Bohrification: From classical concepts to commutative algebras

Abstract

The Bohrification program is an attempt to interpret Bohr's mature doctrine of classical concepts as well as his earlier correspondence principle in the operator-algebraic formulation of quantum theory pioneered by von Neumann. In particular, this involves the study of commutative C*-algebras in relationship to noncommutative ones. This relationship may take the form of either exact Bohrification, in which one studies commutative unital C*-subalgebras of a given noncommutative C*-algebra, or asymptotic Bohrification, which involves deformations of commutative C*-algebras into noncommutative ones. Implementing the doctrine of classical concepts, exact Bohrification is an appropriate framework for the Kochen-Specker Theorem and for (intuitionistic) quantum logic, culminating in the topos-theoretic approach to quantum mechanics that has been developed since 1998. Asymptotic Bohrification was inspired by the correspondence principle and forms the right conceptual and mathematical framework for the explanation of the emergence of the classical world from quantum theory (incorporating the measurement problem and the closely related issue of spontaneous symmetry breaking). The Born rule may be derived from both exact and asymptotic Bohrification, which reflects the Janus faces of probability as applying to individual random events and to relative frequencies, respectively.We review the history, the goals, and the achievements of this program so far.

Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics

“…Meaning of the Limit. In analysing the physical interpretation of the high amplitude limit, we will be guided by two principles, referred to by Landsman [38] as Earman's principle [36] and Butterfield's principle [37]. Earman's principle states that While idealisations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealisations are removed.…”

Control of Quantum Noise: On the Role of Dilations

“…Using [12,Prop. 2.3.24] and [28,Sect. 3], one obtains: Theorem 3.1. ν t is a well-defined state on A.…”

Symmetries in Exact Bohrification

“…As reviewed in Landsman (2016) and explained in detail in Landsman (2017), asymptotic Bohrification provides a mathematical setting for the measurement problem, spontaneous symmetry breaking, the classical limit of quantum mechanics, the thermodynamic limit of quantum statistical mechanics, and the Born rule for probabilities construed as long-run frequencies, whereas exact Bohrification turns out to be an appropriate framework for Gleason's Theorem, the Kadison-Singer conjecture, the Born rule (for single case probabilities), and, initially via the topos-theoretic approach to quantum mechanics, intuitionistic quantum logic. In the context of the present paper it should be mentioned that the poset C (A) we will be concerned with has its origins in the reinterpretation of the Kochen-Specker Theorem in the language of topos theory by Isham & Butterfield (1998).…”