Projective Techniques and Functional Integration

TL;DR

This paper develops a general framework for integrating over infinite dimensional spaces using projective limits, with applications to gauge theories and quantum gravity, providing a pedagogical approach accessible without prior specialized knowledge.

Contribution

It introduces a new, general method for functional integration over infinite dimensional spaces applicable to gauge theories and quantum gravity, using projective techniques.

Findings

Framework applies to Euclidean and Lorentzian approaches

Constructs diffeomorphism-invariant measures

Facilitates integration over gauge connection spaces

Abstract

A general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact Hausdorff spaces. The procedure is then applied to gauge theories to carry out integration over the non-linear, infinite dimensional spaces of connections modulo gauge transformations. This method of evaluating functional integrals can be used either in the Euclidean path integral approach or the Lorentzian canonical approach. A number of measures discussed are diffeomorphism invariant and therefore of interest to (the connection dynamics version of) quantum general relativity. The account is pedagogical; in particular prior knowledge of projective techniques is not assumed. (For the special JMP issue on Functional Integration, edited by C. DeWitt-Morette.)