# Quantum Link Models: A Discrete Approach to Gauge Theories

## TL;DR

Quantum link models offer a discrete, operator-based formulation of gauge theories that could simplify numerical simulations and connect to both condensed matter and particle physics applications.

## Contribution

This paper introduces quantum link models with non-commuting operators, relating them to traditional gauge theories and exploring their potential for simplified numerical treatment and broader physical applications.

## Key findings

- Constructed explicit U(1) and SU(2) quantum link models.
- Quantum link models are related to ordinary gauge theories via dimensional reduction.
- Discrete configuration space may simplify numerical simulations.

## Abstract

We construct lattice gauge theories in which the elements of the link matrices are represented by non-commuting operators acting in a Hilbert space. These quantum link models are related to ordinary lattice gauge theories in the same way as quantum spin models are related to ordinary classical spin systems. Here U(1) and SU(2) quantum link models are constructed explicitly. As Hamiltonian theories quantum link models are nonrelativistic gauge theories with potential applications in condensed matter physics. When formulated with a fifth Euclidean dimension, universality arguments suggest that dimensional reduction to four dimensions occurs. Hence, quantum link models are also reformulations of ordinary quantum field theories and are applicable to particle physics, for example to QCD. The configuration space of quantum link models is discrete and hence their numerical treatment should be simpler than that of ordinary lattice gauge theories with a continuous configuration space.