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A Combinatorial Approach to Diffeomorphism Invariant Quantum Gauge Theories

Jose A. Zapata

TL;DR

The paper presents two graph-based quantum models for diffeomorphism-invariant gauge theories: a piecewise-linear (PL) category tied to a fixed PL structure, and a manifestly combinatorial category built from refining simplicial complexes. Using projective-limit techniques and group averaging, it constructs diffeomorphism-invariant physical Hilbert spaces and shows that the two frameworks yield unitarily equivalent representations of the physical observable algebra in separable spaces. It also demonstrates background-independence of the resulting physics in dimensions up to three and provides a computationally favorable route by embedding continuum physics into a lattice-inspired, combinatorial setting. This work develops a bridge between loop quantum gravity-inspired formalisms and combinatorial/topological methods, offering a concrete, background-free route to quantum geometry with potential pathways to semi-classical limits and macroscopic space-time emergence.

Abstract

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

A Combinatorial Approach to Diffeomorphism Invariant Quantum Gauge Theories

TL;DR

The paper presents two graph-based quantum models for diffeomorphism-invariant gauge theories: a piecewise-linear (PL) category tied to a fixed PL structure, and a manifestly combinatorial category built from refining simplicial complexes. Using projective-limit techniques and group averaging, it constructs diffeomorphism-invariant physical Hilbert spaces and shows that the two frameworks yield unitarily equivalent representations of the physical observable algebra in separable spaces. It also demonstrates background-independence of the resulting physics in dimensions up to three and provides a computationally favorable route by embedding continuum physics into a lattice-inspired, combinatorial setting. This work develops a bridge between loop quantum gravity-inspired formalisms and combinatorial/topological methods, offering a concrete, background-free route to quantum geometry with potential pathways to semi-classical limits and macroscopic space-time emergence.

Abstract

Quantum gauge theory in the connection representation uses functions of holonomies as configuration observables. Physical observables (gauge and diffeomorphism invariant) are represented in the Hilbert space of physical states; physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs. Then a family of graphs embedded in the space manifold (satisfying certain properties) induces a representation of the algebra of physical observables. We construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold, and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes. Even though the two models are different at the kinematical level, they provide unitarily equivalent representations of the algebra of physical observables in separable Hilbert spaces of physical states (their s-knot basis is countable). Hence, the combinatorial framework is compatible with the usual interpretation of quantum field theory.

Paper Structure

This paper contains 9 sections, 12 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. From Quantum Gauge Theory in the Lattice to the Continuum Via the Projective Limit: A Review
  3. PL and combinatoric categories
  4. The PL category
  5. The combinatoric category
  6. Physical observables and physical states
  7. "diffeomorphism" invariance in the PL category
  8. Physical observables and physical states in the combinatorial category
  9. Discussion and Comparison

Key Result

Lemma 1

Given two edges of different graphs $e_1 \in \gamma_1$ and $e_2 \in \gamma_2$, we know that $e_1 \cap e_2$ has finitely many connected components. These connected components are either isolated points or piecewise linear segments.

Theorems & Definitions (12)

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