# Functional Integration on Paracompact Manifolds

## TL;DR

This paper develops a rigorous mathematical framework for functional integration on paracompact manifolds, utilizing Schwartz space techniques, convolution, and gauge theory constructions to address measure and differentiability issues in field theories.

## Contribution

It introduces a novel Taylor-Lagrange scheme for fields in Schwartz-Sobolev spaces, establishing a rigorous foundation for functional integration and gauge theories on curved manifolds.

## Key findings

- Validated the assumption that fields live on differentiable manifolds.
- Defined a sound Laplace-Stieltjes transform in distribution spaces.
- Simplified calculations in non-Abelian gauge theories using specific connections.

## Abstract

In 1948 Feynman introduced functional integration. Long ago the problematic aspect of measures in the space of fields was overcome with the introduction of volume elements in Probability Space, leading to stochastic formulations. More recently Cartier and DeWitt-Morette focused on the definition of a proper integration measure and established a rigorous mathematical formulation of functional integration. Their central observation relates to the distributional nature of fields, for it leads to the identification of distribution functionals with Schwartz space test functions as density measures. This is just the mathematical content of the Taylor-Lagrange Scheme developed by the authors in a recent past. In this scheme fields are living in metric Schwartz-Sobolev spaces, subject to open coverings with subordinate partition-of-unity test functions. In effect these PU, through the convolution operation, lead to smooth field functions on an extended Schwartz space. In this way the basic assumption of differential geometry -- that fields live on differentiable manifolds -- is validated. Next it is shown that convolution in the theory of distributions leads to a sound definition of Laplace-Stieltjes transforms , stemming from the existence of an isometry invariant Hausdorff measure in the space of fields. Turning to gauge theories the construction of smooth vector fields on a curved manifold is established, as required for differentiable vector fibre bundles. The proper choice of a connection further separates in the functional Hausdorff integration measure a finite Gaussian integration over the gauge parameter which factors out and plays no physical role. For non-Abelian Yang-Mills gauge theories the properties of Vilkowisky-DeWitt's connection and of Landau-DeWitt's covariant back-ground gauge result in very simple calculation rules.