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Abstract:
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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. |
