Maslov Classes, Metaplectic Representation and Lagrangian Quantization

The Maslov Classes have been playing an essential role in various parts of applied and pure mathematics, and physics, since the early 70's. Their correct definition is due to V. I. Arnold and J. Leray, in the transversal case, and to P. Dazord and the author in the general case. The aim of this book is to give a thorough treatment of the theory of the Maslov classes and of their relationship with the metaplectic group. It is (among other things) shown that these classes can be reconstructed, modulo 4, using only the analytic properties of the metaplectic group. In the last chapter the author sketches a scheme for geometric quantization by introducing two new concepts, that of metaplectic half-form and that of Lagrangian catalogue, the latter generalizes and simplifies the notion of "Lagrangian function" introduced by J. Leray. A Lagrangian catalogue is a collection of metaplectic half-forms which are themselves "cohomological wave functions", whose definition is made possible by using the combinatorial properties of the Maslov classes. The transformation of Lagrangian catalogues under the metaplectic group and of Hamiltonian flows is studied, and it is shown that one thus recovers very easily the so-called "quasi-classical approximation" to the solutions of Schrödinger equation if one introduces a natural concept, that of projection of a Lagrangian catalogue. An application to geometric phase shifts, including Berry's phase, is given.