Reflection Positivity on the Real Line

Abstract

After providing the conceptual framework for reflection positive representations in the preceding two chapters, we now turn to the fine points of reflection positivity on the additive group $(\mathbb {R},+)$. Although this Lie group is quite trivial, reflection positivity on the real line has many interesting facets and is therefore quite rich. We thus describe its main features in this and the subsequent chapter. As reflection positive functions play a crucial role, we start in Sect. 4.1 with reflection positive functions on intervals $(-a, a)\subseteq \mathbb {R}$. Here we already encounter the main feature of reflection positivity dealing with two different notions of positivity, one related to the group structure on $\mathbb {R}$ and the other related to the $*$-semigroup structure on $\mathbb {R}_+$, resp., the convex structure of intervals. All this is linked to representation theory in Sect. 4.2, where we start our investigation of reflection positive representations of the symmetric semigroup $(\mathbb {R},\mathbb {R}_+,-\mathop {\mathrm{id}}\nolimits _\mathbb {R})$. These are unitary one-parameter groups $(U_t)_{t \in \mathbb {R}}$ on a reflection positive Hilbert space $(\mathscr {E},\mathscr {E}_+,\theta )$ satisfying $U_t \mathscr {E}_+ \subseteq \mathscr {E}_+$ for $t>0$ and $\theta U_t \theta = U_{-t}$ for $t\in \mathbb {R}$. On $\widehat{\mathscr {E}}$ this leads to a semigroup $(\widehat{U}_t)_{t \ge 0}$ of hermitian contractions. The main result in Sect. 4.2 is that the OS transform “commutes with reduction”, where reduction refers to the passage to the fixed points of U and $\widehat{U}$ in $\mathscr {E}$ and $\widehat{\mathscr {E}}$, respectively (Proposition 4.2.6). Reflection positive functions for $(\mathbb {R},\mathbb {R}_+,-\mathop {\mathrm{id}}\nolimits _\mathbb {R})$ are classified in terms of integral representations in Sect. 4.3. We shall see in particular that any hermitian contraction semigroup $(C_t)_{t \ge 0}$ on a Hilbert space $\mathscr {H}$ has a so-called minimal dilation represented by the reflection positive function $\psi (t) := C_{|t|}$. We also provide a concrete model for this dilation on the space $\mathscr {E}= L^2(\mathbb {R},\mathscr {H})$ with $(U_t f)(p) = e^{itp}f(p)$, where $\mathscr {E}_+ = L^2_+(\mathbb {R},\mathscr {H})$ is the positive spectral subspace for the translation group, which is, by the Laplace transform, isomorphic to the $\mathscr {H}$-valued Hardy space $H^2(\mathbb C_+,\mathscr {H})$ on the right half plane $\mathbb C_+ = \mathbb {R}_+ + i \mathbb {R}$. We conclude this chapter by showing that, for any reflection positive one-parameter group for which $\mathscr {E}_+$ is cyclic and fixed points are trivial, the space $\mathscr {E}_+$ is outgoing in the sense of Lax–Phillips scattering theory (Proposition 4.4.2). This establishes a remarkable connection between reflection positivity and scattering theory that leads to a normal form of reflection positive one-parameter groups by translations on spaces of the form $\mathscr {E}= L^2(\mathbb {R},\mathscr {H})$ with $\mathscr {E}_+ = L^2(\mathbb {R}_+,\mathscr {H})$. Applying the Fourier transform to our concrete dilation model leads precisely to this normal form.