Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

Canonical coherent states (CS), discovered by Schr¨odinger in 1926, were popularized by Glauber, Sudarshan and Klauder for the description of lasers, and more generally in quantum optics. Among several equivalent definitions, the group-theoretical one, which links them to the Weyl-Heisenberg group, leads to a considerable extension of the concept of CS. The aim of this talk is to survey the results obtained by this approach. We will treat successively: 1. CS on a locally compact group G, built from a unitary square integrable representation of G. 2. The Gilmore–Perelomov theory, in which CS are indexed by points of the quotient G/H of the group G by the isotropy subgroup H of a given admissible vector. 3. CS on an arbitrary quotient G/H, a generalization due to Ali, Gazeau et the author, which allows to extend the construction to a large class of groups, for instance the relativity groups. 4. Finally, wavelets, which are the CS of the affine groups: the “ax+b ” group in one dimension, the similitude group of the plane in dimension 2; the new element here is the central role of dilations. Moreover, the general CS formalism yields a construction of wavelets on several classes of non-Euclidean manifolds, such as the two-sphere or the two-sheeted hyperboloid.