TY - BOOK

T1 - Bohrification of operator algebras and quantum logic

AU - Heunen, C.

AU - Landsman, N.P.

AU - Spitters, B.A.W.

PY - 2009

Y1 - 2009

N2 - 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 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).

AB - 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 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).

M3 - Report

T3 - arXiv.org [quant-ph]

BT - Bohrification of operator algebras and quantum logic

PB - s.n.

ER -