Canonical quantization of constrained theories on discrete space-time lattices

TL;DR

This paper develops a new canonical quantization framework for constrained systems on discrete space-times, including gauge theories and BF theory, enabling a consistent approach without Euclidean rotation.

Contribution

It introduces a novel canonical formulation and quantization method for constrained systems on discrete space-times, extending to lattice gauge theories and BF theory.

Findings

First lattice treatment of BF theory.

Canonical formulation matches transfer matrix results for lattice gauge theories.

Framework supports Lorentzian signature without Euclidean rotation.

Abstract

We discuss the canonical quantization of systems formulated on discrete space-times. We start by analyzing the quantization of simple mechanical systems with discrete time. The quantization becomes challenging when the systems have anholonomic constraints. We propose a new canonical formulation and quantization for such systems in terms of discrete canonical transformations. This allows to construct, for the first time, a canonical formulation for general constrained mechanical systems with discrete time. We extend the analysis to gauge field theories on the lattice. We consider a complete canonical formulation, starting from a discrete action, for lattice Yang--Mills theory discretized in space and Maxwell theory discretized in space and time. After completing the treatment, the results can be shown to coincide with the results of the traditional transfer matrix method. We then apply the method to BF theory, yielding the first lattice treatment for such a theory ever. The framework presented deals directly with the Lorentzian signature without requiring an Euclidean rotation. The whole discussion is framed in such a way as to provide a formalism that would allow a consistent, well defined, canonical formulation and quantization of discrete general relativity, which we will discuss in a forthcoming paper.