Quantized Abelian principal connections on Lorentzian manifolds

TL;DR

This paper develops a covariant functor framework for quantized Abelian principal connections on Lorentzian manifolds, revealing topological obstructions to locality and introducing a topological generalization of quantum fields.

Contribution

It constructs a new functorial approach to quantized Abelian connections, analyzes gauge invariance issues, and introduces topological quantum field theories related to Chern classes and electric charge.

Findings

Locality is violated due to topological obstructions.

Gauge invariant algebra may not separate gauge classes.

Electric charges can be set to zero, satisfying all axioms.

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.