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Journal of Physics A: Mathematical and Theoretical 43 (2010) 305304
Complex and real Hermite polynomials and related quantizations
N. Cotfas1, J.-P. Gazeau2, K. Górska

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.
1:  Faculty of Physics
2:  APC - UMR 7164 - AstroParticule et Cosmologie
Physics/Mathematical Physics

Mathematics/Mathematical Physics