Bohrification of operator algebras and quantum logic

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-by-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" of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) 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 measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).Comment: 31 page

Bohrification

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.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in "Deep Beauty" (ed. H. Halvorson

Is logic empirical?

The philosophical debate about quantum logic between the late 1960s and the early 1980s was generated mainly by Putnam's claims that quantum mechanics empirically motivates introducing a new form of logic, that such an empirically founded quantum logic is the 'true' logic, and that adopting quantum logic would resolve all the paradoxes of quantum mechanics. Most of that debate focussed on the latter claim, reaching the conclusion that it was mistaken. This chapter will attempt to clarify the possible misunderstandings surrounding the more radical claims about the revision of logic, assessing them in particular both in the context of more general quantum-like theories (in the framework of von Neumann algebras), and against the background of the current state of play in the philosophy and interpretation of quantum mechanics. Characteristically, the conclusions that might be drawn depend crucially on which of the currently proposed solutions to the measurement problem is adopted

Piecewise Boolean Algebras and Their Domains

We characterise piecewise Boolean domains, that is, those domains that arise as Boolean subalgebras of a piecewise Boolean algebra. This leads to equivalent descriptions of the category of piecewise Boolean algebras: either as piecewise Boolean domains equipped with an orientation, or as full structure sheaves on piecewise Boolean domains.Comment: 11 page

A Proof of Specker's Principle

Specker's principle, the condition that pairwise orthogonal propositions must be jointly orthogonal, has been much investigated recently within the programme of finding physical principles to characterise quantum mechanics. It largely appears, however, to lack a transparent justification. In this paper, I provide a derivation of Specker's principle from three assumptions (made suitably precise): the existence of maximal entanglement, the existence of non-maximal measurements, and no-signalling. I discuss these three assumptions and describe canonical examples of non-Specker sets of propositions satisfying any two of them. These examples display analogies with various approaches in the interpretation of quantum mechanics, notably ones based on retrocausation. I also discuss connections with the work of Popescu and Rohrlich. The core of the proof (and the main example violating no-signalling) is illustrated by a variant of Specker's tale of the seer of Nineveh, with which I open the paper

A proof of Specker’s principle

Specker’s principle, the condition that pairwise orthogonal propositions must be jointly orthogonal (or rather, the ‘exclusivity principle’ that follows from it), has been much investigated recently within the programme of finding physical principles to characterize quantum mechanics. Specker’s principle, however, largely appears to lack a physical justification. In this paper, I present a proof of Specker’s principle from three assumptions (made suitably precise): the existence of ‘maximal entanglement’, the existence of ‘non-maximal measurements’ and no-signalling. I discuss these three assumptions and describe canonical examples of non-Specker sets of propositions satisfying any two of them. These examples display analogies with various approaches to the interpretation of quantum mechanics, including retrocausation. I also discuss connections with the work of Popescu & Rohrlich. The core of the proof (and the main example violating no-signalling) is illustrated by a variant of Specker’s tale of the seer of Nineveh, with which I open the paper. This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’

The Many Classical Faces of Quantum Structures

Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey this programme in terms of algebraic quantum theory.Comment: 24 page