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**Poisson geometry and deformation quantisation. An introduction.
(Poisson-Geometrie und Deformationsquantisierung. Eine Einführung.)**
*(German)*
Zbl 1139.53001

Berlin: Springer (ISBN 978-3-540-72517-6/pbk). xii, 612 p. (2007).

The book under review provides a welcome unified exposition on Poisson geometry and deformation quantization. The first chapter is devoted to elementary ideas of Hamiltonian mechanics ranging from the analytical aspects (local dynamical systems associated with ordinary differential equations) to geometrical aspects of Hamiltonian flows and further to algebraic aspects encoded in the Poisson algebra. There follows a chapter in which the differential geometric foundations are laid for the main developments in the book. The third and fourth chapter are devoted to symplectic geometry and Poisson geometry, respectively. Among the topics discussed therein we wish to single out the Marsden-Weinstein reduction, the Poisson cohomology, and the formal Poisson tensors.

One then turns to the circle of ideas related to quantization. Thus, the fifth chapter of the book has the title “Quantization: first steps” and begins by heuristically discussing the problem of quantization and the relationship between classical mechanics and quantum mechanics. An other topic touched on in this chapter is the canonical quantization for polynomial functions, which motivates an elementary exposition on pseudo-differential operators and their symbol calculus. The sixth chapter is devoted to formal deformation quantization: star products on Poisson manifolds, the algebraic deformation theory after Gerstenhaber, and Fedosov’s construction of star products on symplectic manifolds. The final chapter of the book represents quantum states as positive functionals on the algebra of observables of a quantum system. The associated representations obtained by means of the Gelfand-Naĭmark-Segal construction and their relationship to the deformation quantization are discussed as well.

This book will certainly prove to be a useful introductory text for a very active research area where ideas of mathematical physics, differential geometry and operator theory are interacting. The exposition is clear and many comments with a physical flavor are included throughout the text.

One then turns to the circle of ideas related to quantization. Thus, the fifth chapter of the book has the title “Quantization: first steps” and begins by heuristically discussing the problem of quantization and the relationship between classical mechanics and quantum mechanics. An other topic touched on in this chapter is the canonical quantization for polynomial functions, which motivates an elementary exposition on pseudo-differential operators and their symbol calculus. The sixth chapter is devoted to formal deformation quantization: star products on Poisson manifolds, the algebraic deformation theory after Gerstenhaber, and Fedosov’s construction of star products on symplectic manifolds. The final chapter of the book represents quantum states as positive functionals on the algebra of observables of a quantum system. The associated representations obtained by means of the Gelfand-Naĭmark-Segal construction and their relationship to the deformation quantization are discussed as well.

This book will certainly prove to be a useful introductory text for a very active research area where ideas of mathematical physics, differential geometry and operator theory are interacting. The exposition is clear and many comments with a physical flavor are included throughout the text.

Reviewer: Daniel Beltiţă (Bucureşti)

### MathOverflow Questions:

(An introduction to) deformation theory (written) for differential geometers### MSC:

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53D55 | Deformation quantization, star products |

53D05 | Symplectic manifolds (general theory) |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

53D20 | Momentum maps; symplectic reduction |

81S10 | Geometry and quantization, symplectic methods |