Notes on the Wigner Representation Theory of the Poicaré Group , Localization and Statistics

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 framework was recently used by Brunetti, Guido and Longo in order to construct interaction-free nets of local algebras without using non-unique ”free field coordinates”. Here it is shown that this structure preempts among other things properties of localization and braidgroup statistics in low-dimensional QFT. It also sheds some light on string-like localization properties of Wigner’s ”continuous spin” representations.We formulate a constructive nonperturbative program to introduce interactions into such an approach based on the TomitaTakesaki modular theory. The new aspect is the deep relation of the latter with the scattering operator.