Presheaf structures in quantum mechanics
File(s)
Author(s)
Hamilton, John
Type
Thesis
Abstract
This thesis is concerned with the application of topos theory, in particular the theory of presheaves, to the understanding of the structure of quantum mechanics. The topos of presheaves has built into it a contextual, multi-valued logical structure which can be used to model precisely the contextuality of quantum mechanics.
The use of topos theory in quantum mechanics was first introduced by Isham and Butterfield. This thesis first introduces this work, before going on to extend it and discuss some of its implications. This falls into three main topics: 1. An examination of the Kochen-Specker theorem in presheaf-theoretic form, including the relationship between this and existing 'standard' proofs of the theorem; 2. The use of a new category of contexts V over which presheaves may be constructed. This category has as objects commutative von Neumann algebras which arise naturally as contexts if we wish to discuss valuations on propositions and operators which may depend on what other operators are considered simultaneously; 3. The discussion of various types of non-standard valuation which are not prohibited by the Kochen-Specker theorem. These are structures which arise naturally in the topos of presheaves, using the contextual, multi-valued logic which the topics provides.
The use of topos theory in quantum mechanics was first introduced by Isham and Butterfield. This thesis first introduces this work, before going on to extend it and discuss some of its implications. This falls into three main topics: 1. An examination of the Kochen-Specker theorem in presheaf-theoretic form, including the relationship between this and existing 'standard' proofs of the theorem; 2. The use of a new category of contexts V over which presheaves may be constructed. This category has as objects commutative von Neumann algebras which arise naturally as contexts if we wish to discuss valuations on propositions and operators which may depend on what other operators are considered simultaneously; 3. The discussion of various types of non-standard valuation which are not prohibited by the Kochen-Specker theorem. These are structures which arise naturally in the topos of presheaves, using the contextual, multi-valued logic which the topics provides.
Version
Open Access
Date Issued
2000
Date Acceptance
2000
Copyright Statement
Attribution-Non Commercial-No Derivatives 4.0 International Licence (CC BY-NC-ND)