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arXiv:hep-th/9303132 · Journal of Mathematical Physics

Study of quommutators of quantum variables and generalized derivatives

David Fairlie, J. Nuyts

Published Oct 1, 1993• 6 citations
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Abstract

A general deformation of the Heisenberg algebra is introduced with two deformed operators instead of just one. This is generalized to many variables, and permits the simultaneous existence of coherent states, and the transposition of creation operators.
DOI
10.1063/1.530350

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The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.
The qqq-Deformed Wigner Oscillator in Quantum Mechanics
Using a generalization of the q-commutation relations, we develop a formalism in which it is possible to define generalized q-bosonic operators. This formalism includes both types of the usual q-deformed bosons as special cases. The coherent states of these operators show interesting and novel noise reduction properties including simultaneous squeezing in both field components, unlike the conventional case in which squeezing is permitted in only one component. This also contrasts with the usual quantum group deformation which also only permits one component squeezing.