Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 058, 49 pages      arXiv:1510.07445

Reflection Positive Stochastic Processes Indexed by Lie Groups

Palle E.T. Jorgensen a, Karl-Hermann Neeb b and Gestur Ólafsson c
a) Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
b) Department Mathematik, FAU Erlangen-Nürnberg, Cauerstrasse 11, 91058-Erlangen, Germany
c) Department of mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received October 28, 2015, in final form June 09, 2016; Published online June 21, 2016

Reflection positivity originates from one of the Osterwalder-Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary representation of a symmetric Lie group to a unitary representation of its Cartan dual. In this article we continue our investigation of representation theoretic aspects of reflection positivity by discussing reflection positive Markov processes indexed by Lie groups, measures on path spaces, and invariant gaussian measures in spaces of distribution vectors. This provides new constructions of reflection positive unitary representations.

Key words: reflection positivity; stochastic process; unitary representations.

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