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arXiv:hep-th/9406112 · Journal of Physics A Mathematical and General

Parabosons versus supersymmetry in Jahn-Teller systems

E. D. Savage, G E Stedman

Published Jul 7, 1994• 1 citation
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Abstract

The applications of parabosons by Schmutz (1980) and of supersymmetry by Jarvis and Stedman (1984) in Jahn-Teller systems are compared and contrasted. Although a parasupersymmetric Jahn-Teller system has not yet been identified, the method of Schmutz is used here to show that the E* epsilon supersymmetric Jahn-Teller Hamiltonian can be written in terms of paraboson operators.

DOI
10.1088/0305-4470/27/13/039

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The applications of parabosons by Schmutz (1980) and of supersymmetry by Jarvis and Stedman (1984) in Jahn-Teller systems are compared and contrasted. Although a parasupersymmetric Jahn-Teller system has not yet been identified, the method of Schmutz is used here to show that the E* epsilon supersymmetric Jahn-Teller Hamiltonian can be written in terms of paraboson operators.
Coherent states are constructed in p=2-parasupersymmetric quantum mechanics in connection with the two relative para-Bose and para-Fermi sets of trilinear structure relations. They are called parasupercoherent states. Parasupersymmetric operators [such as the Hamiltonian and (two) annihilation operators] are introduced and discussed. In particular, the parasuperspectrum is determined and compared with the recent results obtained by Rubakov and Spiridonov [Mod. Phys. Lett. A 3, 1337 (1988)]. The superalgebra contents subtended by osp(2/2) and osp(3/2) are analyzed and exploited in order to get constants of motion in both contexts through parallel properties on the supersymmetric harmonic oscillator.
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.