# Localization in Nets of Standard Spaces

March 05, 2014

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 compute
their minimal localization length. The analogies and differences to the von
Neumann algebraic situation are discussed.

open access link
Commun. Math. Phys. 336(1), 27-61 (2015)

@article{Lechner:2014mba,
author = "Lechner, Gandalf and Longo, Roberto",
title = "{Localization in Nets of Standard Spaces}",
journal = "Commun. Math. Phys.",
volume = "336",
year = "2015",
number = "1",
pages = "27-61",
doi = "10.1007/s00220-014-2199-2",
eprint = "1403.1226",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1403.1226;%%"
}

Keywords:

standard pairs, modular theory, minimal length, low-dimensional models