Thèse : « topos-theoretical approach to quantum physics »

« The aim of this thesis is to apply concepts and methods from category theory and, in particular, topos theory, within quantum physics. In the resulting theory, “topos physics”, topos theory (the theory of generalized universes of sets and generalizes spaces) is used as a tool with which quantum physics may be constructed by “gluing together” classical perspectives or “snapshots”. The first chapter fills in some of the details which are needed in order to appreciate the physical motivation behind the constructions to follow, with particular attention to the concepts of logic, quantization and space. In chapter 2, the fundamentals of category theory and topos theory are presented as a prelude to the review of the basic method of topos physics: the construction of a state space for quantum mechanics by means of (covariant or contravariant) functors on a category of commutative operator algebras. The approaches known as “neo-realism” (due to Andreas Döring and Chris Isham) and “Bohrification” (due to Chris Heunen, Nicolas P. Landsmaan and Bas Spitters) are given in some detail. Then, in chapter 3, the latter scheme is applied to the theory of loop quantum gravity (LQG). This chapter therefore contains a brief summary of the main results of LQG. We study how LQG may be interpreted within topos physics by appeal to the Weyl algebra version of LQG due to Christian Fleischhack. The topological properties of the state space of LQG within the topos model are investigated, and it is shown how to interpret the gauge and diffeomorphism invariance requirements of LQG within the theory. Finally, using a little-known technique for quantising on general structures (due to Isham), we extend the apparatus of topos physics to a larger category-theoretical framework. We define a measure theory on categories and study the basic entities, the arrow fields on a category, in the category of their representations. It is suggested how this may be applied to the theory of causal sets, and, more radically, to a theory of quantised logic.«

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