Space-Time Symmetries in Noncommutative Gauge Theory: A Hamiltonian Analysis

TL;DR

This paper analyzes space-time symmetries in noncommutative gauge theories using Hamiltonian methods, revealing violations of Lorentz invariance and the non-compliance with Poincare algebra.

Contribution

It constructs Hamiltonian and momentum operators in noncommutative gauge theory and demonstrates the loss of particle Lorentz symmetry and Poincare algebra violations.

Findings

Observer Lorentz invariance is preserved due to translation symmetry.

Particle Lorentz invariance is violated by the noncommutativity tensor.

The Poincare algebra and Lorentz covariant equations do not hold in this framework.

Abstract

We study space-time symmetries in Non-Commutative (NC) gauge theory in the (constrained) Hamiltonian framework. The specific example of NC CP(1) model, posited in \cite{sg}, has been considered. Subtle features of Lorentz invariance violation in NC field theory were pointed out in \cite{har}. Out of the two - Observer and Particle - distinct types of Lorentz transformations, symmetry under the former, (due to the translation invariance), is reflected in the conservation of energy and momentum in NC theory. The constant tensor $\theta_{\mu\nu}$ (the noncommutativity parameter) destroys invariance under the latter. In this paper we have constructed the Hamiltonian and momentum operators which are the generators of time and space translations respectively. This is related to the Observer Lorentz invariance. We have also shown that the Schwinger condition and subsequently the Poincare algebra is not obeyed and that one can not derive a Lorentz covariant dynamical field equation. These features signal a loss of the Particle Lorentz symmetry. The basic observations in the present work will be relevant in the Hamiltonian study of a generic noncommutative field theory.