- Published 1996

It is closely related to the q-deformed oscillator [1] [2] [3], as is shown below. The context in which q is a root of unity, q = exp 2πi n , is also of great interest. It involves θ such that θ = 0 and and can be discussed by truncating the generic case so as to exclude powers of θ higher than the (n−1)-th. However, if we pass with care from the generic case to the limit in which q is a root of unity much more structure can be exposed. The algebraic structure in question is the full algebraic structure of fractional supersymmetry (FSUSY), not only the generalised Grassmann sector of this Zn-graded theory which is the part that where θ enters but also its bosonic sector. The paper shows how both these ‘sectors’ emerge and discusses the representation of the theory in a product Hilbert space. This has an ordinary oscillator factor for the bosonic degree of freedom, and relates the generalised Grassmann sector to the q-deformed oscillator with deformation parameter q, which is exactly what is needed to ensure proper hermiticity properties. We do not here make any extensive discussion of the interplay between the sectors. But some idea of the insights regarding this interplay can be obtained from [4] which is devoted to the case of q = −1, which is that of ordinary (i.e., Z2-graded) supersymmetry in zero space dimension. It seems worthwhile emphasising that the q-deformed oscillators at deformation parameter q emerge as those generalisations from n = 2 to higher

@inproceedings{Dunne1996DepartamentoDF,
title={Departamento de F{\'i}sica Te{\'o}rica and IFIC,},
author={R . S . Dunne and Andrew Macfarlane and J . A . de Azc{\'a}rraga and Joaquı́n P{\'e}rez},
year={1996}
}