Wigner Representation Theory of the Poincare Group, Localization, Statistics and the S-Matrix

TL;DR

This paper explores the Wigner representation theory's modular localization in quantum field theory, linking it to the S-matrix and braid statistics, and proposes a nonperturbative approach to introduce interactions.

Contribution

It introduces a novel nonperturbative framework connecting modular localization with the S-matrix, advancing the understanding of low-dimensional QFT and continuous spin representations.

Findings

Modular localization aids in constructing local algebras without free field coordinates.

The approach reveals deep relations between modular theory and scattering processes.

Insights into string-like localization of continuous spin representations.

Abstract

It has been known that the Wigner representation theory for positive energy orbits permits a useful localization concept in terms of certain lattices of real subspaces of the complex Hilbert -space. This ''modular localization'' is not only useful in order to construct interaction-free nets of local algebras without using non-unique ''free field coordinates'', but also permits the study of properties of localization and braid-group statistics in low-dimensional QFT. It also sheds some light on the string-like localization properties of the 1939 Wigner's ''continuous spin'' representations.We formulate a constructive nonperturbative program to introduce interactions into such an approach based on the Tomita-Takesaki modular theory. The new aspect is the deep relation of the latter with the scattering operator.