Area Preserving Transformations in Non-commutative Space and NCCS Theory

TL;DR

This paper explores area-preserving transformations in non-commutative space, linking them to quantum deformations of classical symplectic diffeomorphisms and formulating the condition as a field equation in non-commutative Chern-Simons theory.

Contribution

It introduces a heuristic rule for non-commutative area transformations, formulates the preservation condition as a gauge field equation, and extends the framework to higher dimensions with an algebraic structure.

Findings

Formulated non-commutative area preservation as a field equation.

Identified the infinite dimensional $ ext{sin}$-Lie algebra structure.

Applied the theory to electrons in the lowest Landau level.

Abstract

We propose an heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformation of the classical symplectic diffeomorphisms. Area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. The higher dimensional generalization is suggested and the corresponding algebraic structure - the infinite dimensional $\sin$-Lie algebra is extracted. As an illustrative example the second-quantized formulation for electrons in the lowest Landau level is considered.