Quantum algebra ofNsuperspace

Abstract

We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincar\'e algebra in the $N>1$ and $D=4$ superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for $D=4$, $N=1$ reported recently. For the case $N=2$ with central charges, we present the equivalent results when the central charge and the mass are different. For the $\ensuremath{\kappa}$-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.