Bosonic andk-fermionic coherent states for a class of polynomial Weyl–Heisenberg algebras
Mohammed Daoud A, M. Kibler
The aim of this paper is to construct coherent states à la Perelomov and à la Barut–Girardello for a polynomial Weyl–Heisenberg algebra. This generalized Weyl–Heisenberg algebra, denoted by , depends on r real parameters and is an extension of the one-parameter algebra (Daoud and Kibler 2010 J. Phys. A: Math. Theor. 43 115303) which covers the cases of the su (1, 1) algebra (for κ > 0), the su (2) algebra (for κ < 0) and the h 4 ordinary Weyl–Heisenberg algebra (for κ = 0). For finite-dimensional representations of and , where is a truncation of order s of in the sense of Pegg–Barnett, a connection is established with k -fermionic algebras (or quon algebras). This connection makes it possible to use generalized Grassmann variables for constructing certain coherent states. Coherent states of the Perelomov type are derived for infinite-dimensional representations of and for finite-dimensional representations of and through a Fock–Bargmann analytical approach based on the use of complex (or bosonic) variables. The same approach is applied for deriving coherent states of the Barut–Girardello type in the case of infinite-dimensional representations of . In contrast, the construction of coherent states à la Barut–Girardello for finite-dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized Grassmann (or k -fermionic) variables. Some of the results are applied to su (2), su (1, 1) and the harmonic oscillator (in a truncated or not truncated form). This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
arXiv:1110.4799•Journal of Physics A Mathematical and Theoretical•May 30, 2012•22 citations