A regularizing commutant duality for a kinematically covariant partial ordered net of observables

Abstract

We consider a net of *-algebras, locally around any point of observation, equipped with a natural partial order related to the isotony property. Assuming the underlying manifold of the net to be a differentiable, this net shall be kinematically covariant under general diffeomorphisms. However, the dynamical relations, induced by the physical state defining the related net of (von Neumann) observables, are in general not covariant under all diffeomorphisms, but only under the subgroup of dynamical symmetries. We introduce algebraically both, IR and UV cutoffs, and assume that these are related by a commutant duality. The latter, having strong implications on the net, allows us to identify a 1-parameter group of the dynamical symmetries with the group of outer modular automorphisms. For thermal equilibrium states, the modular dilation parameter may be used locally to define the notions of both, time and a causal structure.