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Modern Physics Letters A

SUPERSYMMETRIC PARA BOSON-FERMION OSCILLATOR SYSTEMS AND THEIR SPECTRA

Mitsuru Hama, ARATA KOIKE, MAKOTO SAWAMURA, Haruo Suzuki

Published Jul 20, 1991• 3 citations
Publisher

Abstract

Para boson-fermion supersymmetry is exemplified in simple oscillator systems. The parasupercharge satisfies the ordinary supersymmetry algebra. The parabosonic and parafermionic oscillators do not commute and the energy spectra are non-trivial for even the one level system. We calculate the partition functions and compare with those for the non-supersymmetric systems.

DOI
10.1142/s0217732391002189

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Para boson-fermion supersymmetry is exemplified in simple oscillator systems. The parasupercharge satisfies the ordinary supersymmetry algebra. The parabosonic and parafermionic oscillators do not commute and the energy spectra are non-trivial for even the one level system. We calculate the partition functions and compare with those for the non-supersymmetric systems.
We show that the single-mode parafermionic type systems possess supersymmetry, which is based on the symmetry of characteristic functions of the parafermions related to the generalized deformed oscillator of Daskaloyannis et al. The supersymmetry is realized in both unbroken and spontaneously broken phases. As in the case of parabosonic supersymmetry observed recently by one of the authors, the form of the associated superalgebra depends on the order of the parafermion and can be linear or nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems includes usual parafermions, finite-dimensional q-deformed oscillator, q-deformed parafermionic oscillator and parafermionic oscillator with internal Z2 structure.
Recently, we have observed that the bosonized supersymmetric quantum mechanics leads naturally to the R-deformed Heisenberg algebra (RDHA) related to parabosons and parafermions. Here we show that RDHA, in turn, gives rise not only to the bosonized supersymmetry associated with Witten's supersymmetric quantum mechanics but also supplies us with a supersymmetry characterized by a nonlinear superalgebra. Such nonlinear supersymmetry may be specified alternatively by a central extension of the usual N=1 superalgebra with grading operator included nontrivially as an even generator. The systems under consideration are described by the Hamiltonians of the simplest form Hn=a+a−H_n=a^+a^-Hn​=a+a− and Ha=a−a+H_a=a^-a^+Ha​=a−a+ and reveal supersymmetries at the special values of the deformation parameter corresponding to parabosons of odd order. The peculiar nature of parabosonic supersymmetries is encoded in essentially nonlinear structure of their supercharges realized in terms of parabosonic creation-annihilation operators $a^\\pm$. At the values of deformation parameter corresponding to the even order parabosons the spectra of the Hamiltonians HnH_nHn​ and HaH_aHa​ have no supersymmetry but reveal `holes' in comparison with bosonic spectra, whose number is correlated with the parastatistics' order. It is shown that RDHA and the associated linear and nonlinear supersymmetries are realizable in the system of two identical fermions. The nonlinear supersymmetry of pure parabosonic systems possesses the structure of a reduced parasupersymmetry and may also be realized by appropriate modification of the classical analog of Witten's supersymmetric quantum mechanics.
The energy spectra of the systems corresponding to the superposition of (para) bosons and (para)fermions are analyzed. Such a study is subtended by the exploitation of the positive discrete series of the superalgebra osp(2‖2,R) associated with the two relative parabose and parafermi sets of trilinear structure relations. In particular, the spectra corresponding to exact or broken supersymmetry are recovered for specific values of the orders of paraquantization but new ones are also present with their own typical properties.