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Journal of Physics A Mathematical and General

On a hidden dynamical SU(3)-symmetry in parasupersymmetric quantum mechanics

J. Beckers, N. Debergh, A. G. Nikitin

Published Sep 7, 1993• 6 citations
Publisher

Abstract

The superposition of bosons and p=2-parafermions has led to the so-called parasupersymmetric quantum mechanics. The authors show that the structures subtending these developments contain a hidden dynamical SU(3)-symmetry effectively associated with the fundamental irreducible representation 3 (unitarily equivalent to the Gell-Mann one). The representation 3* is also visited. The generalization to arbitrary orders p of paraquantization is briefly discussed.

DOI
10.1088/0305-4470/26/17/017

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arXiv:math-ph/0203036•Modern Physics Letters A•May 10, 2002•6 citations
The superposition of bosons and p=2-parafermions has led to the so-called parasupersymmetric quantum mechanics. The authors show that the structures subtending these developments contain a hidden dynamical SU(3)-symmetry effectively associated with the fundamental irreducible representation 3 (unitarily equivalent to the Gell-Mann one). The representation 3* is also visited. The generalization to arbitrary orders p of paraquantization is briefly discussed.
We revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations, which characterizes the most general multilinear part of the associated parasupersymmetric algebra. We then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric Hamiltonian in terms of the corresponding parasupercharges. The deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed. The complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed. Finally, the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed, based on the parasupersymmetric formalism.
It is well known that Parafermi and Parabose statistics are natural extensions of the usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones. Due to this generalization, positive parameters appear: the so-called orders of paraquantization p (= 1, 2, 3, ...) and h sub 0 (= 1/2, 1, 3/2, ...), respectively, the first value leading in each case to the usual statistics. The superpostion of the parabosonic and parafermionic operators gives rise to parasupermultiplets for which mixed trilinear relations have already been studied leading to two (nonequivalent) sets: the relative Parabose and the relative Parafermi ones. For the specific values p = 1 = 2h sub 0, these sets reduce to the well known supersymmetry. Coherent states associated with this last model have been recently put in evidence through the annihilation operator point of view and the group theoretical approach or displacement operator context. We propose to realize the corresponding studies within the new context p = 2 = 2h sub 0, being then directly extended to any order of paraquantization.