Abstract
We provide a realization of fractional supersymmetry quantum mechanics of order r, where the Hamiltonian and the supercharges involve the fractional Dunkl transform as a Klein type operator. We construct several classes of functions satisfying certain orthogonality relations. These functions can be expressed in terms of the associated Laguerre orthogonal polynomials and have shown that their zeros are the eigenvalues of the Hermitian supercharge. We call them the supersymmetric generalized Hermite polynomials.
Highlights
Supersymmetry relates bosons and fermions on the basis of Z2 -graded superalgebras [1,2], where the fermionic set is realized in terms of matrices of finite dimension or in terms of Grassmann variables [3]
The usual construction of Witten’s supersymmetric quantum mechanics with the superalgebra (1) is performed by introduction of fermion degrees of freedom which commute with bosonic degrees of freedom
We may quote the deformed Heisenberg algebra introduced in connection with parafermionic and parabosonic systems [3,4], the Cλ -extended oscillator algebra developed in the framework of parasupersymmetric quantum mechanics [8], and the generalized Weyl–Heisenberg algebra Wk related to Zk -graded supersymmetric quantum mechanics [3]
Summary
Supersymmetry relates bosons and fermions on the basis of Z2 -graded superalgebras [1,2], where the fermionic set is realized in terms of matrices of finite dimension or in terms of Grassmann variables [3]. The usual construction of Witten’s supersymmetric quantum mechanics with the superalgebra (1) is performed by introduction of fermion degrees of freedom (realized in a matrix form, or in terms of Grassmann variables) which commute with bosonic degrees of freedom. Another realization of supersymmetric quantum mechanics, called minimally bosonized supersymmetric quantum [1,4,5], is built by taking the supercharge as the following Dunkl-type operator:.
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