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.
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.
The Bohrication 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 Bohrication, in which one studies commutative unital C*-subalgebras of a given noncommutative C*-algebra, or asymptotic Bohrication, which involves deformations of commutative C*-algebras into noncommutative ones. Implementing the doctrine of classical concepts, exact Bohrication 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 Bohrication 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 Bohrication, which reects 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.
New foundations for quantum logic and quantum spaces are constructed by merging algebraic quantum theory and topos theory. Interpreting Bohr's "doctrine of classical concepts" mathematically, given a quantum theory described by a noncommutative C*-algebra A, we construct a topos T(A), which contains the "Bohrification" B of A as an internal commutative C*-algebra. Then B has a spectrum, a locale internal to T(A), the external description S(A) of which we interpret as the "Bohrified" phase space of the physical system. As in classical physics, the open subsets of S(A) correspond to (atomic) propositions, so that the "Bohrified" quantum logic of A is given by the Heyting algebra structure of S(A). The key difference between this logic and its classical counterpart is that the former does not satisfy the law of the excluded middle, and hence is intuitionistic. When A contains sufficiently many projections (e.g. when A is a von Neumann algebra, or, more generally, a Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be compared with the traditional quantum logic, i.e. the orthomodular lattice of projections in A. This time, the main difference is that the former is distributive (even when A is noncommutative), while the latter is not. This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.
Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n × n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski–Mulvey) of the “Bohrification” A of A, which is a commutative Rickart C*-algebra in the topos of functors from CA to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns–Lakser completions. Finally, we establish a connection between probability measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).
Following Birkhoff and von Neumann, quantum logic has traditionally been based\non the lattice of closed linear subspaces of some Hilbert space, or, more generally,\non the lattice of projections in a von Neumann algebra A. Unfortunately, the logical\ninterpretation of these lattices is impaired by their nondistributivity and by various\nother problems. We show that a possible resolution of these difficulties, suggested by\nthe ideas of Bohr, emerges if instead of single projections one considers elementary\npropositions to be families of projections indexed by a partially ordered set C(A) of\nappropriate commutative subalgebras of A. In fact, to achieve both maximal generality\nand ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra\nand that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such\nfamilies of projections form a Heyting algebra in a natural way, so that the associated\npropositional logic is intuitionistic: distributivity is recovered at the expense of the\nlaw of the excluded middle.\nSubsequently, generalizing an earlier computation for n×n matrices, we prove that\nthe Heyting algebra thus associated to A arises as a basis for the internal Gelfand\nspectrum (in the sense of Banaschewski–Mulvey) of the "Bohrification" A of A, which\nis a commutative Rickart C*-algebra in the topos of functors from C(A) to the category\nof sets. We explain the relationship of this construction to partial Boolean algebras\nand Bruns–Lakser completions. Finally, we establish a connection between probability\nmeasure on the lattice of projections on a Hilbert space H and probability valuations\non the internal Gelfand spectrum of A for A = B(H).
We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program - notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006) - in replacing infinite tensor products of Hilbert spaces by continuous fields of C*-algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvector-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.
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.