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arXiv:hep-th/9701003

On Generalized Quantum Deformations and Symmetries in Quantum Mechanics

Jules Beckers

Published Jan 2, 1997• 0 citations
arXivHTMLPDF

Abstract

Supersymmetric and parasupersymmetric quantum mechanics are now recognized as two further parts of quantum mechanics containing a lot of new informations enlightening (solvable) physical applications. Both contents are here analysed in connection with generalized quantum deformations. In fact, the parasupersymmetric context is visited when the order of paraquantization p is limited to the first nontrivial value p = 2.

DOI
10.48550/arxiv.hep-th/9701003

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Supersymmetric and parasupersymmetric quantum mechanics are now recognized as two further parts of quantum mechanics containing a lot of new informations enlightening (solvable) physical applications. Both contents are here analysed in connection with generalized quantum deformations. In fact, the parasupersymmetric context is visited when the order of paraquantization p is limited to the first nontrivial value p = 2.
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 revisit parasupersymmetric quantum mechanics of arbitrary order and present a set of nontrivial relations, which characterizes the most general multilinear part of the associated parasupersymmetric algebra. We then show that the formulation of multilinear relations leads immediately to a polynomial of parasupersymmetric Hamiltonian in terms of the corresponding parasupercharges. The deduction of higher derivative supersymmetric quantum mechanics directly via this parasupersymmetric formulation is discussed. The complete degenerate structure of the energy spectrum for parasupersymmetric quantum mechanics of order p is systematically analyzed. Finally, the notion of cyclic symmetry is introduced and the algebra of cyclic charge operators of arbitrary order is developed, based on the parasupersymmetric formalism.
The general features of the degeneracy structure of (p=2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a forth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupersymmetry are shown to be related to a pair of Fredholm operators satisfying two compatibility conditions. An explicit algebraic expression for these invariants is provided.
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.
As with the earlier edition, this book provides an accessible introduction to supersymmetric quantum mechanics and its applications in quantum, statistical and solid state physics. Building on the earlier edition it now encapsulates the substantial developments that have been made in supersymmetric quantum mechanics in recent years including conditionally exactly solvable systems, models of classical field theory and the many new interesting aspects within the area of stochastic dynamics, the Pauli and Dirac Equation, coherent states and non-linear algebras. Aimed at graduate students and scientists this book provides a thorough review on supersymmetric quantum mechanics and now includes problems and solutions.