Complex and real Hermite polynomials and related quantizations

TL;DR

This paper explores alternative coherent state quantizations on the complex plane and real line, revealing non-canonical commutation relations while preserving the harmonic oscillator spectrum, and investigates their statistical and localization properties.

Contribution

It introduces a family of separable Hilbert spaces with overcomplete states that produce non-canonical commutation relations, expanding the understanding of quantization methods beyond the Fock-Bargmann framework.

Findings

Non-canonical commutation relations in new quantizations

Quantum spectrum remains unchanged across different quantizations

Statistical and localization properties are analyzed for these states

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.