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

S. Twareque Ali ; Fabio Bagarello ; Jean Pierre Gazeau

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

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.

Détail

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

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