In this talk it will be explored the relevance of quantization on the construction of polyanalytic function spaces likewise its relevance on the study of class of Toeplitz operators obtained from Berezin quantization. Starting from a group representation for the Heisenberg group <i>\mathbb{H}</i>, it will be shown the relevance of the symmetries of the metaplectic group <i>\widetilde{Sp(2)}</i> (the so-called Shale-Weil representation) and moreover the relevance of the symmetries of <i>\mathbb{H}\rtimes\widetilde{Sp(2)}</i> in the construction of polyanalytic Fock spaces <i>{\bf F}<sup>n(\mathbb{C})</sup></i> of order <i>n</i> on the complex plane <i>\mathbb{C}</i> likewise polyanalytic Bergmann spaces <i>{\bf A}<sup>n</sup><sub>\alpha(\mathbb{D})</sub></i> of order <i>n</i> (<i>\alpha\geq 0</i>) on a Siegel disk <i>\mathbb{D}</i>. Afterwards, we will construct families of coherent states from the polyanalytic function space <i>{\bf F}<sup>n(\mathbb{C})</sup></i> resp. <i>{\bf A}<sup>n</sup><sub>\alpha(\mathbb{D})</sub></i>, and moreover, families of Toeplitz operators regarding these function spaces.
|