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arXiv:math-ph/0212003

Space Structure for the Simplest Parasupersymmetric System

Wei Yang, Si Cong Jing

Published Dec 2, 2002• 1 citation
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Abstract

Structure of the state-vector space for a system consisting of one mode parabose and one mode parafermi degree of freedom with the same parastatistics order $p$ is studied and a complete, orthonormal set of basis vectors in this space is constructed. There is an intrinsic double degeneracy for state vectors with $m$ parabosons and $n$ parafermions, where $m \not= 0$, $n \not= 0$, and $n \not= p$. It is also shown that the degeneracy plays a key role in realization of exact supersymmetry for such a system.

DOI
10.48550/arxiv.math-ph/0212003

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Structure of the state-vector space for a system consisting of one mode parabose and one mode parafermi degree of freedom with the same parastatistics order ppp is studied and a complete, orthonormal set of basis vectors in this space is constructed. There is an intrinsic double degeneracy for state vectors with mmm parabosons and nnn parafermions, where m≠0m \not= 0m=0, n≠0n \not= 0n=0, and n≠pn \not= pn=p. It is also shown that the degeneracy plays a key role in realization of exact supersymmetry for such a system.
This paper researches Fock state space structure and supersymmetry mechanics of parasystem consisting of one degree of freedom of paraboson and one degree of freedom parafermion in the case of the same order p.It is shown that except for the four extreme special states,there is an intrinsic double-degeneracy in the state space structure of a parasystem.Moreover,it is also discovered that the parasystem possesses a hidden intrinsic supersymmetry in addition to the dynamic supersymmetry,and the intrinsic supersymmetry plays a key role in the exact supersymmetry of a parasystem.
The structure of the state-vector space for the two-mode parabose system is investigated and a complete set of state-vectors is constructed. The basis vectors are orthonormal in order p=2p=2p=2. In order p=2p=2p=2, conserved-charge parabose coherent states are constructed and an explicit completeness relation is obtained.
The (p=2) parabose–parafermi supersymmetry is studied in general terms. It is shown that the algebraic structure of the (p=2) parastatistical dynamical variables allows for (symmetry) transformations which mix the parabose and parafermi coordinate variables. The example of a simple parabose-parafermi oscillator is discussed and its symmetries investigated. It turns out that this oscillator possesses two parabose-parafermi supersymmetries. The combined set of generators of the symmetries forms the algebra of supersymmetric quantum mechanics supplemented with an additional central charge. In this sense there is no relation between the parabose–parafermi supersymmetry and the parasupersymmetric quantum mechanics. A precise definition of a quantum system involving this type of parabose-parafermi supersymmetry is offered, thus introducing (p=2) supersymmetric paraquantum mechanics. The spectrum degeneracy structure of general (p=2) supersymmetric paraquantum mechanics is analyzed in detail. The energy eigenvalues and eigenvectors for the parabose–parafermi oscillator are then obtained explicitly. The latter confirms the validity of the results obtained for general supersymmetric paraquantum mechanics.
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.