Hadamard states for quantum Abelian duality

TL;DR

This paper constructs and analyzes Hadamard states within the framework of Abelian duality in quantum field theory, combining differential cohomology and covariant quantization to describe electric and magnetic fluxes.

Contribution

It introduces a novel state construction for the algebra of observables in Abelian duality, including topological sectors, and demonstrates a unitary implementation of duality.

Findings

Existence of a Hadamard state with standard singularity structure

Factorization of the observable algebra into dynamical and topological parts

Unitary implementation of Abelian duality in the quantum setting

Abstract

Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a C$^*$-algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states to such algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three C$^*$-algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors we obtain a state for the full theory, providing ultimately a unitary implementation of Abelian duality.