# Quantum lattice gauge fields and groupoid C*-algebras

## TL;DR

This paper introduces an operator-algebraic framework for quantizing and reducing lattice gauge theories using groupoid C*-algebras, enabling a systematic approach to continuum and thermodynamic limits.

## Contribution

It develops a novel method combining groupoid C*-algebras and Rieffel induction for quantum gauge symmetry implementation in lattice field theories.

## Key findings

- Provides a duality framework between Hilbert spaces and configuration spaces.
- Establishes a limit procedure for continuum and thermodynamic regimes.
- Ensures gauge-equivariant reduction in the limiting process.

## Abstract

We present an operator-algebraic approach to the quantization and reduction of lattice field theories. Our approach uses groupoid C*-algebras to describe the observables and exploits Rieffel induction to implement the quantum gauge symmetries. We introduce direct systems of Hilbert spaces and direct systems of (observable) C*-algebras, and, dually, corresponding inverse systems of configuration spaces and (pair) groupoids. The continuum and thermodynamic limit of the theory can then be described by taking the corresponding limits, thereby keeping the duality between the Hilbert space and observable C*-algebra on the one hand, and the configuration space and the pair groupoid on the other. Since all constructions are equivariant with respect to the gauge group, the reduction procedure applies in the limit as well.