# A $C^*$-algebra for quantized principal $U(1)$-connections on globally hyperbolic Lorentzian manifolds

July 11, 2013

The aim of this work is to complete our program on the quantization of
connections on arbitrary principal $U(1)$-bundles over globally hyperbolic
Lorentzian manifolds. In particular, we show that one can assign via a
covariant functor to any such bundle an algebra of observables which separates
gauge equivalence classes of connections. The $C^*$-algebra we construct
generalizes the usual CCR-algebras since, contrary to the standard
field-theoretic models, it is based on a presymplectic Abelian group instead of
a symplectic vector space. We prove a no-go theorem according to which neither
this functor, nor any of its quotients, satisfy the strict axioms of general
local covariance. Yet, if we fix any principal $U(1)$-bundle, there exists a
suitable category of sub-bundles for which a quotient of our functor yields a
quantum field theory in the sense of Haag and Kastler. We shall provide a
physical interpretation of this feature and we obtain some new insights
concerning electric charges in locally covariant quantum field theory.

Keywords:

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