119 research outputs found
Topos Quantum Logic and Mixed States
The topos approach to the formulation of physical theories includes a new
form of quantum logic. We present this topos quantum logic, including some new
results, and compare it to standard quantum logic, all with an eye to
conceptual issues. In particular, we show that topos quantum logic is
distributive, multi-valued, contextual and intuitionistic. It incorporates
superposition without being based on linear structures, has a built-in form of
coarse-graining which automatically avoids interpretational problems usually
associated with the conjunction of propositions about incompatible physical
quantities, and provides a material implication that is lacking from standard
quantum logic. Importantly, topos quantum logic comes with a clear geometrical
underpinning. The representation of pure states and truth-value assignments are
discussed. It is briefly shown how mixed states fit into this approach.Comment: 25 pages; to appear in Electronic Notes in Theoretical Computer
Science (6th Workshop on Quantum Physics and Logic, QPL VI, Oxford, 8.--9.
April 2009), eds. B. Coecke, P. Panangaden, P. Selinger (2010
The physical interpretation of daseinisation
We provide a conceptual discussion and physical interpretation of some of the
quite abstract constructions in the topos approach to physics. In particular,
the daseinisation process for projection operators and for self-adjoint
operators is motivated and explained from a physical point of view.
Daseinisation provides the bridge between the standard Hilbert space formalism
of quantum theory and the new topos-based approach to quantum theory. As an
illustration, we will show all constructions explicitly for a three-dimensional
Hilbert space and the spin-z operator of a spin-1 particle. This article is a
companion to the article by Isham in the same volume.Comment: 39 pages; to appear in "Deep Beauty", ed. Hans Halvorson, Cambridge
University Press (2010
Kochen-Specker theorem for von Neumann algebras
The Kochen-Specker theorem has been discussed intensely ever since its
original proof in 1967. It is one of the central no-go theorems of quantum
theory, showing the non-existence of a certain kind of hidden states models. In
this paper, we first offer a new, non-combinatorial proof for quantum systems
with a type factor as algebra of observables, including .
Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von
Neumann algebra without summands of types and ,
using a known result on two-valued measures on the projection lattice
. Some connections with presheaf formulations as proposed by
Isham and Butterfield are made.Comment: 22 pages, no figure
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