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arXiv · 1001.3248

Complex and real Hermite polynomials and related quantizations

Katarzyna Gorska·Jean Pierre Gazeau·Nicolae Cotfas
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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 the present 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 states quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

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Katarzyna Gorska, Jean Pierre Gazeau, Nicolae Cotfas. 2010-01-19. Complex and real Hermite polynomials and related quantizations. https://arxiv.org/abs/1001.3248

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