A categorical approach to the foundations of Quantum theory

Student thesis: Doctoral Thesis

Abstract

In this work we pursue two goals: the first is to bridge a gap between various projects within the field of categorical quantum theory, namely: topos quantum theory as initiated by Butterfield and Isham; monoidal quantum theory, as initiated by Abramsky and Coecke; and the sheaf theoretic approach to contextuality and non-locality, as initiated by Abramsky and Brandenburger. We show connections between these projects on the level of mathematical formalisms, and also on the level of physical interpretations. The central element is a generalisation of the topos theoretic structures of Butterfield and Isham, which incorporates the categorical structures considered within monoidal quantum theory, resulting in structures that can be embedded into the general sheaf-formalism of Abramsky and Brandenburger. The second main thrust of this work is in providing a framework within which to consider the foundations of quantum theory, specifically within the context of a pragmatic metaphysical interpretation. This is done using similar mathematical structures as topos quantum theory, but using quite different metaphysical presuppositions. In particular, we consider presheaves on posets of commutative subsemialgebras which can be naturally associated with a physical system as represented in categorical quantum mechanics. The main distinction between our work and topos quantum theory is that we do not pursue a realist metaphysical interpretation of these structures. Rather we pursue a pragmatic interpretation of metaphysics which is consistent with the sheaf-formalism of Abramsky and Brandenburger.From this categorical perspective, we consider some of the fundamental features of quantum foundations, including a derivation of the Born rule, contextuality/non-locality, the meaning of quantum probabilities, and model toy quantum-like theories. We also present a new outlook on quantum metaphysics which seeks to lend clarity to some of the conceptual problems which exist in quantum foundations and which draws a connection with the Bohrification programme of Heunen, Spitters and Landsman.
Date of Award19 Sep 2019
Original languageEnglish
Awarding Institution
  • University Of Strathclyde
SponsorsEPSRC (Engineering and Physical Sciences Research Council)
SupervisorRoss Duncan (Supervisor) & Neil Ghani (Supervisor)

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