$\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral 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 $\mathrm{GL}(2,\mathbb{C})$ 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.

References in corpus (3)

• POVM Quantization
• Classes of Bivariate Orthogonal Polynomials
• Deformed Complex Hermite Polynomials

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• A concise review on pseudo-bosons, pseudo-fermions and their relatives
• Coherence, squeezing and entanglement -- an example of peaceful coexistence
• On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank
• Radial operators on polyanalytic weighted Bergman spaces
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