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arXiv:hep-th/9603163 · International Journal of Modern Physics A

Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators

Ali Mostafazadeh

Published Jun 20, 1997• 13 citations
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Abstract

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.

DOI
10.1142/s0217751x9700150x

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Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators

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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.
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.
The (p=2) parabose–parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the (p=2) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose-parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose–parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose-parafermi supersymmetry is offered, thus introducing (p=2) supersymmetric paraquantum mechanics. The spectrum degeneracy structure of general (p=2) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose–parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.