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Projective Techniques and Functional Integration

Abhay Ashtekar, Jerzy Lewandowski

TL;DR

The paper addresses how to perform functional integration over the non-linear, infinite-dimensional space of connections modulo gauge transformations. It develops a general projective-limit framework built from compact Hausdorff spaces, introduces cylindrical $C^*$-algebras, and shows the Gel'fand spectrum of these algebras coincides with the projective limit, enabling measure construction via consistent finite-level marginals. It then applies this machinery to gauge theories through the hoop group, identifying the continuum configuration space ${ar{ rak A}}/{ rak G}$ with a projective limit of ${ m Hom}(S,G)$ and characterizing the holonomy algebra spectrum as ${ m Hom}({ rak HG},G)/{ m Ad}(G)$, while constructing several diffeomorphism-invariant measures (Haar-based, Baez-type, knot-based, homotopy, heat-kernel) and discussing 2D Yang–Mills. The framework unifies non-linear, non-perturbative functional integration with lattice-inspired approaches, offering tools that may be relevant for quantum gravity contexts such as loop quantum gravity and diffeomorphism-invariant quantization.

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.)

Projective Techniques and Functional Integration

TL;DR

The paper addresses how to perform functional integration over the non-linear, infinite-dimensional space of connections modulo gauge transformations. It develops a general projective-limit framework built from compact Hausdorff spaces, introduces cylindrical -algebras, and shows the Gel'fand spectrum of these algebras coincides with the projective limit, enabling measure construction via consistent finite-level marginals. It then applies this machinery to gauge theories through the hoop group, identifying the continuum configuration space with a projective limit of and characterizing the holonomy algebra spectrum as , while constructing several diffeomorphism-invariant measures (Haar-based, Baez-type, knot-based, homotopy, heat-kernel) and discussing 2D Yang–Mills. The framework unifies non-linear, non-perturbative functional integration with lattice-inspired approaches, offering tools that may be relevant for quantum gravity contexts such as loop quantum gravity and diffeomorphism-invariant quantization.

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.)

Paper Structure

This paper contains 11 sections, 12 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Projective Techniques: The general framework
  3. Projective family and the associated $C^{\star}$ algebra of cylindrical functions
  4. The Gel'fand spectrum of ${\overline {\rm Cyl}}({\overline {\cal X}})$.
  5. Regular Borel measures on the projective limit
  6. Quotient of a projective family.
  7. Application to gauge-theories
  8. The projective family
  9. $C^{\star}$-algebras of cylindrical functions
  10. Measures on $\overline{{\cal A}/{\cal G}}$
  11. Discussion

Key Result

Lemma 1

: Given any $f,g\in {\rm Cyl}({\overline {\cal X}})$, there exists $S\in L$ and $f_S, g_S\in C^0({\cal X}_S)$ such that

Theorems & Definitions (12)

  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • ...and 2 more