## Maslov classes, metaplectic representation and Lagrangian quantization.(English)Zbl 0872.58031

Mathematical Research. 95. Berlin: Akademie Verlag. 186 p. (1997).
The book is an introduction to the theory of metaplectic groups and Arnold-Leray-Maslov index theory and their applications to quantum mechanics. The material is divided up in five chapters, as follows: Introduction to symplectic geometry, Maslov classes, The metaplectic representation of $$\text{Sp}_2$$, Lagrangian quantization, and Quantum mechanics.
Chapter one is more or less a review of some well known results from symplectic geometry and Hamiltonian mechanics. Chapter two introduces the notions of signature and index of inertia of a triple of Lagrangian planes in arbitrary positions and Maslov index on the coverings of the Lagrangian Grassmannian and on the symplectic group and points out some of their cohomological properties.
Chapter 3 is devoted to the study of the metaplectic group which is the unitary representation in the space of square integrable functions of the double covering of the symplectic group. The metaplectic group plays a central role in problems related to quantization, and this is mainly due to the fact that there is a one-to-one correspondence between quadratic Hamiltonians and the one parameter subgroups of the metaplectic group.
In Chapter 4 the machinery developed in the previous chapters is applied to study the notion of Lagrangian catalogue on a Lagrangian manifold. Finally in the last chapter all the above results are applied to give answers at some concrete problems in quantum mechanics.
The book is interesting and will be useful to all those interested in quantization.

### MSC:

 53D50 Geometric quantization 37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 81S10 Geometry and quantization, symplectic methods 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)