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**Lectures on the geometry of quantization.**
*(English)*
Zbl 1049.53061

Berkeley Mathematics Lecture Notes 8. Providence, RI: American Mathematical Society; Berkeley, CA: Berkeley Center for Pure and Applied Mathematics (ISBN 0-8218-0798-6/pbk). vi, 137 p. (1997).

These notes are based on a course entitled “Symplectic Geometry and Geometric Quantization” taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (Hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.

The first chapter gives a concrete example, the harmonic oscillator, the second is concerned with the WKB method and the geometric interpretation of semi-classical solutions to the Schrödinger equation. Then the third chapter offers the tools needed for extending the Hamiltonian viewpoint from \(\mathbb R^{2n}\) to general cotangent bundles and consider the semiclassical approximation. The fourth chapter supplies a systematic method based on Maslov’s technique for quantizing Lagrangian submanifolds of cotangent bundles. The quantization condition for Lagrangian submanifolds is given using the Maslov class.

Chapter 5 opens with a discussion on symplectic reduction and concludes with a general formulation of the quantization problem. Chapter 6 contains the first example of a quantization theory which provides a certain functorial relation. The authors also consider WKB quantization in cotangent bundles and the symbol calculus of Fourier integral operators as a concrete case of the classical-quantum correspondence. Chapter 7 introduces geometric quantization, starting with the prequantum bundle and its geometric properties, a discussion on the need for polarizations and the pairing of polarizations and the Blattner-Kostant-Sternberg kernel a.o.

The last chapter containss two approaches to algebraic quantization, deformation quantization and the method of symplectic groupoids.

Appendices supply fundamentals in (A) Densities; (B) The method of stationary phase; (C) Čech cohomology; (D) Principal \(\mathbb{T}_{\hslash}\) bundles. According to the authors these lectures show ”how symplectic geometry arises from the study of semiclassical solutions to the Schrödinger equation, and in turn provides a geometric foundation for the further analysis of this and other formulations of quantum mechanics”.

The first chapter gives a concrete example, the harmonic oscillator, the second is concerned with the WKB method and the geometric interpretation of semi-classical solutions to the Schrödinger equation. Then the third chapter offers the tools needed for extending the Hamiltonian viewpoint from \(\mathbb R^{2n}\) to general cotangent bundles and consider the semiclassical approximation. The fourth chapter supplies a systematic method based on Maslov’s technique for quantizing Lagrangian submanifolds of cotangent bundles. The quantization condition for Lagrangian submanifolds is given using the Maslov class.

Chapter 5 opens with a discussion on symplectic reduction and concludes with a general formulation of the quantization problem. Chapter 6 contains the first example of a quantization theory which provides a certain functorial relation. The authors also consider WKB quantization in cotangent bundles and the symbol calculus of Fourier integral operators as a concrete case of the classical-quantum correspondence. Chapter 7 introduces geometric quantization, starting with the prequantum bundle and its geometric properties, a discussion on the need for polarizations and the pairing of polarizations and the Blattner-Kostant-Sternberg kernel a.o.

The last chapter containss two approaches to algebraic quantization, deformation quantization and the method of symplectic groupoids.

Appendices supply fundamentals in (A) Densities; (B) The method of stationary phase; (C) Čech cohomology; (D) Principal \(\mathbb{T}_{\hslash}\) bundles. According to the authors these lectures show ”how symplectic geometry arises from the study of semiclassical solutions to the Schrödinger equation, and in turn provides a geometric foundation for the further analysis of this and other formulations of quantum mechanics”.

Reviewer: Olaf Ninnemann (Berlin)