Reflection positivity on spheres

Neeb KH, Ólafsson G (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Analysis and Mathematical Physics Springer

Book Volume: 10

Article Number: 9

Journal Issue: 1

DOI: 10.1007/s13324-019-00353-3

Abstract

In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe–Ritter to the sphere Sⁿ. We determine the resulting Osterwalder–Schrader Hilbert space, a construction that can be viewed as the step from euclidean to relativistic quantum field theory. We show that this process gives rise to an irreducible unitary spherical representation of the orthochronous Lorentz group G^c= O 1 , n(R) ^↑ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain Ξ , known as the crown of the hyperboloid, containing a half-sphere S+n and the hyperboloid Hⁿ as totally real submanifolds. This domain provides a bridge between those two manifolds when we study unitary representations of G^c in spaces of holomorphic functions on Ξ. We connect this analysis with the boundary components which are the de Sitter space and a bundle over the space of future pointing lightlike vectors.

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., & Ólafsson, G. (2020). Reflection positivity on spheres. Analysis and Mathematical Physics, 10(1). https://dx.doi.org/10.1007/s13324-019-00353-3

MLA:

Neeb, Karl Hermann, and Gestur Ólafsson. "Reflection positivity on spheres." Analysis and Mathematical Physics 10.1 (2020).

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