A Combinatorial Approach to Diffeomorphism Invariant Quantum Gauge Theories

TL;DR

This paper develops a combinatorial framework for diffeomorphism invariant quantum gauge theories, constructing equivalent models using graphs on manifolds and simplicial complexes, ensuring compatibility with quantum field theory principles.

Contribution

It introduces a new combinatorial approach to quantum gauge theories that yields unitarily equivalent representations of physical observables.

Findings

Models are constructed on piecewise linear graphs and simplicial complexes.

Both models produce unitarily equivalent representations of the algebra of physical observables.

The resulting physical state spaces are separable and have a countable s-knot basis.

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.