Quantized Abelian principal connections on Lorentzian manifolds

Marco Benini, Claudio Dappiaggi, Alexander Schenkel
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.
open access link doi:10.1007/s00220-014-1917-0
@article{Benini:2013tra, author = "Benini, Marco and Dappiaggi, Claudio and Schenkel, Alexander", title = "{Quantized Abelian principal connections on Lorentzian manifolds}", journal = "Commun. Math. Phys.", volume = "330", year = "2014", pages = "123-152", doi = "10.1007/s00220-014-1917-0", eprint = "1303.2515", archivePrefix = "arXiv", primaryClass = "math-ph", reportNumber = "DESY-13-045, BUW-IMACM-13-04", SLACcitation = "%%CITATION = ARXIV:1303.2515;%%" }

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