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Differential Geometry on the Space of Connections via Graphs and Projective Limits

Abhay Ashtekar, Jerzy Lewandowski

TL;DR

The paper develops a metric-free differential geometry on the quantum configuration space $\overline{{\cal A}/{\cal G}}$, obtained as a projective limit of graphs labeling compact manifolds. It constructs differential forms, vector fields, volume forms, and momentum operators on the upstairs space $\overline{\cal A}$ and descends invariantly to the quotient $\overline{{\cal A}/{\cal G}}$, all without a background metric. By introducing edge-metrics $l$ and natural co-metrics $k_{o}$, it defines Laplacians $\Delta_{(l)}$ and $\Delta^{o}$ and associated heat-kernel measures that respect diffeomorphism invariance. The framework also handles non-compact structure groups and discusses extensions to quantum gravity via diffeomorphism-invariant measures and connections to non-perturbative loop-quantum gravity concepts. Overall, the work provides a robust algebraic-geometric toolkit for analyzing gauge theories in a background-independent setting, with concrete operators and measures that can underpin rigorous treatments of quantum gravity.

Abstract

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. $\agb$ is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, $\agb$ is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on $\agb$: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although $\agb$ is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity.

Differential Geometry on the Space of Connections via Graphs and Projective Limits

TL;DR

The paper develops a metric-free differential geometry on the quantum configuration space , obtained as a projective limit of graphs labeling compact manifolds. It constructs differential forms, vector fields, volume forms, and momentum operators on the upstairs space and descends invariantly to the quotient , all without a background metric. By introducing edge-metrics and natural co-metrics , it defines Laplacians and and associated heat-kernel measures that respect diffeomorphism invariance. The framework also handles non-compact structure groups and discusses extensions to quantum gravity via diffeomorphism-invariant measures and connections to non-perturbative loop-quantum gravity concepts. Overall, the work provides a robust algebraic-geometric toolkit for analyzing gauge theories in a background-independent setting, with concrete operators and measures that can underpin rigorous treatments of quantum gravity.

Abstract

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, , of the space of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on : differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity.

Paper Structure

This paper contains 25 sections, 21 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Projective Techniques: general framework
  3. Projective families for spaces of connections.
  4. Set of labels.
  5. The projective family for ${\cal A}$.
  6. The projective family for ${\cal G}$.
  7. The quotient ${\overline {\cal A}}/{\overline {\cal G}}$ and the projective family for $\overline{{\cal A}/{\cal G}}$.
  8. Elements of the differential geometry on ${\overline {\cal A}}$
  9. Differential forms
  10. Volume forms.
  11. Vector fields.
  12. Vector fields as momentum operators
  13. Elements of differential geometry on $\overline{{\cal A}/{\cal G}}$
  14. Forms and volume forms
  15. Vector fields on $\overline{{\cal A}/{\cal G}}$
  16. ...and 10 more sections

Key Result

Theorem 1

: Let $({\cal X}_\gamma, p_{\gamma \gamma'})_{\gamma\gamma' \in L}$ be a compact, Hausdorff projective family and ${\overline {\cal X}}$ be its projective limit; (a) Suppose $\mu$ is a regular Borel, probability measure on ${\overline {\cal X}}$. Then $\mu$ defines a consistent family of regular, Bo (b) Suppose $(\mu_\gamma)_{\gamma,\gamma'\in L}$ is a consistent family of regular, Borel, probabil

Theorems & Definitions (21)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Lemma 1
  • Theorem 2
  • ...and 11 more