Deformed Heisenberg algebra, fractional spin fields and supersymmetry without fermions

TL;DR

This paper develops a group-theoretical framework using deformed Heisenberg algebra to describe fractional spin fields and constructs a bosonized supersymmetric quantum mechanics model in (2+1) dimensions.

Contribution

It introduces a minimal covariant set of equations for anyons based on DHA and extends this to supersymmetric models without fermions, linking to Calogero systems.

Findings

Established connection with universal anyon equations

Realized supersymmetry via bosonization in quantum mechanics

Related DHA applications to OSp(2|2) supersymmetry

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), $[a^{-},a^{+}]=1+\nu K$, involving the Klein operator $K$, $\{K,a^{\pm}\}=0$, $K^{2}=1$. 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$\vert$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.