On Fractional Supersymmetric Quantum Mechanics: The Fractional Supersymmetric Oscillator

Abstract

The Hamiltonian for a fractional supersymmetric oscillator is derived from three approaches. The first one is based on the Q-uon → boson + k-fermion decomposition. The second one starts from a generalized Weyl-Heisenberg algebra. Finally, the third one relies on the quantum algebra Uq(sl 2 ) where q is a root of unity. Generalities about supersymmetryWhat is supersymmetry ? Roughly speaking SUperSYmmetry or SUSY can be defined as a symmetry between bosons and fermions (as considered as elementary particles or simply as degrees of freedom). In other words, SUSY is based on the postulated existence of operators Q α which transform a bosonic field into a fermionic field and vice versa. In the context of quantum mechanics, such symmetry operators induce a Z 2 -grading of the Hilbert space of quantum states. In a more general way, fractional SUSY corresponds to a Z k -grading for which the Hilbert space involves both bosonic degrees of freedom (associated to bosons) and k-fermionic degrees of freedom (associated to k-fermions to be described below) with k ∈ N \ {0, 1} ; the case k = 2 corresponds to ordinary SUSY.Where does SUSY come from ? The concept of SUSY goes back to the sixtees when many attemps were done in order to unify external symmetries (described by the Poincaré group) and internal symmetries (described by gauge groups) for elementary particles. These attempts led to a no-go theorem by Coleman and Mandula 1 in 1967 which states that, under reasonable assumptions concerning the S-matrix, the unification of internal and external symmetries can be achieved solely through the introduction of the direct product of 1

Reducibility and Bosonization of Parasupersymmetric and Orthosupersymmetric Quantum Mechanics

“…. , p. Each of the p + 1 sets of operators {Q µ , Q † µ , H µ } satisfies the RSK PSSQM algebra (14) - (16) and is written in terms of a single bosonic degree of freedom through the operators N, a † , a of A(G(N)). We have therefore proved that RSK PSSQM is fully reducible and, in addition, we have obtained a minimal bosonization thereof.…”

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

“…Thus, the algebra A κ can be referred to as a generalized oscillator algebra. In fact, the algebra A κ represents a particular case of the generalized Weyl-Heisenberg algebra W k introduced in [12][13][14][15] to describe a fractional supersymmetric oscillator. A similar algebra, namely the C λ -extended oscillator algebra, was studied in connection with a generalized oscillator [8][9][10][11].…”

“…The use of a generalized oscillator algebra for characterizing a dynamical system gave rise to a great deal of papers. Among many works, we may quote the polynomial Heisenberg algebra worked out in the context of supersymmetry [1][2][3], the deformed Heisenberg algebra introduced in connection with parafermionic and parabosonic systems [4][5][6][7], the C λ -extended oscillator algebra developed in the framework of parasupersymmetric quantum mechanics [8][9][10][11], and the generalized Weyl-Heisenberg algebra W k related to Z k -graded supersymmetric quantum mechanics [12][13][14][15][16]. In this direction, the construction of a truncated generalized oscillator algebra was developed by several authors.…”

On the Statistical Origin of Topological Symmetries

“…The idea of finding a realization of FSQM using an oscillator consisting of a boson and another particle has been pursued in Refs. [12,22].…”