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

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.