Foundations of Quantum Theory:

Abstract

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 orig inally 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 Theo rem, the theory of symmetry (including Wigner’s Theorem and its relatives, culmi nating in a recent theorem of Hamhalter’s), Bell’s Theorem(s) and the like, quantiza tion theory, indistinguishable particle, large systems, spontaneous symmetry break ing, 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. All of this is held together by what we call Bohrification, i.e., the mathematical interpretation of Bohr’s classical concepts by commutative C*-algebras, which in turn are studied in their quantum habitat of noncommutative C*-algebras. Thus the book is mostly written in mathematical physics style, but its real subject is natural philosophy. Hence its intended readership consists not only of mathemati cal physicists, but also of philosophers of physics, as well as of theoretical physicists whowishtodomorethan‘shutupandcalculate’,andfinallyofmathematicians who are interested in the mathematical and conceptual structure of quantum theory. To serve all these groups, the native mathematical language (i.e. of C*-algebras) is introduced slowly, starting with finite sets (as classical phase spaces) and finite dimensional Hilbert spaces. In addition, all advanced mathematical background that is necessary but may distract from the main development is laid out in extensive appendices on Hilbert spaces, functional analysis, operator algebras, lattices and logic, and category theory and topos theory, so that the prerequisites for this book are limited to basic analysis and linear algebra (as well as some physics). These appendices not only provide a direct route to material that otherwise most readers would have needed to extract from thousands of pages of diverse textbooks, but they also contain some original material, and may be of interest even to mathematicians