Coherent and squeezed states of quantum Heisenberg algebras

Abstract

Starting from deformed quantum Heisenberg Lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator. Then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states. They are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers. Some properties of these deformed states also are analyzed. * email address: alvarez@dms.umontreal.ca

On Linear Differential Equations Involving a Para-Grassmann Variable

“…Remark 7.1. As noted already in the introduction, a certain differential equation involving a para-Grassmann variable was solved in [48] (by writing f (x, θ) = p k=0 f k (x)θ k , inserting this into the differential equation and comparing coefficients of θ k ). Adapting the notation to the one used in the present paper, the differential equation considered in [48] can be written as…”

“…Before turning to a more detailed description of the present paper we would like to point out that in [48] a differential equation containing a para-Grassmann variable is solved in connection with deformed coherent states. To the best of our knowledge this is the only paper where such a differential equation is solved explicitly (more details can be found below).…”

“…In particular in this approach the geometrical point of view of merging the ordinary variables with the para-Grassmann variables into a set of coordinates on a parasuperspace and considering the fractional supersymmetric fields as functions on this space has been used in close analogy to the well-known superspace and superfields. From a different point of view a connection to deformed algebras or quantum groups was reported [42,43,44,45,46,47,48,49]. Of course, para-Grassmann variables have also been considered in their own right [50,51,52,53].…”