arXiv subjects

Karl-Hermann Neeb

Publications and source records attributed to Karl-Hermann Neeb.

At least 37 records · Page 2Linked to original sources

arXiv2022

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.

math-ph↗

arXiv2022

From Euler elements and 3-gradings to non-compactly causal symmetric spaces

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.

math-ph↗

arXiv2022

A family of non-modular covariant AQFTs

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.

math.OA↗

arXiv2021

Induced differential characters on nonlinear Graßmannians

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from $M$ to the nonlinear Graßmannians $\mathrm{Gr}^S(M)$ of submanifolds of $M$ of a fixed type $S$. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden-Weinstein symplectic form. The associated Kostant-Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

math.DG↗

arXiv2021

Elements in pointed invariant cones in Lie algebras and corresponding affine pairs

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.

math.RT↗

arXiv2021

Positive energy representations of gauge groups I: Localization

This is the first in a series of papers on projective positive energy representations of gauge groups. Let $Ξ\rightarrow M$ be a principal fiber bundle, and let $Γ_{c}(M,\mathrm{Ad}(Ξ))$ be the group of compactly supported (local) gauge transformations. If $P$ is a group of `space-time symmetries' acting on $Ξ\rightarrow M$, then a projective unitary representation of $Γ_{c}(M,\mathrm{Ad}(Ξ))\rtimes P$ 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 $Γ_{c}(M,\mathrm{Ad}(Ξ))_0$ 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.

math-ph↗

arXiv2021

Ground state representations of topological groups

Let $α: {\mathbb R} \to Aut(G)$ define a continuous ${\mathbb R}$-action on the topological group $G$. A unitary representation $π^\flat$ of the extended group $G^\flat := G \rtimes_α{\mathbb R}$ is called a ground state representation if the unitary one-parameter group $π^\flat(e,t) = e^{itH}$ 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 $π^\flat$ and the representation of the subgroup $G^0 := Fix(α)$ 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.

math.RT↗

arXiv2021

Wedge domains in compactly causal symmetric spaces

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.

math.RT↗

arXiv2021

Standard subspaces of Hilbert spaces of holomorphic functions on tube domains

In this article we study standard subspaces of Hilbert spaces of vector-valued holomorphic functions on tube domains E + i C^0, where C \subeq E is a pointed generating cone invariant under e^{R h} for some endomorphism h \in \End(E), diagonalizable with the eigenvalues 1,0,-1 (generalizing a Lorentz boost). This data specifies a wedge domain W(E,C,h) \subeq E and one of our main results exhibits corresponding standard subspaces as being generated using test functions on these domains. We also investigate aspects of reflection positivity for the triple (E,C,e^{πi h}) and the support properties of distributions on E, arising as Fourier transforms of operator-valued measures defining the Hilbert spaces H. For the imaginary part of these distributions, we find similarities to the well known Huygens' principle, relating to wedge duality in the Minkowski context. Interesting examples are the Riesz distributions associated to euclidean Jordan algebras.

math.RT↗

arXiv2021

Reflection positivity and Hankel operators -- the multiplicity free case

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 $(G,S,τ)$, where $G$ is a group, $τ$ an involutive automorphism of $G$ and $S \subseteq G$ a subsemigroup with $τ(S) = S^{-1}$. 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 $μ_H$ 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 $H$.

math.FA↗

arXiv2021

Central extensions of Lie groups preserving a differential form

Let $M$ be a manifold with a closed, integral $(k+1)$-form $ω$, and let $G$ be a Fréchet-Lie group acting on $(M,ω)$. As a generalization of the Kostant-Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of $\mathfrak{g}$ by $\mathbb{R}$, indexed by $H^{k-1}(M,\mathbb{R})^*$. We show that the image of $H_{k-1}(M,\mathbb{Z})$ in $H^{k-1}(M,\mathbb{R})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group $\mathbb{T}$. The idea is to represent a class in $H_{k-1}(M,\mathbb{Z})$ by a weighted submanifold $(S,β)$, where $β$ is a closed, integral form on $S$. We use transgression of differential characters from $S$ and $ M $ to the mapping space $ C^\infty(S, M) $, and apply the Kostant-Souriau construction on $ C^\infty(S, M) $.

math.DG↗

arXiv2020

Covariant homogeneous nets of standard subspaces

Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti-Guido-Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $\mathbb Z_2$-graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. This allows us to exhibit an analog of the Haag-Kastler one-particle net axioms for such general Lie groups without referring to any specific spacetime. This set of axioms supports a first quantization net obtained by generalizing the BGL construction. The construction is possible for a large family of Lie groups and provides several new models. We further comment on orthogonal wedges and extension of symmetries.

math-ph↗

arXiv2020

Semigroups in 3-graded Lie groups and endomorphisms of standard subspaces

Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular group of V and that the modular involution J_V normalizes U(G). We want to determine the semigroup $S_V = \{ g\in G : U(g)V \subseteq V\}.$ In previous work we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup S_V under the assumption that ad h defines a 3-grading. Concretely, we show that the ad h-eigenspaces for the eigenvalue $\pm 1$ contain closed convex cones $C_\pm$, such that $S_V = exp(C_+) G_V exp(C_-)$, where $G_V$ is the stabilizer of V in G. To obtain this result we compare several subsemigroups of G specified by the grading and the positive cone $C_U$ of U. In particular, we show that the orbit U(G)V, endowed with the inclusion order, is an ordered symmetric space covering the adjoint orbit $Ad(G)h$, endowed with the partial order defined by~$C_U$.

math.FA↗

arXiv2020

Crossed products of C^*-algebras for singular actions with spectrum conditions

We analyze existence of crossed product constructions of Lie group actions on C^*-algebras which are singular. These are actions where the group need not be locally compact, or the action need not be strongly continuous. In particular, we consider the case where spectrum conditions are required for the implementing unitary group in covariant representations of such actions. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, we prove that the existence of cross representations is stable with respect to a large set of perturbations of the action, and we fully analyze the structure of cross representations of inner actions on von Neumann algebras. For one-parameter automorphism groups we study the cross property for covariant representations, where the generator of the implementing unitary group is positive. In particular, we find that if the Borchers-Arveson minimal implementing group is cross, then so are all other implementing groups. For higher dimensional Lie group actions, we consider a class of spectral conditions which include the ones occurring in physics, and is sensible also for non-abelian or for infinite dimensional Lie groups. We prove that the cross property of a covariant representation is fully determined by the cross property of a certain one-parameter subsystem. This greatly simplifies the analysis of the existence of cross representations, and it allows us to prove the cross property for several examples of interest to physics. We also consider non-abelian extensions of the Borchers-Arveson theorem. There is a full extension in the presence of a cyclic invariant vector, but otherwise one needs to determine the vanishing of lifting obstructions.

math.OA↗

arXiv2020

Covariant representations for possibly singular actions on C*-algebras

Singular actions on C*-algebras are automorphic group actions on C*-algebras, where the group need not be locally compact, or the action need not be strongly continuous. We study the covariant representation theory of such actions. In the usual case of strongly continuous actions of locally compact groups on C*-algebras, this is done via crossed products, but this approach is not available for singular C*-actions (this was our path in a previous paper). The literature regarding covariant representations for singular actions is already large and scattered, and in need of some consolidation. We collect in this survey a range of results in this field, mostly known. We improve some proofs and elucidate some interconnections. These include existence theorems by Borchers and Halpern, Arveson spectra, the Borchers-Arveson theorem, standard representations and Stinespring dilations as well as ground states, KMS states and ergodic states and the spatial structure of their GNS representations.

math.OA↗

arXiv2020

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 $τ_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{πi ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_τ= G \rtimes \{e,τ_G\}$ 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 $U(φ)E$, where $φ\in C^\infty_c(O,\mathbb{R})$ 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 $J_V = U(τ_G)$ is the modular conjugation and $Δ_V^{-it/2π} = U(\exp th)$ 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.

math-ph↗

arXiv2019

Symmetric spaces with dissecting involutions

An involutive diffeomorphism $σ$ of a connected smooth manifold $M$ is called dissecting if the complement of its fixed point set is not connected. Dissecting involutions on a complete Riemannian manifold are closely related to constructive quantum field theory through the work of Dimock and Jaffe/Ritter on the construction of reflection positive Hilbert spaces. In this article we classify all pairs $(M,σ)$, where $M$ is an irreducible symmetric space, not necessarily Riemannian, and $σ$ is a dissecting involutive automorphism. In particular, we show that the only irreducible $1$-connected Riemannian symmetric spaces are $S^n$ and $H^n$ with dissecting isometric involutions whose fixed point spaces are $S^{n-1}$ and $H^{n-1}$, respectively.

math.DG↗

arXiv2019

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 $Ξ$, 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 $Ξ$. We connect this analysis with the boundary components which are the de Sitter space and the Lorentz cone of future pointing light like vectors.

math.FA↗