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arXiv · 1509.03822

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

S. Twareque Ali·Fabio Bagarello·Jean Pierre Gazeau
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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.

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S. Twareque Ali, Fabio Bagarello, Jean Pierre Gazeau. 2015-10-01. ${\mathcal D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization. https://doi.org/10.3842/sigma.2015.078

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