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Journal of Physics A Mathematical and General

Hierarchy of the parasupersymmetric Hamiltonians in quantum mechanics

Vyacheslav Pavlovich Spiridonov

Published May 21, 1991• 11 citations
Publisher

Abstract

The simplest parasupersymmetric model found in one-dimensional quantum mechanics is considered. It contains one bosonic and one parafermionic degree of freedom. By successive restriction of the superpotentials the author shows how the algebra of symmetry is simplified. Some discussion of the general parasupersymmetry algebras is presented.

DOI
10.1088/0305-4470/24/10/007

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The simplest parasupersymmetric model found in one-dimensional quantum mechanics is considered. It contains one bosonic and one parafermionic degree of freedom. By successive restriction of the superpotentials the author shows how the algebra of symmetry is simplified. Some discussion of the general parasupersymmetry algebras is presented.
The general features of the degeneracy structure of (p=2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a forth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for these invariants is provided.
The general features of the degeneracy structure of (p = 2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a fourth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for the topological invariants of a class of parasupersymmetric systems is provided.
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.