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Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

Jerzy Lewandowski, Thomas Thiemann

TL;DR

This work extends the loop-quantized, diffeomorphism-invariant framework for gravity from piecewise analytic to fully smooth paths by leveraging Baez–Sawin webs and a Master Theorem that guarantees holonomic independence. It introduces spin-web states as a generalization of spin-networks, develops a robust diffeomorphism averaging procedure on nondegenerate webs, and shows that a natural, diffeomorphism-invariant measure can be defined on the extended quantum configuration space ${\overline{\cal A}}$. The authors further analyze the action of diffeomorphism-invariant operators, identify sector preservation by Hamiltonian-constraint and geometric operators, and demonstrate the well-definedness of the dual Hamiltonian constraint on averaged states. Together, these results provide a comprehensive, diffeomorphism-invariant quantization scheme for gravitational connections in the smooth category, with concrete extensions of the Ashtekar–Isham algebra and gravitational observables.

Abstract

In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we $(i)$ characterize the spectrum of the Ashtekar-Isham configuration space, $(ii)$ introduce spin-web states, a generalization of the spin-network states, $(iii)$ extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally $(iv)$ extend the 3-geometry operators and the Hamiltonian operator.

Diffeomorphism invariant Quantum Field Theories of Connections in terms of webs

TL;DR

This work extends the loop-quantized, diffeomorphism-invariant framework for gravity from piecewise analytic to fully smooth paths by leveraging Baez–Sawin webs and a Master Theorem that guarantees holonomic independence. It introduces spin-web states as a generalization of spin-networks, develops a robust diffeomorphism averaging procedure on nondegenerate webs, and shows that a natural, diffeomorphism-invariant measure can be defined on the extended quantum configuration space . The authors further analyze the action of diffeomorphism-invariant operators, identify sector preservation by Hamiltonian-constraint and geometric operators, and demonstrate the well-definedness of the dual Hamiltonian constraint on averaged states. Together, these results provide a comprehensive, diffeomorphism-invariant quantization scheme for gravitational connections in the smooth category, with concrete extensions of the Ashtekar–Isham algebra and gravitational observables.

Abstract

In the canonical quantization of gravity in terms of the Ashtekar variables one uses paths in the 3-space to construct the quantum states. Usually, one restricts oneself to families of paths admitting only finite number of isolated intersections. This assumption implies a limitation on the diffeomorphisms invariance of the introduced structures. In this work, using the previous results of Baez and Sawin, we extend the existing results to a theory admitting all the possible piecewise smooth finite paths and loops. In particular, we characterize the spectrum of the Ashtekar-Isham configuration space, introduce spin-web states, a generalization of the spin-network states, extend the diffeomorphism averaging to the spin-web states and derive a large class of diffeomorphism invariant states and finally extend the 3-geometry operators and the Hamiltonian operator.

Paper Structure

This paper contains 35 sections, 17 theorems, 177 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The algebra of cylindrical functions on the space of connections
  3. Preliminaries
  4. The Baez-Sawin framework
  5. Holonomic independence of the curves in a Baez-Sawin web.
  6. Immediate implications of the Master Theorem
  7. Group Identities
  8. Spectrum of the Ashtekar -- Isham algebra
  9. A natural measure
  10. Spin-webs and diffeomorphism averaging
  11. Spin-webs
  12. Averaging over the diffeomorphism group
  13. Diffeomorphism invariant operators
  14. Ranges of diffeomorphism-invariant operators
  15. Strongly degenerate labelled webs
  16. ...and 20 more sections

Key Result

Theorem 2.1

Let $\{p_1,...,p_n\}$ be a finite family of paths which satisfies the properties $(a)$ and $(b)$ above. Then the paths are holonomically independent.

Figures (3)

  • Figure 1: The edge $e_1$ is the horizontal line whereas the edge $e_2$ consists of the infinitely many bumps up the horizontal line connected with the horizontal segments (only four bumps of $e_2$ are visible in the figure.) The edges $e'_1$ and $e'_2$ are obtained by replacing the solid segment $s$ by the dashed segment $s'$.
  • Figure 2: The web $\tilde{w}$ consists of three tassels. The first tassel is based at the point $p$ and consists of the segments connecting $p$ with the points $q'_5$ and $q'_6$. The second tassel is based at the point $p'_1$. It consists of the segments connecting $p'_1$ with the points $q'_3, q'_4, q'_5, q'_6$. The third tassel is the set of the segments connecting the point $p'_2$ with the points $q_1, q_2, q'_3, q'_4$.
  • Figure :

Theorems & Definitions (19)

  • Theorem 2.1: Master Theorem
  • Definition 4.1
  • Theorem 4.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • ...and 9 more