# Quantized Abelian principal connections on Lorentzian manifolds

March 11, 2013

We construct a covariant functor from a category of Abelian principal bundles
over globally hyperbolic spacetimes to a category of *-algebras that describes
quantized principal connections. We work within an appropriate differential
geometric setting by using the bundle of connections and we study the full
gauge group, namely the group of vertical principal bundle automorphisms.
Properties of our functor are investigated in detail and, similar to earlier
works, it is found that due to topological obstructions the locality property
of locally covariant quantum field theory is violated. Furthermore, we prove
that, for Abelian structure groups containing a nontrivial compact factor, the
gauge invariant Borchers-Uhlmann algebra of the vector dual of the bundle of
connections is not separating on gauge equivalence classes of principal
connections. We introduce a topological generalization of the concept of
locally covariant quantum fields. As examples, we construct for the full
subcategory of principal U(1)-bundles two natural transformations from singular
homology functors to the quantum field theory functor that can be interpreted
as the Euler class and the electric charge. In this case we also prove that the
electric charges can be consistently set to zero, which yields another quantum
field theory functor that satisfies all axioms of locally covariant quantum
field theory.

Keywords:

locally covariant quantum field theory, quantum field theory on curved spacetimes, gauge theory on principal bundles