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.
open access link Commun. Math. Phys. 332, 477 (2014)
@article{Benini:2013ita, author = "Benini, Marco and Dappiaggi, Claudio and Hack, Thomas-Paul and Schenkel, Alexander", title = "{A C*-algebra for quantized principal U(1)-connections on globally hyperbolic Lorentzian manifolds}", journal = "Commun. Math. Phys.", volume = "332", year = "2014", pages = "477-504", doi = "10.1007/s00220-014-2100-3", eprint = "1307.3052", archivePrefix = "arXiv", primaryClass = "math-ph", reportNumber = "BUW-IMACM-13-12", SLACcitation = "%%CITATION = ARXIV:1307.3052;%%" }

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