Communications in Mathematical Physics

, Volume 336, Issue 1, pp 27–61 | Cite as

Localization in Nets of Standard Spaces

  • Gandalf Lechner
  • Roberto Longo


Starting from a real standard subspace of a Hilbert space and a representation of the translation group with natural properties, we construct and analyze for each endomorphism of this pair a local, translationally covariant net of standard subspaces, on the lightray and on two-dimensional Minkowski space. These nets share many features with low-dimensional quantum field theory, described by corresponding nets of von Neumann algebras.

Generalizing a result of Longo and Witten to two dimensions and massive multiplicity free representations, we characterize these endomorphisms in terms of specific analytic functions. Such a characterization then allows us to analyze the corresponding nets of standard spaces, and in particular to compute their minimal localization length. The analogies and differences to the von Neumann algebraic situation are discussed.


Half Plane Blaschke Product Double Cone Standard Space Standard Pair 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly

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