Bohrification: From classical concepts to commutative algebras
arXiv:1601.02794
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.
26 pages, 2 figures. To appear in "Niels Bohr in the 21st Century", eds. J. Faye and H. Folse
References in corpus (12)
- Macroscopic quantum resonators (MAQRO): 2015 Update
- Niels Bohr on the wave function and the classical/quantum divide
- Intuitionistic quantum logic of an n-level system
- Niels Bohr as Philosopher of Experiment: Does Decoherence Theory Challenge Bohr's Doctrine of Classical Concepts?
- The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
- Ramanujan Graphs and the Solution of the Kadison-Singer Problem
- Quantum measurements without macroscopic superpositions
- Many worlds and the emergence of probability in quantum mechanics
- Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras
- Quantum Mechanics: From Realism to Intuitionism
- Matter-wave interferometry with composite quantum objects
- Domains of commutative C*-subalgebras