Résumé : In this talk, we consider the normal ordering of operators of the type $$\Omega=\sum_{\alpha+\beta=r}c_{\alpha,\beta}(a^+)^\alpha a(a^+)^\beta,\ \ \alpha,\beta,r\textrm{ integers}$$ where $a$ (resp. $a^{+}$) is a boson annihilation (resp. creation) operator; these satisfy $[a,a^{+}]\equiv a a^{+}-a^{+}a=1$, and for the purposes of this presentation may be thought of as $a\equiv d/dx$ and $a^{+}\equiv x$. We discuss the integration of the one-parameter groups $e^{\lambda\Omega}$ and their combinatorial by-products. In particular we show how these groups can be realized as groups of substitutions with prefactor functions. To end with, we provide a recent application of the concept one-parameter groups to arithmetics.

Dernière modification : Tuesday 11 February 2025

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