Local Observables in $\operatorname{SU}_q(2)$ Lattice Gauge Theory

TL;DR

This paper develops a $q$-deformed lattice gauge theory framework using local observables based on spinors, applicable to quantum gravity and topological models, with explicit algebraic structures at classical and quantum levels.

Contribution

It introduces a novel set of local observables in $SU_q(2)$ lattice gauge theory using spinors, and establishes their algebraic structure at both classical and quantum levels.

Findings

Classical spinors define invariant local observables on lattice vertices.

Quantum spinors become operators with algebra governed by the quantum R-matrix.

The quantum algebra is a $q$-deformation of $rak{so}^*(2n)$.

Abstract

We consider a deformation of 3D lattice gauge theory in the canonical picture, first classically, based on the Heisenberg double of $\operatorname{SU}(2)$, then at the quantum level. We show that classical spinors can be used to define a fundamental set of local observables. They are invariant quantities which live on the vertices of the lattice and are labelled by pairs of incident edges. Any function on the classical phase space, e.g. Wilson loops, can be rewritten in terms of these observables. At the quantum level, we show that spinors become spinor operators. The quantization of the local observables then requires the use of the quantum $\mathcal{R}$-matrix which we prove to be equivalent to a specific parallel transport around the vertex. We provide the algebra of the local observables, as a Poisson algebra classically, then as a $q$-deformation of $\mathfrak{so}^*(2n)$ at the quantum level. This formalism can be relevant to any theory relying on lattice gauge theory techniques such as topological models, loop quantum gravity or of course lattice gauge theory itself.