This bachelor thesis has been submitted in August 2011:
Bohrification of local nets of observables
Proceedings of QPL 2011
EPTCS 95, 2012, pp. 211-218
on the order-theoretic structure in quantum mechanics and its application to algebraic quantum field theory.
Recent results by Spitters et. al. suggest that quantum phase space can usefully be regarded as a ringed topos via a process called Bohrification. They show that quantum kinematics can then be interpreted as classical kinematics, internal to this ringed topos.
We extend these ideas from quantum mechanics to algebraic quantum field theory: from a net of observables we construct a presheaf of quantum phase spaces. We can then naturally express the causal locality of the net as a descent condition on the corresponding presheaf of ringed toposes: we show that the net of observables is local, precisely when the presheaf of ringed toposes satifies descent by a local geometric morphism.