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Modern Physics Letters A

q-DEFORMED ALGEBRAS AND THE TWO-ANYON PROBLEM

Pinaki Roy

Published Jun 14, 1996• 4 citations
Publisher

Abstract

It is shown that the two-anyon system in a harmonic oscillator potential and in a constant magnetic field admits certain q-deformed symmetry algebras realized over the space of wave functions of the respective systems.
DOI
10.1142/s021773239600148x

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From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated su(2)su(2)su(2) algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator potential.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.
Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e., the spectrum of one can be obtained from another (with possible exception of the lowest level) by the q2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra su q(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.