### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 078, 23 pages      arXiv:1509.03822      https://doi.org/10.3842/SIGMA.2015.078

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

S. Twareque Ali a, Fabio Bagarello b and Jean Pierre Gazeau cd
a) Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada H3G 1M8
b) Dipartimento di Energia, ingegneria dell'Informazione e modelli Matematici, Scuola Politecnica, Università di Palermo, I-90128 Palermo, and INFN, Torino, Italy
c) APC, UMR 7164, Univ Paris Diderot, Sorbonne Paris-Cité, 75205 Paris, France
d) Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, 22290-180 Rio de Janeiro, Brazil

Received March 28, 2015, in final form September 21, 2015; Published online October 01, 2015

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.

Key words: pseudo-bosons; coherent states; quantization; complex Hermite polynomials; finite group representation.

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