Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative $U(1)$ gauge theory

TL;DR

This paper develops a nonlocal Lagrangian formalism and extends Noether's theorem for such systems, applying these to non-commutative U(1) gauge theory to explore its Hamiltonian structure and quantization.

Contribution

It introduces an extended second Noether theorem for nonlocal Lagrangians and applies it to non-commutative gauge theories, demonstrating new theoretical tools.

Findings

Extended second Noether theorem for nonlocal Lagrangians

Application to non-commutative U(1) gauge theory

Insights into Hamiltonian structure and quantization

Abstract

This article focuses on three main contributions. Firstly, we provide an in-depth overview of the nonlocal Lagrangian formalism. Secondly, we introduce an extended version of the second Noether's theorem tailored for nonlocal Lagrangians. Finally, we apply both the formalism and the extended theorem to the context of non-commutative U(1) gauge theory, including its Hamiltonian and quantization, showcasing their practical utility.