On a generalized Uncertainty Principle, coherent states, and the moment map

It is shown that, under general circumstances, symplectic G-orbits in a hamiltonian manifold acted on (symplectically) by a Lie group G provide critical points for the norm squared of the moment map. This fact yields a “variational” interpretation of the symplectic orbits appearing in the projective space attached to an irreducible representation of a compact simple Lie group (according to work of Kostant and Sternberg and of Giavarini and Onofri),where the previous function is also related to the invariant uncertainty considered by Delbourgo and Perelomov. A notion of generalized canonical conjugate variables (in the Ka"hler case) is also presented and used in the framework of a Ka"hIer geometric interpretation of the Heisenberg uncertainty relations (building on the analysis given by Cirelli, Mania and Pizzocchero and by Provost and Vallee); it is proved, in particular, that the generalized coherent states of Rawnsley minimize the uncertainty relationsfor any pair of generalized canonically conjugate variables.