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Modern Physics Letters A

PARASUPERSYMMETRIC QUANTUM MECHANICS

V. A. Rubakov, Vyacheslav Pavlovich Spiridonov

Published Oct 1, 1988• 188 citations
Publisher

Abstract

We construct the second order parasupersymmetric quantum mechanics of one boson and one parafermion degrees of freedom. The parasupersymmetry algebra is Q2 Q†+ QQ†Q+Q†Q2=4QH, Q3=0, [H, Q]=0, where H is the Hamiltonian and Q is the para-supercharge. The spectra of this class of systems are 3-fold degenerate, at least above the second level.

DOI
10.1142/s0217732388001616

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We construct the second order parasupersymmetric quantum mechanics of one boson and one parafermion degrees of freedom. The parasupersymmetry algebra is Q2 Q†+ QQ†Q+Q†Q2=4QH, Q3=0, [H, Q]=0, where H is the Hamiltonian and Q is the para-supercharge. The spectra of this class of systems are 3-fold degenerate, at least above the second level.
We propose an alternative formulation of parasupersymmetric quantum mechanics (PSQM) of arbitrary order which is simpler than the earlier versions and which leads to an explicit expression for the Hamiltonian in terms of the supercharge Q: [Formula: see text] We also use it to construct the PSQM of infinite order whose algebra corresponds to that of Greenberg’s infinite statistics. A model of conformal PSQM in which dilatation and conformal generators are analogously expressed in terms of the conformal supercharges is constructed.
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.
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.