Realizations of the Lie superalgebra q(2) and applications

TL;DR

This paper explores explicit realizations of the Lie superalgebra q(2) as differential operators, demonstrating their application in analyzing the spectra of physical models like the Jaynes-Cummings model.

Contribution

It provides new differential operator realizations of q(2) and applies these to solve for spectra in specific quantum models.

Findings

Realizations of q(2) as differential operators are explicitly constructed.

These realizations facilitate spectral analysis of certain physical Hamiltonians.

Applications include the sphaleron, Moszkowski, and Jaynes-Cummings models.

Abstract

The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.