Casimir effect in 2+1 dimensional noncommutative theories

TL;DR

This paper investigates the Casimir effect for a scalar field in a 2+1 dimensional noncommutative space, revealing how noncommutativity influences the energy for different geometries and boundary conditions.

Contribution

It introduces a method to impose boundary conditions in noncommutative space and analyzes the Casimir energy for specific geometries, highlighting the effects of noncommutativity.

Findings

Casimir energy for parallel lines matches the commutative case regardless of noncommutativity.

For a circular boundary, noncommutative effects alter the energy dependence on radius.

Energy becomes regular as radius approaches zero due to noncommutative corrections.

Abstract

We study the Dirichlet Casimir effect for a complex scalar field on two noncommutative spatial coordinates plus a commutative time. To that end, we introduce Dirichlet-like boundary conditions on a curve contained in the spatial plane, in such a way that the correct commutative limit can be reached. We evaluate the resulting Casimir energy for two different curves: (a) Two parallel lines separated by a distance $L$, and (b) a circle of radius $R$. In the first case, the resulting Casimir energy agrees exactly with the one corresponding to the commutative case, regardless of the values of $L$ and of the noncommutativity scale $\theta$, while for the latter the commutative behaviour is only recovered when $R >> \sqrt{\theta}$. Outside of that regime, the dependence of the energy with $R$ is substantially changed due to noncommutative corrections, becoming regular for $R \to 0$.