Reflection Positivity and Conformal Symmetry

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory and duality between unitary representations of the euclidean motion group and the Poincare group. On the mathematical side this duality can be made precise as follows. If \g is a Lie algebra with an involutive automorphism $\tau$. Decompose \g = \fh \oplus \fq = \ker(\tau - \1) \oplus \ker(\tau + \1) into $\tau$-eigenspaces and let \g^c := \fh \oplus i \fq. At the core of the notion of reflection positivity is the idea that this duality can sometimes be implemented on the level of unitary representations. The idea is simple on the Lie algebra level: Let (\pi,\cH^0) be a representation of \g where $\pi$ acts by skew-symmetric operators. Assume that there exists a unitary operator $J$ of order two such that $J \pi J = \pi\circ \tau$ and a \g-invariant subspace \cK^0 which is {\it $J$-positive}. Then complex linear extension leads to a representation of \g^c on \cK^0 by operators which are skew-symmetric with respect to $h_J$, so that we obtain a "unitary" representation of \g^c on the pre-Hilbert space \cK_J^0 := \cK_J/{v \: \cK^0 \: h_J(v,v)=0}. The aim of this article is twofold. First we discuss reflection positivity in an abstract setting using {\it reflection positive distributions} on the Lie group $G_\tau =G\rtimes {1,\tau}$ and {\it reflection positive distribution vectors} of a unitary representation of $G_\tau$. Then we apply these ideas to the conformal group \OO_{1,n+1}^+(\R) of the sphere \bS^n as well as the the half-space picture mostly used in physics

Reflection positive one-parameter groups and dilations

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincare group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect reflection positivity with the outgoing realization of unitary one-parameter groups by Lax and Phillips. We further show that our results provide effective tools to construct reflection positive representations of general symmetric Lie groups, including the ax+b-group, the Heisenberg group, the euclidean motion group and the euclidean conformal group.Comment: Final version. To appear in Complex Analysis and Operator Theor

Integrability of unitary representations on reproducing kernel spaces

Let g be a Banach Lie algebra and \tau : g ---> g an involution. Write g=h+q for the eigenspace decomposition of g with respect to \tau and g^c := h+iq for the dual Lie algebra. In this article we show the integrability of two types of infinitesimally unitary representations of g^c. The first class of representation is determined by a smooth positive definite kernel K on a locally convex manifold M. The kernel is assumed to satisfying a natural invariance condition with respect to an infinitesimal action \beta : g \to V(M) by locally integrable vector fields that is compatible with a smooth action of a connected Lie group $H$ with Lie algebra h. The second class is constructed from a positive definite kernel corresponding to a positive definite distribution K \in C^{-\infty}(M \times M) on a finite dimensional smooth manifold M which satisfies a similar invariance condition with respect to a homomorphism \beta : g \to V(M). As a consequence, we get a generalization of the Luscher--Mack Theorem which applies to a class of semigroups that need not have a polar decomposition. Our integrability results also apply naturally to local representations and representations arising in the context of reflection positivity

Nets of standard subspaces on Lie groups

Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $\tau_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{\pi i ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_\tau = G \rtimes \{e,\tau_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspace $H_E(O) \subseteq H$, generated by the subspaces $U(\varphi)E$, where $\varphi \in C^\infty_c(O,\mathbb{R})$ is a real-valued test function on $O$. Then $H_E(O)$ is dense in $H_E(G)$ for every non-empty open subset $O \subseteq G$ (Reeh--Schlider property). For the real standard subspace $V \subseteq H$, for which $J_V = U(\tau_G)$ is the modular conjugation and $\Delta_V^{-it/2\pi} = U(\exp th)$ is the modular group, we obtain sufficient conditions to be of the form $H_E(S)$ for an open subsemigroup $S \subseteq G$. If $\mathfrak{g}$ is semisimple with simple hermitian ideals of tube type, we verify these criteria and obtain nets of cyclic subspacs $H_E(O)$, $O \subseteq G$, satisfying the Bisognano--Wichman property for some domains O. Our construction also yields such nets on simple Jordan space-times and compactly causal symmetric spaces of Cayley type. By second quantization, these nets lead to free quantum fields in the sense of Haag--Kastler on causal homogeneous spaces whose groups are generated by modular groups and conjugations.Comment: 50 pages; error in Thm. 5.3 has been correcte

Reflection positivity on spheres

In this article we specialize a construction of a reflection positive Hilbert space due to Dimock and Jaffe--Ritter to the sphere $\mathbb{S}^n$. 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 $G^c = \mathrm{O}_{1,n}(\mathbb{R})^{\uparrow}$ and that the representations thus obtained are the irreducible unitary spherical representations of this group. A key tool is a certain complex domain $\Xi$, known as the crown of the hyperboloid, containing a half-sphere $\mathbb{S}^n_+$ and the hyperboloid $\mathbb{H}^n$ 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 $\Xi$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.Comment: Final versio

Algebraic Quantum Field Theory and Causal Symmetric Spaces

In this article we review our recent work on the causal structure of symmetric spaces and related geometric aspects of Algebraic Quantum Field Theory. Motivated by some general results on modular groups related to nets of von Neumann algebras,we focus on Euler elements of the Lie algebra, i.e., elements whose adjoint action defines a 3-grading. We study the wedge regions they determine in corresponding causal symmetric spaces and describe some methods to construct nets of von Neumann algebras on causal symmetric spaces that satisfy abstract versions of the Reeh--Schlieder and the Bisognano-Wichmann condition

Wedge domains in non-compactly causal symmetric spaces

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory (AQFT), modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric space $G/H$. This class contains the de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of \fg. We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of those seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of those spaces has a natural extension to a non-compactly causal symmetric space of the form G_\C/G^c where G^c is certain real form of the complexification G_\ of G. As G_\C/G^c is non-compactly causal it also comes with the three types of wedge domains. Our results says that the intersection of those domains with G/H$ agrees with the wedge domains in G/H.Comment: Minor changes and clarification