Fractional Supersymmetric Quantum Mechanics Research Articles

Fractional supersymmetric quantum mechanics and lacunary Hermite polynomials

We consider a realization of fractional supersymmetric of quantum mechanics of order r, where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has r-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.

Miscellaneous Applications of Quons

This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We motivate why such algebras are interesting for fractional supersymmetric quantum mechanics, angular momentum theory and quantum information. More precisely, quon algebras are used for (i) a realization of a generalized Weyl-Heisenberg algebra from which it is possible to associate a fractional supersymmetric dynamical system, (ii) a polar decomposition of SU2 and (iii) a construction of mutually unbiased bases in Hilbert spaces of prime dimension. We also briefly discuss (symmetric informationally complete) positive operator valued measures in the spirit of (iii).

Fractional supersymmetry and hierarchy of shape invariant potentials

Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Fractional supersymmetry as a superposition of ordinary supersymmetry

It is shown how to derive fractional supersymmetric quantum mechanics of order k as a superposition of k-1 copies of ordinary supersymmetric quantum mechanics.

Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics

A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this Letter.

FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS, TOPOLOGICAL INVARIANTS AND GENERALIZED DEFORMED OSCILLATOR ALGEBRAS

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.

Topological generalizations of supersymmetry

We introduce a class of generalizations of supersymmetry that involve a set of integer-valued topological invariants. These symmetries are defined in terms of certain conditions on the spectral properties of the corresponding quantum systems. We identify the operator algebras of the generators of these symmetries and show that the algebras of ordinary and fractional supersymmetric quantum mechanics and parasupersymmetric quantum mechanics of order two are special cases of these operator algebras.

Fractional supersymmetry and quantum mechanics

We present a set of quantum-mechanical Hamiltonians which can be written as the F th power of a conserved charge: H = Q F with [ H, Q] = 0 and F = 2,3,…. This new construction, which we call fractional supersymmetric quantum mechanics, is realized in terms of paraGrassmann variables satisfying θ F = 0. Furthermore, in a pseudo-classical context, we describe fractional supersymmetry transformations as the F th roots of time translations, and provide an action invariant under such transformations.

EXTENDED FRACTIONAL SUPERSYMMETRIC QUANTUM MECHANICS

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.