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arXiv:1005.5607 · Journal of Mathematical Physics

Aspects of coherent states of nonlinear algebras

T. Shreecharan, K. V. S. Shiv Chaitanya

Published Dec 1, 2010• 2 citations
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Abstract

Various aspects of coherent states of nonlinear su(2) and su(1, 1) algebras are studied. It is shown that the nonlinear su(1, 1) Barut–Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.

DOI
10.1063/1.3514118

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Various aspects of coherent states of nonlinear su(2) and su(1, 1) algebras are studied. It is shown that the nonlinear su(1, 1) Barut–Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.
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In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su (2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su (1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su (1, 1) algebra which contains the Barut–Girardello set and the Perelomov set of the SU (1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.
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'.
It is the aim of this paper to show how to construct á la Perelomov and á la Barut-Girardello coherent states for a polynomial Weyl-Heisenberg algebra. This algebra depends on r parameters. For some special values of the parameter corresponding to r = 1, the algebra covers the cases of the su(1,1) algebra, the su(2) algebra and the ordinary Weyl-Heisenberg or oscillator algebra. For r arbitrary, the generalized Weyl-Heisenberg algebra admits finite or infinite-dimensional representations depending on the values of the parameters. Coherent states of the Perelomov type are derived in finite and infinite dimensions through a Fock-Bargmann approach based on the use of complex variables. The same approach is applied for deriving coherent states of the Barut-Girardello type in infinite dimension. In contrast, the construction of á la Barut-Girardello coherent states in finite dimension can be achieved solely at the price to replace complex variables by generalized Grassmann variables. Finally, some preliminary developments are given for the study of Bargmann functions associated with some of the coherent states obtained in this work.