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

TL;DR

This work develops a covariant quantization framework for Abelian gauge theories on globally hyperbolic Lorentzian manifolds by modelling observables as a presymplectic Abelian group and promoting them to CCR-algebras. It shows that strict locality cannot be achieved in the full category of principal $U(1)$-bundles due to topological obstructions encoded in de Rham cohomology, and proves a precise criterion linking locality to injectivity of certain cohomology maps. While no global quotient restores locality, the authors construct a Haag–Kastler-type quantum field theory on suitable subcategories or fixed spacetimes via a quotient by the equation of motion, plus a Ker-based refinement that yields locality, causality, and time-slice properties. They also connect Aharonov–Bohm observables and the Chern class to the theory's topological content, offering physical interpretations of charges within locally covariant QFT. Overall, the paper provides a topology-aware, covariant quantization scheme for electromagnetic-like gauge theories on curved spacetimes and clarifies when locality can be recovered in a region-based Haag–Kastler sense.

Abstract

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, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We then show that fixing 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.

Theorems & Definitions (59)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Proposition 3.4
• proof
• Lemma 3.5
• proof
• Proposition 3.6