Reflection Positive Representations

  • Karl-Hermann NeebEmail author
  • Gestur Ólafsson
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 32)


In this chapter we turn to operators on reflection positive (real or complex) Hilbert spaces and introduce the Osterwalder–Schrader transform to pass from operators on \(\mathscr {E}_{+}\) to operators on \(\widehat{\mathscr {E}}\) (Sect. 3.1). The objects represented in reflection positive Hilbert spaces \((\mathscr {E},\mathscr {E}_+,\theta )\) are symmetric Lie groups \((G,\tau )\), i.e., a Lie group G, endowed with an involutive automorphism \(\tau \). A typical example in physics arises from the euclidean motion group and time reversal. There are several ways to specify compatibility of a unitary representation \((U,\mathscr {E})\) of \((G,\tau )\) with \(\mathscr {E}_+\) and \(\theta \) and thus to define reflection positive representations (Sect. 3.3). One is to specify a subset \(G_{+}\subseteq G\) and assume that \(\mathscr {E}_{+}\) is generated by applying \(G_{+}^{-1}\) to a suitable subspace of \(\mathscr {E}_{+}\). The other simpler one applies if \(S := G_{+}^{-1}\) is a subsemigroup of G invariant under the involution \(s \mapsto s^{\sharp } = \tau (s)^{-1}\). Then we simply require \(\mathscr {E}_{+}\) to be S-invariant. In both cases we can use the integrability results in Chap.  7 to obtain unitary representations of the 1-connected Lie group \(G^{c}\) with Lie algebra \({\mathfrak g}^{c} = {\mathfrak h}+ i {\mathfrak q}\) on \(\widehat{\mathscr {E}}\). As reflection positive unitary representations are mostly constructed by applying a suitable Gelfand–Naimark–Segal (GNS) construction to reflection positive functions, we discuss this correspondence in some detail in Sect. 3.4. In particular, we discuss the Markov condition in this context (Proposition 3.4.9).


Positive Reflection Unitary Representation Suitable Subspace Subsemigroup Euclidean Motion Group 
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© The Author(s) 2018

Authors and Affiliations

  1. 1.Department MathematikUniversität Erlangen-NürnbergErlangenGermany
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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