Deformed Heisenberg Algebra, Fractional Spin Fields, and Supersymmetry without Fermions

Abstract

Within a group-theoretical approach to the description of (2+1)-dimensional anyons, the minimal covariant set of linear differential equations is constructed for the fractional spin fields with the help of the deformed Heisenberg algebra (DHA), [a−, a+]=1+νK, involving the Klein operatorK, {, a±}=0,K2=1. The connection of the minimal set of equations with the earlier proposed “universal” vector set of anyon equations is established. On the basis of this algebra, a bosonization of supersymmetric quantum mechanics is carried out. The construction comprises the cases of exact and spontaneously brokenN=2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin- 12 representation of SU(2) group in terms of one bosonic oscillator. The construction admits an extension to the case of OSp(2∣2) supersymmetry, and, as a consequence, both applications of the DHA turn out to be related. The possibility of “superimposing” the two applications of the DHA for constructing a supersymmetric (2+1)-dimensional anyon system is discussed. As a consequential result we point out that theosp(2∣2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model.