Study on Noncommutative Representations of Galilean Generators

TL;DR

This paper develops noncommutative representations of Galilean symmetry generators, constructs an invariant dynamical model, and explores their algebraic structure, highlighting implications for electrons in magnetic fields and Berry curvature.

Contribution

It introduces a consistent framework for noncommutative Galilean generators and models their algebraic structure in phase space.

Findings

Reproduces noncommutative algebra via Dirac brackets

Constructs invariant dynamical model in noncommutative phase space

Highlights role of Jacobi identities in physical contexts

Abstract

The representations of Galilean generators are constructed on a space where both position and momentum coordinates are noncommutating operators. A dynamical model invariant under noncommutative phase space transformations is constructed. The Dirac brackets of this model reproduce the original noncommutative algebra. Also, the generators in terms of noncommutative phase space variables are abstracted from this model in a consistent manner. Finally, the role of Jacobi identities is emphasised to produce the noncommuting structure that occurs when an electron is subjected to a constant magnetic field and Berry curvature.