Reflection positive one-parameter groups and dilations

Neeb KH, Olafsson G (2015)

Publication Type: Journal article, Original article

Publication year: 2015

Journal

Complex Analysis and Operator Theory Springer Verlag (Germany)

Publisher: Springer Verlag (Germany)

Book Volume: 9

Pages Range: 653 - 721

Journal Issue: 3

DOI: 10.1007/s11785-014-0402-2

Abstract

The concept of reflection positivity has its origins in the work of Osterwalder–Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincaré group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary one-parameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the $a x + b$ -group, the Heisenberg group, the euclidean motion group and the euclidean conformal group

Authors with CRIS profile

Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

Involved external institutions

Louisiana State University

United States (USA) (US)

How to cite

APA:

Neeb, K.H., & Olafsson, G. (2015). Reflection positive one-parameter groups and dilations. Complex Analysis and Operator Theory, 9(3), 653 - 721. https://doi.org/10.1007/s11785-014-0402-2

MLA:

Neeb, Karl Hermann, and Gestur Olafsson. "Reflection positive one-parameter groups and dilations." Complex Analysis and Operator Theory 9.3 (2015): 653 - 721.

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