Symplectic geometry and quantum mechanics.(English)Zbl 1098.81004

Operator Theory: Advances and Applications 166. Advances in Partial Differential Equations. Basel: Birkhäuser (ISBN 3-7643-7574-4/hbk). xx, 367 p. (2006).
The book is devoted to the techniques and topics intervening in the symplectic treatment of quantum mechanics, in both its semi-classical and its operator-theoretical formulation, with a special emphasis on phase-space techniques.
The first part contains a rigorous presentation of the of symplectic geometry and of its multiply-oriented extension “$$q$$-symplectic geometry”. Complete proofs are given and some new results are included. The basic tool for the study of $$q$$-symplectic geometry is the Arnold-Leray-Maslov (ALM) index and its topological and combinatorial properties. Then one studies and extends to the degenerate case diverses Lagrangian and symplectic intersection indices, with a special emphasis on the Conley-Zehnder index. The latter is essential in the theory of the metaplectic group and its applications to the study of quantum systems with chaotic classical counterpart.
The second part of the book begins with study of the notion of phase of a Lagrangian manifold which, together with the properties of the ALM index, allows to define and describe the quantized Lagrangian manifolds. Then, one gives the geometric definition of the Heisenberg group and algebra, the Weyl operator and Wigner-Moyal transform, the metaplectic group and the associated Maslov indices.
The third part starts with a rigorous geometric treatment of the uncertainty principle of quantum mechanics. It is shown that this principle can be expressed in terms of the notion of symplectic capacity, which is closely related to Williamson’s diagonalization theorem in the linear case, and to Gromov’s non-squeezing theorem. The Hilbert-Schmidt method and the trace-class operators are used to give a rigorous mathematical treatment of the density matrix notion. The Weyl pseudo-differential calculus is extended to phase space using Stone and von Neumann’s theorem on the irreducible representations of the Heisenberg group. The Schrodinger equation in phase space is obtained and it is shown that its solutions are related with those of the usual Schrodinger equation by a “wave-packet transform” generalizing the physicist’s Bargmann transform.
The book is self-contained and can be used as a basic reference for graduate courses. Its topics are also of a great interest for mathematicians and scientists working in geometry and topology or looking to the applications of symplectic geometry in quantum mechanics.

MSC:

 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics 81Q50 Quantum chaos 53D50 Geometric quantization 53D05 Symplectic manifolds (general theory) 35S99 Pseudodifferential operators and other generalizations of partial differential operators 34B20 Weyl theory and its generalizations for ordinary differential equations 47N50 Applications of operator theory in the physical sciences 81S10 Geometry and quantization, symplectic methods