N. P. Landsman Libri

This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*-algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its „spontaneous“ breaking), the measurement problem, the Kochen-Specker, Free Will, and Bell Theorems, the Kadison-Singer conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory. This book is Open Access under a CC BY licence. 

InhaltsverzeichnisSome comments on the history, theory, and applicationsof symplectic reduction.Homology of complete symbols and non-commutative geometry.Cohomology of the Mumford quotient.Poisson sigma models and symplectic groupoids.Pseudo-differential operators and deformation quantization.Singularities and Poisson geometry of certainrepresentation spaces.Quantized reduction as a tensor product.Analysis of geometric operator on open manifolds: a groupoid approach.Smooth structures on stratified spaces.Singular projective varieties and quantization.Poisson structure and quantization of Chern-Simons theory.Combinatorial quantization of Euclidean gravityin three dimensions.The Yang—Mills measure and symplectic structureover spaces of connections.

This monograph draws on two traditions: the algebraic formulation of quantum mechanics as well as quantum field theory, and the geometric theory of classical mechanics. These are combined in a unified treatment of the theory of Poisson algebras of observables and pure state spaces with a transition probability, which leads on to a discussion of the theory of quantization and the classical limit from this perspective. A prototype of quantization comes from the analogy between the C*- algebra of a Lie groupoid and the Poisson algebra of the corresponding Lie algebroid. The parallel between reduction of symplectic manifolds in classical mechanics and induced representations of groups and C*- algebras in quantum mechanics plays an equally important role. Examples from physics include constrained quantization, curved spaces, magnetic monopoles, gauge theories, massless particles, and $theta$- vacua. Accessible to mathematicians with some prior knowledge of classical and quantum mechanics, and to mathematical physicists and theoretical physicists with some background in functional analysis.