JournalsprimsVol. 22 , No. 1DOI 10.2977/prims/1195178376

Reflection Positivity for the Complementary Series of <i>SL</i>(2<i>n</i>, <i>ℂ</i>)

  • Robert Schrader

    Freie Universität Berlin, Germany
Reflection Positivity for the Complementary Series of <i>SL</i>(2<i>n</i>, <i>ℂ</i>) cover

Abstract

We apply the concept of reflection positivity in euclidean quantum field theory to the complementary series of S_SL_(2_n_, as given by Gelfand and Neumark for n=1 and by Stein for n>1. The result is a virtual representation in the sense of Fröhlich, Osterwalder and Seiler or equivalently a strongly continuous representation of a closed subsemigroup by contractions on a new Hilbert space. Analytic continuation gives a unitary representation of a certain dual group of SL(2_n_, . The possible relation to the theory of noncommuting monodromy matrices appearing in the theory of integrable quantum systems is briefly discussed.