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arXiv:math-ph/0203036 · Modern Physics Letters A

REDUCIBILITY AND BOSONIZATION OF PARASUPERSYMMETRIC AND ORTHOSUPERSYMMETRIC QUANTUM MECHANICS

C. Quesne, N. Vansteenkiste

Published May 10, 2002• 6 citations
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Abstract

Order-p parasupersymmetric and orthosupersymmetric quantum mechanics are shown to be fully reducible when they are realized in terms of the generators of a generalized deformed oscillator algebra and a ℤp+1-grading structure is imposed on the Fock space. The irreducible components provide p + 1 sets of bosonized operators corresponding to both unbroken and broken cases. Such a bosonization is minimal.

DOI
10.1142/s021773230200717x

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Order-p parasupersymmetric and orthosupersymmetric quantum mechanics are shown to be fully reducible when they are realized in terms of the generators of a generalized deformed oscillator algebra and a ℤp+1-grading structure is imposed on the Fock space. The irreducible components provide p + 1 sets of bosonized operators corresponding to both unbroken and broken cases. Such a bosonization is minimal.
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Fractional supersymmetric quantum mechanics of order λ is realized in terms of the generators of a generalized deformed oscillator algebra and a ℤλ-grading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing λ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of ℤλ-graded uniform topological symmetry of type (1, 1, …, 1) with topological invariants generalizing the Witten index.
Fractional supersymmetric quantum mechanics of order λ is realized in terms of the generators of a generalized deformed oscillator algebra and a Zλgrading structure is imposed on the Fock space of the latter. This realization is shown to be fully reducible with the irreducible components providing λ sets of minimally bosonized operators corresponding to both unbroken and broken cases. It also furnishes some examples of Zλ-graded uniform topological symmetry of type (1, 1,..., 1) with topological invariants generalizing the Witten index.
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.