Relative Haag Duality for the Free Field in Fock Representation

TL;DR

This paper extends Haag duality to localized observable algebras in quantum field theory, proving a relative form for free scalar and electromagnetic fields in higher dimensions, with implications for superselection sectors.

Contribution

It introduces and proves a relative Haag duality for non-irreducible observable algebras in free quantum fields, broadening the understanding of duality in localized regions.

Findings

Proves relative Haag duality for free scalar and electromagnetic fields in dimensions d>1.

Establishes conditions under which the duality holds in non-irreducible cases.

Provides a foundation for developing superselection sector theories for electromagnetic fields.

Abstract

We consider a natural generalization of Haag duality to the case in which the observable algebra is restricted to a subset of the space-time and is not irreducible: the commutant and the causal complement have to be considered relatively to the ambient space. We prove this relative form of Haag duality under quite general conditions for the free scalar and electromagnetic field of space dimension d>1 in the vacuum representation. Such property is interesting in view of a theory of superselection sectors for the electromagnetic field.