# Analytic Elements Haag-Araki theory

Why the elements of a local von Neumann algebra cannot be analytic elements of the timelike translation?

• Could you please state the definition of analytic element with respect to timelike translations? – Valter Moretti Aug 22 '19 at 17:23
• The definition is as found in Brattelli Robinson I: – val 72 Aug 22 '19 at 18:47
• I cannot check it now, but I guess that $X$ is the local von Neumann algebra and the indicated topology is one of the natural weak topologies interesting for von Neumann algebras. I think that one is also assuming some spectral condition on the representation of the group of four-translations implying that the generator of time displacements is bounded below. I expect that this asymmetry is responsible for the lack of analiticy in any set of the form $I_\lambda$. – Valter Moretti Aug 22 '19 at 19:22
• Yes, you are right! The question is about whether one, given the usual axioms of a Haag - Araki theory (Isotony, weak additivity, Poincare Covariance, Microcausality, Spectrum Condition) can think of an analytic element as belonging in a local algebra associated with an open bounded region, say, a double cone. – val 72 Aug 22 '19 at 19:26
• In general, X is the global von Neumann Algebra, so as to have the unitary implementation of the translations. However, can we consider the observable to belong in a local algebra? – val 72 Aug 22 '19 at 19:29