Complex and real Hermite polynomials and related quantizations

N. Cotfas 1 J.-P. Gazeau 2 K. Górska
2 APC - THEORIE
APC - UMR 7164 - AstroParticule et Cosmologie
Abstract : It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock-Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock-Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.
Document type :
Journal articles
Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2010, 43, pp.305304. <10.1088/1751-8113/43/30/305304>


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Submitted on : Sunday, October 7, 2012 - 9:37:55 PM
Last modification on : Tuesday, October 28, 2014 - 6:18:12 PM

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N. Cotfas, J.-P. Gazeau, K. Górska. Complex and real Hermite polynomials and related quantizations. Journal of Physics A: Mathematical and Theoretical, Institute of Physics: Hybrid Open Access, 2010, 43, pp.305304. <10.1088/1751-8113/43/30/305304>. <hal-00739323>

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