XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Stefan Berceanu

Coherent states associated to the Jacobi group - a variation on a theme by Erich Kaehler

In my talk I shall present some data about Erich Kaehler (1906 - 2000): his life, his papers. Then I shall recall what the squeezed states are. Mathematically, the squeezed states are realized by the Jacobi group, i.e. the semidirect product of the Heisenberg group and the real symplectic group of an adequate dimension. Among others, the Jacobi group was considered by Kaehler in his paper Raum-Zeit-Individuum (1992). In my papers arXiv: math.DG/0408219 and DG/0604381 I have introduced coherent states associated to the Jacobi group. I have obtained a holomorphic representation of the Jacobi algebra in differential operators based on the manifold M which is the product of the unit disc times the complex plane. In my talk I shall show that when expressed in adequate (Eichler-Zagier) coordinates, the Kaehler two form on M is identical with the Kaehler two-form considered by Erich Kaehler and Rolf Berndt.