Notes on Wigner's Representation Theory of the Poincare Group, Localization and Statistics

It has been known that the Wigner representation theory for pos-itive energy orbits permits a useful localization concept in terms of certain lattices of real subspaces of the complex Hilbert-space. This framework was recently used by Brunetti, Guido and Longo in or-der to construct interaction-free nets of local algebras without using non-unique "free eld coordinates". Here it is shown that this struc-ture preempts among other things properties of localization and braid-group statistics in low-dimensional QFT. It also sheds some light on string-like localization properties of Wigner's "continuous spin " rep-resentations.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.