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 ${\mathcal{C}(A)}$ 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 ${\mathcal{C}(A)}$ 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” ${\underline A}$ of A, which is a commutative Rickart C*-algebra in the topos of functors from ${\mathcal{C}A}$ 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 measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of ${\underline A}$ for A = B(H).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 $${\mathcal{C}(A)}$$ 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 $${\mathcal{C}(A)}$$ 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” $${\underline A}$$ of A, which is a commutative Rickart C*-algebra in the topos of functors from $${\mathcal{C}A}$$ 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 measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of $${\underline A}$$ for A = B(H).

arXiv:0905.2275SyntheseMar 28, 201130 citations

Relative State Quantum Logic

M. P. Vaughan

A projective quantum logic in terms of relative states is developed, emphasizing the importance of information transfer between a system under study and its environment. The need for accounting for the historical evolution of system is highlighted and it is found that the conjunction of observations involving conjugate variables can be consistently defined but is found to be non-commutative. It is shown that the Birkhoff and von Neumann approach to quantum logic is unable to deal with such conjunctions. It is found that whilst the proposed scheme is still not distributive in general, the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment. It is argued that the probabilities associated with projections be mapped to an orthocomplemented ternary logic, in which it is shown that the law of the excluded middle still holds.A projective quantum logic in terms of relative states is developed, emphasizing the importance of information transfer between a system under study and its environment. The need for accounting for the historical evolution of system is highlighted and it is found that the conjunction of observations involving conjugate variables can be consistently defined but is found to be non-commutative. It is shown that the Birkhoff and von Neumann approach to quantum logic is unable to deal with such conjunctions. It is found that whilst the proposed scheme is still not distributive in general, the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment. It is argued that the probabilities associated with projections be mapped to an orthocomplemented ternary logic, in which it is shown that the law of the excluded middle still holds.

arXiv:2203.06695Mar 13, 20220 citations

Quantum Logic and Quantum Probability

Enrico G. Beltrametti

Springer eBooksJan 1, 20044 citations

{\mathcal{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

{\mathcal{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”

{\underline A}

of A, which is a commutative Rickart C*-algebra in the topos of functors from

{\mathcal{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 measures on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of

{\underline A}

for A = B(H).