Wigner representation theory of the Poincaré group, localization, statistics and the S-matrix Original Research Article

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 Wignerʹs “continuous spin” representations. We formulate a constructive non-perturbative 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.