An algebraic Haag's theorem

TL;DR

This paper proves that the algebraic structure of local von Neumann algebras in a quantum field theory model uniquely determines the model, extending Haag's theorem algebraically and applicable to conformal chiral QFT.

Contribution

It establishes an algebraic version of Haag's theorem showing the unitary equivalence of local algebras determines the entire QFT model without needing vacuum or symmetry state connections.

Findings

Unitary equivalence of local algebras implies model equivalence.

Applicable to conformal chiral quantum field theories.

Discusses related conjectures on modular inclusions.

Abstract

Under natural conditions (such as split property and geometric modular action of wedge algebras) it is shown that the unitary equivalence class of the net of local (von Neumann) algebras in the vacuum sector associated to double cones with bases on a fixed space-like hyperplane completely determines an algebraic QFT model. More precisely, if for two models there is unitary connecting all of these algebras, then --- without assuming that this unitary also connects their respective vacuum states or spacetime symmetry representations --- it follows that the two models are equivalent. This result might be viewed as an algebraic version of the celebrated theorem of Rudolf Haag about problems regarding the so-called "interaction-picture" in QFT. Original motivation of the author for finding such an algebraic version came from conformal chiral QFT. Both the chiral case as well as a related conjecture about standard half-sided modular inclusions will be also discussed.