Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

The aim of this book is to analyze the foundations of quantum theory from the point of view of classical-quantum duality, using the mathematical formalism of operator algebras on Hilbert space (and, more generally, C*-algebras) that was originally created by von Neumann (followed by Gelfand and Naimark). In support of this analysis, but also as a matter of independent interest, the book covers many of the traditional topics one might expect to find in a treatise on the foundations of quantum mechanics, like pure and mixed states, observables, the Born rule and its relation to both single-case probabilities and long-run frequencies, Gleason's Theorem, the theory of symmetry (including Wigner's Theorem and its relatives, culminating in a recent theorem of Hamhalter's), Bell's Theorem(s) and the like, quantization theory, indistinguishable particle, large systems, spontaneous symmetry breaking, the measurement problem, and (intuitionistic) quantum logic. One also finds a few idiosyncratic themes, such as the Kadison–Singer Conjecture, topos theory (which naturally injects intuitionism into quantum logic), and an unusual emphasis on both conceptual and mathematical aspects of limits in physical theories.

This book is Open Access under a CC BY licence.