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

Marco Benini, Claudio Dappiaggi, Thomas-Paul Hack, Alexander Schenkel
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.

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