locally compact, then there exists a unique (up to positive scalar multiplication) Haar measure. If $G$ is compact, then it is unimodular.

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Definition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Corollary 2.10
  • ...and 61 more