167 research outputs found
Topos theory and `neo-realist' quantum theory
Topos theory, a branch of category theory, has been proposed as mathematical
basis for the formulation of physical theories. In this article, we give a
brief introduction to this approach, emphasising the logical aspects. Each
topos serves as a `mathematical universe' with an internal logic, which is used
to assign truth-values to all propositions about a physical system. We show in
detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop
"Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200
Topos quantum theory with short posets
Topos quantum mechanics, developed by Isham et. al., creates a topos of
presheaves over the poset V(N) of abelian von Neumann subalgebras of the von
Neumann algebra N of bounded operators associated to a physical system, and
established several results, including: (a) a connection between the
Kochen-Specker theorem and the non-existence of a global section of the
spectral presheaf; (b) a version of the spectral theorem for self-adjoint
operators; (c) a connection between states of N and measures on the spectral
presheaf; and (d) a model of dynamics in terms of V(N). We consider a
modification to this approach using not the whole of the poset V(N), but only
its elements of height at most two. This produces a different topos with
different internal logic. However, the core results (a)--(d) established using
the full poset V(N) are also established for the topos over the smaller poset,
and some aspects simplify considerably. Additionally, this smaller poset has
appealing aspects reminiscent of projective geometry.Comment: 14 page
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