Reflection Positive Representations

Abstract

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).