Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of complex plane produces both canonical commutation rule and quantum spectrum the harmonic oscillator (up to the addition of a constant). In the present, we show that these two issues are not necessarily coupled: there exists a of separable Hilbert spaces, including the usual Fock-Bargmann space, in each element in this family there exists an overcomplete set of-norm states resolving the unity. With the exception of the Fock-Bargmann, they all produce non-canonical commutation relation whereas the quantum of the harmonic oscillator remains the same up to the addition of a. The statistical aspects of these non-equivalent coherent states are investigated. We also explore the localization aspects in the line yielded by similar quantizations based on real Hermite polynomials.