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arXiv:math-ph/0101009 · International School on Symmetry and Structural Properties of Condensed Matter (SSPCM)

ON FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS: THE FRACTIONAL SUPERSYMMETRIC OSCILLATOR

Mohammed Daoud A, M. Kibler

Published May 1, 2001• 6 citations
arXivHTMLPDF

Abstract

The Hamiltonian for a fractional supersymmetric oscillator is derived from three approaches. The first one is based on a decomposition in which a Q-uon gives rise to an ordinary boson and a k-fermion (a k-fermion being an object interpolating between boson and fermion). The second one starts from a generalized Weyl-Heisenberg algebra. Finally, the third one relies on the quantum algebra Uq(sl(2)) where q is a root of unity.

DOI
10.1142/9789812811479_0046

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The Hamiltonian for a fractional supersymmetric oscillator is derived from three approaches. The first one is based on a decomposition in which a Q-uon gives rise to an ordinary boson and a k-fermion (a k-fermion being an object interpolating between boson and fermion). The second one starts from a generalized Weyl-Heisenberg algebra. Finally, the third one relies on the quantum algebra Uq(sl(2)) where q is a root of unity.
Two complementary approaches of N = 2 fractional supersymmetric quantum mechanics of order k are studied in this article. The first one, based on a generalized Weyl-Heisenberg algebra W(k) (that comprizes the affine quantum algebra Uq(sl(2)) with q to k = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second one uses generalized bosonic and k-fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetric oscillator of order k.
Abstract Two complementary approaches of 𝒩 = 2 fractional supersymmetrical quantum mechanics of order k are studied in this article. The first, based on a generalized Weyl–Heisenberg algebra Wk (which comprises the affine quantum algebra Uq(sl2) with qk = 1 as a special case), apparently contains solely one bosonic degree of freedom. The second uses generalized bosonic and k‐fermionic degrees of freedom. As an illustration, a particular emphasis is put on the fractional supersymmetrical oscillator of order k. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
We review some basic elements on k-fermions, which are objects interpolating between bosons and fermions. In particular, we define k-fermionic coherent states and study some of their properties. The decomposition of a Q-uon into a boson and a k-fermion leads to a definition of fractional supercoherent states. Such states involve bosonic coherent states and k-fermionic coherent states. We construct an Hamiltonian which generalizes the ordinary (or Z_2-graded) supersymmetric oscillator Hamiltonian. Our Hamiltonian describes a fractional (or Z_k-graded) supersymmetric oscillator for which the fractional supercoherent states are coherent states.
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.
Recently, we presented a new class of quantum-mechanical Hamiltonians which can be written as the Fth power of a conserved charge: H=QF with F=2, 3,…. This construction, called fractional supersymmetric quantum mechanics, was realized in terms of a paragrassmann variable θ of order F, which satisfies θF=0. Here, we present an alternative realization of such an algebra in which the internal space of the Hamiltonians is described by a tensor product of two paragrassmann variables of orders F and F−1 respectively. In particular, we find q-deformed relations (where q are roots of unity) between different conserved charges.