Noncommutative Geometry and Anyonic Field Theory in the Magnetic Field

TL;DR

This paper explores how a magnetic field induces noncommutative geometry in Minkowski space and studies the thermodynamic properties of anyons within this framework, highlighting the role of Moyal phase factors.

Contribution

It presents a straightforward method to realize noncommutative spacetime via magnetic fields and analyzes the impact of Moyal phases on anyonic thermodynamics.

Findings

Moyal phase factors influence the thermodynamic potential.

Projection to the lowest Landau level simplifies calculations.

Noncommutative geometry affects the thermodynamic behavior of anyons.

Abstract

We consider an easy way to get the noncommutative spacetime in Minkowski space. This corresponds to introducing a magnetic field ${\rm\bf B} = B \hat {\rm\bf k}$ in the plane. We construct a green's function in coordinate space which includes a Moyal phase factor. The projection to the lowest Landau level(LLL) is necessary for a simple calculation. Using this green's function and a second quantized formalism, we study the thermodynamic property of the anyons on the noncommutative geometry. It turns out that the Moyal phase factors contribute to the thermodynamic potential $\Omega$ as opposed to the free-particle nature.