Foundations of quantum physics in smooth toposes
Abstract
The usual analytic approach to differential geometry was explored in the early 18th century using the defining notion of limit.With the advent of topos theory in the late 20th century, it has been possible to rediscover without the use of limits the differential and integral calculus, and its development into Synthetic Differential Geometry (SDG), in mathematical universes other than that represented by set theory.These universes obey a subtle but well understood generalisation of classical logic: intuitionistic logic.This thesis presents an exploration of the construction of quantum mechanics in the context of SDG.Chapter 1 introduces SDG, intuitionistic logic, topos theory and a particular topos used in the thesis.Chapter 2 explores the new logical perspective afforded by topos theory and SDG in describing the quantum world from an intuitionistic viewpoint.We build a working quantum-mechanical model using topos theory and analyse the physical and philosophical relevance of the logical steps taken, and account for the consistency of the model with its set-theoretic predecessor by suggesting a certain physical interpretation for the mathematics.In chapter 3, we analyse the extension of this quantum mechanics to state spaces with an infinite number of dimensions, and the associated use of various mathematical constructions that are taken for granted in set theory.Chapter 4 reaches an important proof of the lack of validity of the Spectral 4 The Spectral Theorem 108 4.1 Proof…
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