Deformed Heisenberg Algebra, Fractional Spin Fields and Supersymmetry without Fermions

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 operator K, {K, a±} = 0, K = 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 broken N = 2 supersymmetry allowing us to realize a Bose-Fermi transformation and spin-1/2 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. A 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 osp(2|2) superalgebra is realizable as an operator algebra for a quantum mechanical 2-body (nonsupersymmetric) Calogero model. On leave from the Institute for High Energy Physics, Protvino, Moscow Region, Russia; E-mail: [email protected]