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The author makes a valuable contribution in presenting the basic ideas and methods of geometric quantization, and developing them far enough to give the reader a good sense of the scope of the subject. Since it came into its own as a mathematical topic approximately fifteen years ago in work of B. Kostant and J.-L Souriau, it has found wide application, for example in group representation theory and the general theory of pseudodifferentials and Fourier integral operators. The book is organized in a way which will appeal both to mathematicians and mathematical physicists, in that the generality of treatment and the level of mathematical rigor are sufficient to indicate the direction needed to complete the arguments (if they can be further developed with present knowledge), and yet not so abstract as to discourage a reader who prefers more concreteness.
The first two chapters constitute a (generally coordinate-free) review of symplectic mechanics. Chapter 3 treats symmetry in Hamiltonian systems, including the Marsden-Weinstein reduction method, and presents several examples. Chapter 4 introduces real and complex polarizations and contains a fairly detailed study of Hamilton-Jacobi theory.
Chapter 5 begins the treatment of quantization. The prequantization bundle is introduced with the necessary cohomology conditions, the need for polarizations is discussed, the space of wave functions and the required densities, along with the classical → quantum correspondence of observables. Also discussed is the pairing of polarizations and its relation with the Bargmann transform. Several examples are given of this Blattner-Kostant-Sternberg method.
In Chapter 6 the discrepancies with known answers obtained by the methods of Chapter 5 are pointed out, and the need to use half-forms instead of half-densities for correcting these is explained. This essentially amounts to using the metaplectic group, the double cover of the symplectic group, as the basic structure; heuristically, it resolves sign ambiguities inherent in the previous treatment. The method followed is due to Rawnsley, which involves his and Kostant's idea of cohomological wave functions. Chapter 7 gives an excellent account of spinors and relativistic spinning particles in the context of Poincaré invariance. Their quantization is given in great detail. The link with twistor theory and the massless field equations is briefly treated. The book concludes with a section on spin coupled to a (passive) gravitational field and one on proposals of how to do quantum field theory in the framework of geometric quantization, incorporating ideas of I. E. Segal.
In summary, the author brings the reader to the forefront of current research, and indicates the directions likely to be followed in the future. (MathSciNet)

Bibliogr. p. [309]-312. Index

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