Mathematical quantization (Q2748519)

The idea of quantization of mathematical structures goes back to von Neumann, Birkhoff and Murray, and led to the introduction of fields such as the quantum measure theory in terms of von Neumann algebras, or quantum logic. In recent years there have appeared several mathematical structures which are quantizations of classical objects. Concrete examples of Hopf algebras obtained by quantization of Poisson structures on classical Lie groups, which are now known as quantum groups, or noncommutative algebras viewed as quantization of classical manifolds, and hence termed quantum (topological) spaces, are but a few examples of quantized mathematical structures, which all have appeared within the last twenty years.NEWLINENEWLINENEWLINEThe aim of the reviewed monograph is to give basic mathematical grounding for quantization of mathematical structures viewed as a single phenomenon. The main observation is that such a unification can be achieved by studying the role of Hilbert spaces. This leads to the underlying principle of the monograph that the quantization is achieved by replacing sets with Hilbert spaces, and complex functions with operators on Hilbert spaces. This point of view should be contrasted with other approaches to quantization, such as for example formal deformation quantization, in which the noncommutativity of underlying `algebras of functions' is considered to be the main feature characterizing quantized mathematical structures. This can be best illustrated by the author's selection of quantum analogs of the classical theories, in which the Riemannian geometry is quantized to noncommutative geometry in the sense of Connes, and topological groups are quantized to quantum (matrix) groups in the sense of Woronowicz.NEWLINENEWLINENEWLINEThe monograph is intended as a broad introductory survey of mathematical topics in quantization. The author starts with the basic description of quantum mechanics, which is contrasted with the classical theory in Chapter 1. Chapter 2 introduces Hilbert spaces and their basic properties such as duals, direct sums and tensor products. The role played by Hilbert spaces in quantum logic is also indicated. Chapter 3 is devoted to the introduction of operators on Hilbert spaces. Special classes of operators such as unitarities and projections are described. Borel functional calculus is introduced, and so are spectral measures, leading to spectral theorems (for bounded and unbounded operators). The chapter concludes with Stone's theorem. In Chapter 4 the author introduces the first and the simplest example of a quantized space, termed the quantum plane. The quantum plane is understood here as the structure underlying momentum and position operators, and should not be confused with the algebraic quantum plane or Manin's plane which appears in the context of quantum groups (as a symmetry space of the quantized linear group). Chapter 5 deals with \(C^{*}\)-algebras, starting from the algebra of continuous functions on a topological space. The noncommutative examples include the quantum plane and tori. In Chapter 6 the author describes von Neumann algebras. The trace class operators and the algebras of bounded operators on a Hilbert space are studied. Chapter 7 gives applications of operator algebras in quantum field theory. Chapter 8 is devoted to the theory of operator spaces, in particular of matrix-valued functions and operator systems. In Chapter 9 the author describes the quantization of Hilbert bundles in terms of Hilbert modules. In consecutive sections various types of such modules are introduced, in particular Hilbert \(L^{\infty}\)-, \(C^{*}\)- and \(W^{*}\)-modules. Also crossed products and Hilbert \(*\)-bimodules are described. In Chapter 10 Lipschitz algebras are introduced, including quantum Markov semigroups. Finally in Chapter 11 the author describes quantum groups by means of a \(C^*\)-algebraic approach. The analysis of such topological quantum groups is restricted to finite and compact quantum groups, and culminates in the proof of Woronowicz's theorem on existence of the Haar measure on a compact quantum group.NEWLINENEWLINENEWLINEThe book certainly fulfills the author's aim ``to write a sort of broad introductory survey, including some deep results but keeping the whole account as nontechnical as possible'' (quoted from the preface). The general theory is well illustrated by detailed examples, although at some places the presentation is somewhat dry and sketchy. This probably is the price one has to pay in order to cover a broad range of topics, while avoiding technicalities. Overall it is an extremely valuable survey of recent developments in our understanding of quantization, and can be recommended to those mathematicians and mathematical physicists who would like to be quickly guided through quantization of mathematical structures. After getting acquainted with mathematical quantization through the reviewed book, they can find more detailed analysis of similar topics for example in the recent excellent monograph by \textit{N. P. Landsman} [ Mathematical topics between classical and quantum mechanics, Springer-Verlag, New York (1998; Zbl 0923.00008)].