Bose-Einstein condensation theory for any integer spin: approach based in noncommutative quantum mechanics

TL;DR

This paper develops a Bose-Einstein condensation theory for integer spins using noncommutative quantum mechanics, deriving an effective potential and a generalized Gross-Pitaevskii equation with non-local dipolar effects.

Contribution

It introduces a novel approach applying noncommutative quantum mechanics to Bose-Einstein condensation for any integer spin, linking noncommutativity to dipolar interactions.

Findings

Effective potential derived as a multipolar expansion in noncommutativity parameter

At second order, recovers standard dipole-dipole interaction

For $^{52}$Cr, noncommutative effects are indistinguishable from dynamical interactions

Abstract

A Bose-Einstein condensation theory for any integer spin using noncommutative quantum mechanics methods is considered. The effective potential is derived as a multipolar expansion in the non-commutativity parameter ($\theta$) and, at second order in $\theta$, our procedure yields to the standard dipole-dipole interaction with $\theta^2$ playing the role of the strength interaction parameter. The generalized Gross-Pitaevskii equation containing non-local dipolar contributions is found. For $^{52}$Cr isotopes $\theta = C_{dd}/4\pi$ becomes $\sim 10^{-11}$ cm and, thus for this value of $\theta$ one cannot distinguish interactions coming from non-commutativity or those of dynamical origin.