Modular localization and Wigner particles

TL;DR

This paper introduces a new framework for constructing local observable algebras in quantum field theory using modular localization, which overcomes limitations of the Wigner formalism and applies to various spacetime symmetries.

Contribution

It develops an intrinsic spacetime localization approach based on the Tomita-Takesaki theory, extending to continuous spin representations and different spacetime geometries.

Findings

Constructs local algebras using Bisognano-Wichmann relations and Poincare' group representations.

Establishes equivalence between energy positivity and net isotony.

Extends the framework to conformal and de Sitter spacetimes.

Abstract

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano-Wichmann relations and a representation of the Poincare' group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita-Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh-Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and de Sitter spacetime.