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Quantized Abelian Principal Connections on Lorentzian Manifolds

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Abstract

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 category of principal U(1)-bundles two natural transformations from singular homology functors to the quantum field theory functor that can be interpreted as the Chern 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.

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Pavia & INFN, sezione di Pavia-Via Bassi 6, 27100, Pavia, Italy

    Marco Benini & Claudio Dappiaggi

  2. Fachgruppe Mathematik, Bergische Universität Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany

    Alexander Schenkel

  3. II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761, Hamburg, Germany

    Marco Benini

Authors
  1. Marco Benini
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  2. Claudio Dappiaggi
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  3. Alexander Schenkel
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Correspondence to Claudio Dappiaggi.

Additional information

Communicated by Y. Kawahigashi

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Benini, M., Dappiaggi, C. & Schenkel, A. Quantized Abelian Principal Connections on Lorentzian Manifolds. Commun. Math. Phys. 330, 123–152 (2014). https://doi.org/10.1007/s00220-014-1917-0

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