Field Theory with Coordinate Dependent Noncommutativity

TL;DR

This paper develops a framework for classical field theories on n-dimensional noncommutative space-time using associative star products, deriving conservation laws and exploring implications for free and interacting fields, including curved space.

Contribution

It introduces a simple method to derive conservation laws in noncommutative field theories with coordinate-dependent noncommutativity, extending the understanding of energy-momentum conservation.

Findings

Derived local conservation laws for charge and energy-momentum in noncommutative space.

Established an analogy between free fields and damped harmonic oscillators.

Addressed formulation of field theories on curved noncommutative space.

Abstract

We discuss the formulation of classical field theoretical models on $n$-dimensional noncommutative space-time defined by a generic associative star product. A simple procedure for deriving conservation laws is presented and applied to field theories in noncommutative space-time to obtain local conservation laws (for the electric charge and for the energy-momentum tensor of free fields) and more generally an energy-momentum balance equation for interacting fields. For free field models an analogy with the damped harmonic oscillator in classical mechanics is pointed out, which allows us to get a physical understanding for the obtained conservation laws. To conclude, the formulation of field theories on curved noncommutative space is addressed.