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An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic elements appear naturally in the geometry of causal symmetric spaces and in representation theory. We characterize stably elliptic elements in terms of the fixed point algebra of $\mathrm{Ad}(g)$ and show that the connected components of the set $G^{se}$ of stably elliptic elements can be described in terms of the Weyl group action on a compactly embedded Cartan subalgebra. In the case of simple hermitian Lie groups we relate stably elliptic elements to maximal invariant cones and the associated subsemigroups. In particular we show that the basic connected component $G^{se}(0)$ can be characterized in terms of the compactness of order intervals and that $G^{se}(0)$ is globally hyperbolic with respect to the induced biinvariant causal structure., Comment: 39 pages

Phantom systems consisting of liposome suspensions are widely employed to investigate quantitative MRI parameters mimicking cellular membranes. The proper physical understanding of the measurement results, however, requires proper models for liposomes and their interaction with the surrounding water molecules. Here, we present an MD-based approach for the theoretical prediction of R1=1/T1, the dependence of R1 on water concentration and the magnetization exchange between lipids and interacting water layer in lipids and lipid mixtures. Moreover, a new parameter is introduced which quantitatively measures the amount of hydration water (hydration water fraction, f_HW) based on conventional spoiled gradient echo MR acquisitions. Both f_HW and the magnetisation exchange rate between lipids and hydration water were determined quantitatively from spoiled gradient echo data. We observed that liposome systems behaved similarly, apart from PLPC which showed both lower hydration water fraction and lower exchange rate. The extracted parameters accurately predicted the measured water fraction-dependent R1 rates and allowed for a theoretical understanding of MR parameters in liposomes of different composition.

We show that the Fr\'echet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi--Metzner--Sachs group of symmetries of an asymptotically flat space-time is not locally exponential., Comment: 26 pages, v2: Major revision, corrected a mistake in earlier version. Main results are still valid but were substantially expanded. New coauthor joined. v3: Corrected typos

We develop a novel perspective on reflection positivity (RP) on the strip by systematically developing the analogies with the unit disc and the upper half plane in the complex plane. These domains correspond to the three conjugacy classes of one-parameter groups in the M\"obius group (elliptic for the disc, parabolic for the upper half plane and hyperbolic for the strip). In all cases, reflection positive functions correspond to positive functionals on $H^\infty$ for a suitable involution. For the strip, reflection positivity naturally connects with Kubo--Martin--Schwinger (KMS) conditions on the real line and further to standard pairs, as they appear in Algebraic Quantum Field Theory. We also exhibit a curious connection between Hilbert spaces on the strip and the upper half plane, based on a periodization process.

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations: they are well-known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half Lie groups that reduces matters to the case where a ``maximal torus'' is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from ``bounded extremal weights'', and for these the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists., Comment: 52 pages

This paper builds on our previous work in which we showed that, for all connected semisimple linear Lie groups $G$ acting on a non-compactly causal symmetric space $M = G/H$, every irreducible unitary representation of $G$ can be realized by boundary value maps of holomorphic extensions in distributional sections of a vector bundle over $M$. In the present paper we discuss this procedure for the connected Lorentz group $G = SO_{1,d}(R)_e$ acting on de Sitter space $M = dS^d$. We show in particular that the previously constructed nets of real subspaces satisfy the locality condition. Following ideas of Bros and Moschella from the 1990's, we show that the matrix-valued spherical function that corresponds to our extension process extends analytically to a large domain $G_C^{cut}$ in the complexified group $G_C = \SO_{1,d}(C)$, which for $d = 1$ specializes to the complex cut plane $C \setminus (-\infinity, 0]$. A number of special situations is discussed specifically: (a) The case $d = 1$, which closely corresponds to standard subspaces in Hilbert spaces, (b) the case of scalar-valued functions, which for $d > 2$ is the case of spherical representations, for which we also describe the jump singularities of the holomorphic extensions on the cut in de Sitter space, (c) the case $d = 3$, where we obtain rather explicit formulas for the matrix-valued spherical functions., Comment: 53pp

Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincar\'e group can be studied for general Lie groups, whose Lie algebra contains an Euler element, i.e., ad h is diagonalizable with eigenvalues in {-1,0,1}. This has been explored by the authors and their collaborators during recent years. A key property in this construction is the Bisognano-Wichmann property (thermal property for wedge region algebras) concerning the geometric implementation of modular groups of local algebras. In the present paper we prove that under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano-Wichmann property, are always generated by Euler elements. We also show the converse, namely that in presence of Euler elements and the Bisognano-Wichmann property, regularity and localizability hold in a quite general setting. Lastly we show that, in this generalized AQFT, in the vacuum representation, under analogous assumptions (regularity and Bisognano-Wichmann), the von Neumann algebras associated to wedge regions are type III_1 factors, a property that is well-known in the AQFT context., Comment: 65 pages

We continue our investigation of the interplay between causal structures on symmetric spaces and geometric aspects of Algebraic Quantum Field Theory. We adopt the perspective that the geometric implementation of the modular group is given by the flow generated by an Euler element of the Lie algebra (an element defining a 3-grading). Since any Euler element of a semisimple Lie algebra specifies a canonical non-compactly causal symmetric space M = G/H, we turn in this paper to the geometry of this flow. Our main results concern the positivity region W of the flow (the corresponding wedge region): If G has trivial center, then W is connected, it coincides with the so-called observer domain, specified by a trajectory of the modular flow which at the same time is a causal geodesic, it can also be characterized in terms of a geometric KMS condition, and it has a natural structure of an equivariant fiber bundle over a Riemannian symmetric space that exhibits it as a real form of the crown domain of G/K. Among the tools that we need for these results are two observations of independent interest: a polar decomposition of the positivity domain and a convexity theorem for G-translates of open $H$-orbits in the minimal flag manifold specified by the 3-grading., Comment: 51pp

We study holomorphic extensions of one-parameter groups on locally convex spaces with a view to applications to KMS boundary conditions. In the first part we deal with analytic extensions of one-parameter groups of operators on locally convex spaces and in the second part we apply our results to spaces of distribution vectors of unitary representations of Lie groups. This leads to new tools that can be used to construct, from unitary Lie group representations, nets of standard subspaces, as they appear in Algebraic Quantum Field Theory. We also show that these methods fail for spaces of analytic vectors, and this in turn leads to new maximality results for domains of analytic extensions of orbit maps for unitary representations., Comment: 48 pages; accepted for publication in the journal Pure and Applied Functional Analysis

Let G be a connected simple linear Lie group and H in G a symmetric subgroup such that the corresponding symmetric space G/H is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of standard subspaces on G/H that is isotone, covariant and has the Reeh--Schlieder and the Bisognano--Wichmann property. We also show that this result extends to the universal covering group of SL(2,R) which has some interesting application to intersections of standard subspaces associated to representations of such groups. For this a detailed study of hyperfunction and distribution vectors is needed. In particular we show that every H-finite hyperfunction vector is in fact a distribution vector., Comment: Final version, 75 pages

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.

In this article we discuss the interplay between causal structures of symmetric spaces and geometric aspects of Algebraic Quantum Field Theory (AQFT). The central focus is the set of Euler elements in a Lie algebra, i.e., elements whose adjoint action defines a 3-grading. In the first half of this article we survey the classification of reductive causal symmetric spaces from the perspective of Euler elements. This point of view is motivated by recent applications in AQFT. In the second half we obtain several results that prepare the exploration of the deeper connection between the structure of causal symmetric spaces and AQFT. In particular, we explore the technique of strongly orthogonal roots and corresponding systems of sl_2-subalgebras. Furthermore, we exhibit real Matsuki crowns in the adjoint orbits of Euler elements and we describe the group of connected components of the stabilizer group of Euler elements., Comment: 61 pages

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 clarifications

Based on the construction provided in our paper "Covariant homogeneous nets of standard subspaces", Comm. Math. Phys. 386 (2021), 305-358, we construct non-modular covariant one-particle nets on the two-dimensional de Sitter spacetime and on the three-dimensional Minkowski space.

In this note we study in a finite dimensional Lie algebra ${\mathfrak g}$ the set of all those elements x for which the closed convex hull of the adjoint orbit contains no affine lines; this contains in particular elements whose adjoint orbits generates a pointed convex cone~$C_x$. Assuming that ${\mathfrak g}$ is admissible, i.e., contains a generating invariant convex subset not containing affine lines, we obtain a natural characterization of such elements, also for non-reductive Lie algebras. Motivated by the concept of standard (Borchers) pairs in QFT, we also study pairs $(x,h)$ of Lie algebra elements satisfying $[h,x]=x$ for which $C_x$ pointed. Given $x$, we show that such elements $h$ can be constructed in such a way that ${\rm ad} h$ defines a $5$-grading, and characterize the cases where we even get a $3$-grading., Comment: 30 pages

This is the first in a series of papers on projective positive energy representations of gauge groups. Let $\Xi \rightarrow M$ be a principal fiber bundle, and let $\Gamma_{c}(M,\mathrm{Ad}(\Xi))$ be the group of compactly supported (local) gauge transformations. If $P$ is a group of `space-time symmetries' acting on $\Xi\rightarrow M$, then a projective unitary representation of $\Gamma_{c}(M,\mathrm{Ad}(\Xi))\rtimes P$ is of positive energy if every `timelike generator' $p_0 \in \mathfrak{p}$ gives rise to a Hamiltonian $H(p_0)$ whose spectrum is bounded from below. Our main result shows that in the absence of fixed points for the cone of timelike generators, the projective positive energy representations of the connected component $\Gamma_{c}(M,\mathrm{Ad}(\Xi))_0$ come from 1-dimensional $P$-orbits. For compact $M$ this yields a complete classification of the projective positive energy representations in terms of lowest weight representations of affine Kac-Moody algebras. For noncompact $M$, it yields a classification under further restrictions on the space of ground states. In the second part of this series we consider larger groups of gauge transformations, which contain also global transformations. The present results are used to localize the positive energy representations at (conformal) infinity., Comment: 121 pages. Part I of a series of papers

Let $\alpha : {\mathbb R} \to Aut(G)$ define a continuous ${\mathbb R}$-action on the topological group $G$. A unitary representation $\pi^\flat$ of the extended group $G^\flat := G \rtimes_\alpha {\mathbb R}$ is called a ground state representation if the unitary one-parameter group $\pi^\flat(e,t) = e^{itH}$ has a non-negative generator $H \geq 0$ and the subspace $\ker H$ of ground states generates the Hilbert space under $G$. In this paper we introduce the class of strict ground state representations, where $\pi^\flat$ and the representation of the subgroup $G^0 := Fix(\alpha)$ on $\ker H$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of $G^0$. This is particularly effective if the occurring representations of $G^0$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations., Comment: 51 pages

Motivated by construction in Algebraic Quantum Field Theory we introduce wedge domains in compactly causal symmetric spaces M=G/H, which includes in particular anti de Sitter space in all dimensions and its coverings. Our wedge domains generalize Rindler wedges in Minkowski space. The key geometric structure we use is the modular flow on M defined by an Euler element in the Lie algebra of G. Our main geometric result asserts that three seemingly different characterizations of these domains coincide: the positivity domain of the modular vector field; the domain specified by a KMS like analytic extension condition for the modular flow; and the domain specified by a polar decomposition in terms of certain cones. In the second half of the article we show that our wedge domains share important properties with wedge domains in Minkowski space. If G is semisimple, there exist unitary representations of G and isotone covariant nets of real subspaces defined for any open subset of M, which assign to connected components of the wedge domains a standard subspace whose modular group corresponds to the modular flow on M. This corresponds to the Bisognano--Wichmann property in Quantum Field Theory. We also show that the set of G-translates of the connected components of the wedge domain provides a geometric realization of the abstract wedge space introduced by the first author and V. Morinelli., Comment: 62 pp

We analyze reflection positive representations in terms of positive Hankel operators. This is motivated by the fact that positive Hankel operators are described in terms of their Carleson measures, whereas the compatibility condition between representations and reflection positive Hilbert spaces is quite intricate. This leads us to the concept of a Hankel positive representation of triples $(G,S,\tau)$, where $G$ is a group, $\tau$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $\tau(S) = S^{-1}$. For the triples $(\mathbb Z,\mathbb N,-id_{\mathbb Z})$, corresponding to reflection positive operators, and $(\mathbb R,\mathbb R_+,-id_{\mathbb R})$, corresponding to reflection positive one-parameter groups, we show that every Hankel positive representation can be made reflection positive by a slight change of the scalar product. A key method consists in using the measure $\mu_H$ on $\mathbb R_+$ defined by a positive Hankel operator $H$ on $H^2(\mathbb C_+)$ to define a Pick function whose imaginary part, restricted to the imaginary axis, provides an operator symbol for $H$., Comment: 36 pages