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

TL;DR

This paper develops a new C*-algebra framework for quantizing U(1)-connections on globally hyperbolic Lorentzian manifolds, addressing gauge invariance and locality issues in quantum field theory.

Contribution

It introduces a covariant functor assigning a C*-algebra to principal U(1)-bundles, generalizing CCR-algebras with a presymplectic Abelian group structure, and explores locality and charge interpretations.

Findings

Constructed a C*-algebra that separates gauge classes of connections.

Proved a no-go theorem on local covariance of the functor.

Identified conditions under which a quotient yields a Haag-Kastler quantum field theory.

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.