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Journal of Mathematical Physics

Parasupercoherent states and color supergroups

N. Debergh

Published Jan 1, 1992• 0 citations
Publisher

Abstract

Parabosonic coherent states are illustrated from the displacement operator point of view. Such a construction requires the use of the Green–Cusson Ansätze, a specific representation of parastatistics, leading to color supergroups. A comparison with a previous result is shown through the annihilation operator and the minimum uncertainty contexts. The extension to the parasupersymmetric case is also considered.

DOI
10.1063/1.529920

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Parabosonic coherent states are illustrated from the displacement operator point of view. Such a construction requires the use of the Green–Cusson Ansätze, a specific representation of parastatistics, leading to color supergroups. A comparison with a previous result is shown through the annihilation operator and the minimum uncertainty contexts. The extension to the parasupersymmetric case is also considered.
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.
It is well known that Parafermi and Parabose statistics are natural extensions of the usual Fermi and Bose ones, enhancing trilinear (anti)commutation relations instead of bilinear ones. Due to this generalization, positive parameters appear: the so-called orders of paraquantization p (= 1, 2, 3, ...) and h sub 0 (= 1/2, 1, 3/2, ...), respectively, the first value leading in each case to the usual statistics. The superpostion of the parabosonic and parafermionic operators gives rise to parasupermultiplets for which mixed trilinear relations have already been studied leading to two (nonequivalent) sets: the relative Parabose and the relative Parafermi ones. For the specific values p = 1 = 2h sub 0, these sets reduce to the well known supersymmetry. Coherent states associated with this last model have been recently put in evidence through the annihilation operator point of view and the group theoretical approach or displacement operator context. We propose to realize the corresponding studies within the new context p = 2 = 2h sub 0, being then directly extended to any order of paraquantization.
New properties on parabosonic and parafermionic Cusson’s realizations of the Green’s Anzätze are first obtained. Then paraproperties are shown to have a specific interest for our real world, where bosons and fermions have to be superposed if Nature is supersymmetric, these results being obtained on supersymmetric harmonic oscillators. In particular, from the Green–Cusson Ansätze, we construct a class of supersymmetric Hamiltonians that are more general than the ones issued from standard or spin–orbit coupling considerations. We explicitly consider the simplest (previously unknown) super-Hamiltonian constructed from paraproperties in quantum mechanics and establish its invariance Lie superalgebra.
A general approach is presented for constructing coherent states for supersymmetric systems. It uses Rogers's supermanifold formulation of supergroups to extend the group-theoretic method. Supercoherent states are explicitly obtained for the supersymmetric harmonic oscillator. They are shown to be eigenstates of the supersymmetric annihilation operator and to be minimum-uncertainty states. Two more-complex situations with extended physical supersymmetries are also considered: an electron moving in a constant magnetic field, and the electron-monopole system. The supercoherent states for these systems are found using super Baker-Campbell-Hausdorff relations and their interpretation is elucidated.
The general parasupersymmetric annihilation operator of arbitrary order does not reduce to the Kornbluth–Zypman general supersymmetric annihilation operator for the first order. In this paper, we introduce an annihilation operator for a parasupersymmetric harmonic oscillator that in the first order matches with the Kornblouth–Zypman results. Then, using the latter operator, we obtain the parasupercoherent states and calculate their entanglement, uncertainties, and statistics. We observe that these states are entangled for any arbitrary order of parasupersymmetry and their entanglement goes to zero for the large values of the coherency parameter. In addition, we find that the maximum of the entanglement of parasupercoherent states is a decreasing function of the parasupersymmetry order. Moreover, these states are minimum uncertainty states for large and also small values of the coherency parameter. Furthermore, these states show squeezing in one of the quadrature operators for a wide range of the coherency parameter, while no squeezing in the other quadrature operator is observed at all. In addition, using the Mandel parameter, we find that the statistics of these new states are subPoissonian for small values of the coherency parameter.