The quantum algebra of superspace

TL;DR

This paper develops the complete set of quantum algebras for massive superparticles in four-dimensional superspace, providing explicit solutions and revealing non-commuting space-time variables linked to internal angular momentum.

Contribution

It introduces explicit realizations of N=1, D=4 quantum superalgebras for massive superparticles using superfield projectors, including a new superparticle action for superspin 1/2.

Findings

Quantum algebras are realized with unconstrained operators on superfield Hilbert spaces.

Space-time variables are non-commutative, with commutators tied to internal angular momentum.

A new superparticle action for superspin 1/2 is proposed, with wave functions corresponding to tensor multiplets.

Abstract

We present the complete set of $N=1$, $D=4$ quantum algebras associated to massive superparticles. We obtain the explicit solution of these algebras realized in terms of unconstrained operators acting on the Hilbert space of superfields. These solutions are expressed using the chiral, anti-chiral and tensorial projectors which define the three irreducible representations of the supersymmetry on the superfields. In each case the space-time variables are non-commuting and their commutators are proportional to the internal angular momentum of the representation. The quantum algebra associated to the chiral or the anti-chiral projector is the one obtained by the quantization of the Casalbuoni-Brink-Schwarz (superspin 0) massive superparticle. We present a new superparticle action for the (superspin 1/2) case and show that their wave functions are the ones associated to the irreducible tensor multiplet.