Group Cohomology, Modular Theory and Space-time Symmetries

TL;DR

This paper demonstrates that the Bisognano-Wichmann property ensures the existence of a covariant Poincaré group action in quantum field theory, providing an intrinsic characterization of certain conformal structures using group cohomology.

Contribution

It establishes an intrinsic criterion for Poincaré covariance based on modular theory and analyzes the triviality of central extensions of the Poincaré group via group cohomology.

Findings

Bisognano-Wichmann property implies Poincaré covariance.

Universal covering of Poincaré group has only trivial central extensions.

Characterization of positive-energy conformal pre-cosheaves.

Abstract

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincar\'e group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincar\'e group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.