${\mathcal D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

TL;DR

This paper explores the ${ m D}$-pseudo-boson formalism through examples involving complex Hermite polynomials and operator-valued functions, revealing new insights into their mathematical structure and quantum interpretation.

Contribution

It introduces a pseudo-bosonic framework for complex Hermite polynomials and operator functions, linking them to quantum phase space and integral quantization.

Findings

Deformed complex Hermite polynomials linked to ${ m GL}(2,{ m C})$ representations

Pseudo-bosonic operator functions provide a quantum version of the complex plane

New insights into the structure of pseudo-bosonic systems and their quantization

Abstract

The ${\mathcal D}$-pseudo-boson formalism is illustrated with two examples. The first one involves deformed complex Hermite polynomials built using finite-dimensional irreducible representations of the group ${\rm GL}(2,{\mathbb C})$ of invertible $2 \times 2$ matrices with complex entries. It reveals interesting aspects of these representations. The second example is based on a pseudo-bosonic generalization of operator-valued functions of a complex variable which resolves the identity. We show that such a generalization allows one to obtain a quantum pseudo-bosonic version of the complex plane viewed as the canonical phase space and to understand functions of the pseudo-bosonic operators as the quantized versions of functions of a complex variable.