##
**Mathematical topics between classical and quantum mechanics.**
*(English)*
Zbl 0923.00008

Springer Monographs in Mathematics. New York, NY: Springer. xix, 529 p. (1999).

The book is devoted, generally speaking, to the mathematical structures of classical and quantum mechanics. According to the author’s own words there are three central themes: the relations between observables and pure states; the analogy between the \(C^*\)-algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid; the parallel between symplectic reduction in classical mechanics and Rieffel induction in quantum mechanics.

The book consists of four chapters and an introductory overview. The first one deals with the structure of algebras of observables and spaces of pure states and relations between them. Poisson manifolds are introduced and their foliation by symplectic subspaces is described. The \(C^*\)-algebras, Jordan-Lie algebras and von Neumann algebras are briefly discussed. The GNS construction is described. These mathematical notions are used to analyse the relation between pure states and observables.

Chapter II is devoted to the study of quantization procedures. The main ideas of Berezin quantization are presented and applied first to flat space systems. Then the case of Riemannian manifold is treated in some detail.

In Chapter III the Poisson algebras and \(C^*\)-algebras are constructed from Lie groups and Lie algebras and related by a quantization procedure. The theory of Lie groupoids and algebroids is then discussed as providing a perspective unifying a large class of examples in quantization theory.

The last chapter deals with the concepts of symplectic reduction which provides the construction of new symplectic manifolds from the old and induction – an analogous technique in the representation theory of \(C^*\)-algebras.

As an application constrained and relativistic quantum systems are discussed.

The book gives a clear account of mathematical aspects of quantization procedures. The reviewer finds it interesting; especially Chapter IV gives a nice presentation.

The book consists of four chapters and an introductory overview. The first one deals with the structure of algebras of observables and spaces of pure states and relations between them. Poisson manifolds are introduced and their foliation by symplectic subspaces is described. The \(C^*\)-algebras, Jordan-Lie algebras and von Neumann algebras are briefly discussed. The GNS construction is described. These mathematical notions are used to analyse the relation between pure states and observables.

Chapter II is devoted to the study of quantization procedures. The main ideas of Berezin quantization are presented and applied first to flat space systems. Then the case of Riemannian manifold is treated in some detail.

In Chapter III the Poisson algebras and \(C^*\)-algebras are constructed from Lie groups and Lie algebras and related by a quantization procedure. The theory of Lie groupoids and algebroids is then discussed as providing a perspective unifying a large class of examples in quantization theory.

The last chapter deals with the concepts of symplectic reduction which provides the construction of new symplectic manifolds from the old and induction – an analogous technique in the representation theory of \(C^*\)-algebras.

As an application constrained and relativistic quantum systems are discussed.

The book gives a clear account of mathematical aspects of quantization procedures. The reviewer finds it interesting; especially Chapter IV gives a nice presentation.

Reviewer: P.Kosiński (Łódź)

### MSC:

00A79 | Physics |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

81S10 | Geometry and quantization, symplectic methods |

46N50 | Applications of functional analysis in quantum physics |

53D50 | Geometric quantization |

46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |

81R15 | Operator algebra methods applied to problems in quantum theory |

70Hxx | Hamiltonian and Lagrangian mechanics |

37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |

53D20 | Momentum maps; symplectic reduction |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

17B63 | Poisson algebras |

00A05 | Mathematics in general |