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 1970s. This volume intends to provide a thorough treatment of the Maslov classes and of their relationship with the metaplectic group. It is shown that these classes can be reconstructed, modulo 4, using only analytic properties of the metaplectic group. In the last chapter, the author sketches a scheme for geometric quantatization by introducing two new concepts, that of metaplectic half-form and that of Lagrangrian catalogue, the latter generalizes and simplifies the notion of "Lagrangrian function" introduced by J. Leray. A Langrangrian 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 the 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 Schrodinger equation if a natural concept is introduced, that of natural projection of a Lagrangian catalogue. An application to geometric phase shifts, including Berry's phase is given.