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On virtual representations of symmetric spaces and their analytic continuation. (English) Zbl 0537.22017

The motivation of the paper stems from models of relativistic quantum fields based on Euclidean field theory. In this theory one introduces fields based on the Euclidean group ISO(n,\({\mathbb{R}})\) and then by analytic continuation to the Poincaré group ISO(n-1,1,\({\mathbb{R}})\) recovers a relativistic field theory [cf. K. Osterwalder and R. Schrader [Commun. Math. Phys. 31, 83-112 (1973; Zbl 0274.46047), ibid. 42, 281-305 (1975; Zbl 0303.46034)]. To generalize the Euclidean approach, the authors formulate analytic continuation of representations in terms of a symmetric Lie group (G,K,\(\sigma)\) and its real symmetric Lie algebra (g,k,\(\sigma)\) with the decomposition \(g=k\oplus m\) and an involutive automorphism \(\sigma\). The group \(G^*\) is defined as the simply connected Lie group with the dual symmetric Lie algebra \(g^*\). A virtual representation of G is defined as a local homomorphism \(\pi\) from G into linear operators densely defined on a separable Hilbert space \({\mathcal H}\) with the properties: \(\pi\) restricted to K is continuous and unitary, there exists a neighbourhood U of e in G invariant under right translation by K and a linear subspace \({\mathcal D}\) related to U and dense in \({\mathcal H}\) where \(\pi\) fulfills three local conditions. Examples of virtual representations and the possibility of their extension into global representations of G are discussed. For two classes of symmetric Lie groups G, one of them being the Euclidean group, the authors prove that a virtual representation of G admits an analytic continuation into a unitary representation of \(G^*\).
Reviewer: P.Kramer

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
81T08 Constructive quantum field theory
22E43 Structure and representation of the Lorentz group
20C35 Applications of group representations to physics and other areas of science
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