Realizations of the Lie superalgebra q(2) and applications

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.