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31	Premonoidal -Categories and Algebraic Quantum Field Theory Comeau, Marc A* 16 March 2012 (has links) Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered. category premonoidal category -category von Neumann category monoidal category algebraic quantum field theory DHR analysis Haag duality dagger compact closed category categorical physics causality quantum protocols premonoidal -categories category theory and physics reconstruction theorem Tannaka-Krein reconstruction Doplicher-Roberts reconstruction theorem C*-categories categorified functional analysis categorical GNS theorems
32	Premonoidal -Categories and Algebraic Quantum Field Theory Comeau, Marc A* January 2012 (has links) Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered. category premonoidal category -category von Neumann category monoidal category algebraic quantum field theory DHR analysis Haag duality dagger compact closed category categorical physics causality quantum protocols premonoidal -categories category theory and physics reconstruction theorem Tannaka-Krein reconstruction Doplicher-Roberts reconstruction theorem C*-categories categorified functional analysis categorical GNS theorems

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Premonoidal *-Categories and Algebraic Quantum Field Theory

Premonoidal *-Categories and Algebraic Quantum Field Theory