arXiv subjects

Vincenzo Morinelli

Publications and source records attributed to Vincenzo Morinelli.

At least 19 recordsLinked to original sources

arXiv2026

Orthogonal pairs of Euler elements II: Geometric Bisognano--Wichmann and Spin--Statistics Theorems

Models in Algebraic Quantum Field Theory (AQFT) may be generalized including Lie groups of symmetries whose Lie algebras admit an Euler element $h$, characterized by the property that $ad h$ is diagonalizable with eigenvalues in $\{-1, 0, 1\}$. These elements becomes fundamental to the formal description of wedge localization. In this paper, we extend the geometric analysis of Euler wedges and investigate their applications within the AQFT framework. We call a pair of Euler elements $(h, k)$ orthogonal if $e^{i \pi \operatorname{ad} h}(k) = -k.$ Using the geometric framework established in our previous work, we derive both a Bisognano--Wichmann Theorem and a Spin--Statistics Theorem for nets of standard subspaces and von Neumann algebras. Our results {show} how this generalized approach recovers classical results in the AQFT literature while providing a deeper structural understanding of the underlying geometry in established models.

math-ph↗

arXiv2025

Orthogonal pairs of Euler elements I. Classification, fundamental groups and twisted duality

The current article continues our project on representation theory, Euler elements, causal homogeneous spaces and Algebraic Quantum Field Theory (AQFT). We call a pair (h,k) of Euler elements orthogonal if $e^{πi \ad h} k = -k$. We show that, if (h,k) and (k,h) are orthogonal, then they generate a 3-dimensional simple subalgebra. We also classify orthogonal Euler pairs in simple Lie algebras and determine the fundamental groups of adjoint Euler elements in arbitrary finite-dimensional Lie algebras. Causal complements of wedge regions in spacetimes can be related to so-called twisted complements in the space of abstract Euler wedges, defined in purely group theoretic terms. We show that any pair of twisted complements can be connected by a chain of successive complements coming from $3$-dimensional subalgebras.

math.RT↗

arXiv2025

A geometric perspective on Algebraic Quantum Field Theory

In this paper we give a streamlined overview of some of the recent constructions provided with K.-H. Neeb, G. Ólafsson and collaborators for a new geometric approach to Algebraic Quantum Field Theory (AQFT). Motivations, fundamental concepts and some of the relevant results about the abstract structure of these models are here presented.

math.OA↗

arXiv2025

The modular Hamiltonian in asymptotically flat spacetime conformal to Minkowski

We consider a four-dimensional globally hyperbolic spacetime $(M,g)$ conformal to Minkowski spacetime, together with a massless, conformally coupled scalar field. Using a bulk-to-boundary correspondence, one can establish the existence of an injective $*$-homomorphism $Υ_M$ between $\mathcal{W}(M)$, the Weyl algebra of observables on $M$ and a counterpart which is defined intrinsically on future null infinity $\Im^+\simeq\mathbb{R}\times\mathbb{S}^2$, a component of the conformal boundary of $(M,g)$. Using invariance under the asymptotic symmetry group of $\Im^+$, we can individuate thereon a distinguished two-point correlation function whose pull-back to $M$ via $Υ_M$ identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider $\mathsf{V}^+_x$, a future light cone stemming from $x\in M$ as well as $\mathcal{W}(\mathsf{V}^+_x)=\mathcal{W}(M)|_{\mathsf{V}^+_x}$, its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in $\mathsf{K}_x$, a positive half strip on $\Im^+$. To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to $\mathsf{K}_x$. We extend such correspondence replacing $\mathsf{K}_x$ and $\mathsf{V}^+_x$ with deformed counterparts, denoted by $\mathsf{S}_C$ and $\mathsf{V}_C$. In addition, since the one particle Hilbert space at the boundary decomposes as a direct integral on the sphere of $U(1)$-currents defined on the real line, we prove that also the generator of the modular group associated to the standard subspace of $\mathsf{V}_C$ decomposes as a suitable direct integral. This result allows us to study the relative entropy between coherent states of the algebras associated to the deformed cones $\mathsf{V}_C$ establishing the quantum null energy condition.

math-ph↗

arXiv2024

An entropy bound due to symmetries

Let $H$ be a local net of real Hilbert subspaces of a complex Hilbert space on the family of double cones of the spacetime $\mathbb{R}^{d+1}$, covariant with respect to a positive energy, unitary representation $U$ of the Poincaré group, with the Bisognano-Wichmann property for the wedge modular group. We set an upper bound on the local entropy $S_H(ϕ|\! | C)$ of a vector in a region $C$ that depends only on $U$ and the PCT anti-unitary canonically associated with $H$. A similar result holds for local, Möbius covariant nets of standard subspaces on the circle. We compute the entropy increase and illustrate this bound for the nets associated with the $U(1)$-current derivatives.

math.OA↗

arXiv2023

From local nets to Euler elements

Various aspects of the geometric setting of Algebraic Quantum Field Theory (AQFT) models related to representations of the Poincaré 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.

math.OA↗

arXiv2023

Modular geodesics and wedge domains in non-compactly causal symmetric spaces

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.

math.DG↗

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

Scaling limits of lattice quantum fields by wavelets

We present a rigorous renormalization group scheme for lattice quantum field theories in terms of operator algebras. The renormalization group is considered as an inductive system of scaling maps between lattice field algebras. We construct scaling maps for scalar lattice fields using Daubechies' wavelets, and show that the inductive limit of free lattice ground states exists and the limit state extends to the familiar massive continuum free field, with the continuum action of spacetime translations. In particular, lattice fields are identified with the continuum field smeared with Daubechies' scaling functions. We compare our scaling maps with other renormalization schemes and their features, such as the momentum shell method or block-spin transformations.

math-ph↗

arXiv2021

Operator-algebraic renormalization and wavelets

We report on a rigorous operator-algebraic renormalization group scheme and construct the continuum free field as the scaling limit of Hamiltonian lattice systems using wavelet theory. A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets. Causality follows from Lieb-Robinson bounds for harmonic lattice systems. The scheme is related with the multi-scale entanglement renormalization ansatz and augments the semi-continuum limit of quantum systems.

math-ph↗

arXiv2021

Modular operator for null plane algebras in free fields

We consider the algebras generated by observables in quantum field theory localized in regions in the null plane. For a scalar free field theory, we show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres, and the modular operator decomposes accordingly. This implies that a certain form of QNEC is valid in free fields involving the causal completions of half-spaces on the null plane (null cuts). We also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states.

math-ph↗

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

Spacelike deformations: Higher-helicity fields from scalar fields

In contrast to Hamiltonian perturbation theory which changes the time evolution, "spacelike deformations" proceed by changing the translations (momentum operators). The free Maxwell theory is only the first member of an infinite family of spacelike deformations of the complex massless Klein-Gordon quantum field into fields of higher helicity. A similar but simpler instance of spacelike deformation allows to increase the mass of scalar fields.

math-ph↗

arXiv2019

The Bisognano-Wichmann property for asymptotically complete massless QFT

We prove the Bisognano-Wichmann property for asymptotically complete Haag-Kastler theories of massless particles. These particles should either be scalar or appear as a direct sum of two opposite integer helicities, thus, e.g., photons are covered. The argument relies on a modularity condition formulated recently by one of us (VM) and on the Buchholz' scattering theory of massless particles.

math-ph↗

arXiv2019

Split property for free massless finite helicity fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincaré representations extend to the conformal group and the conformal covariance plays an essential role in the argument. The split property is ensured by the trace class condition: Tr (exp(-s L_0)) is finite for all s>0 where L_0 is the conformal Hamiltonian of the Möbius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in [7]. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity.

math-ph↗

arXiv2018

Conformal covariance and the split property

We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, Möbius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.

math-ph↗

arXiv2018

Scale and Möbius covariance in two-dimensional Haag-Kastler net

Given a two-dimensional Haag-Kastler net which is Poincaré-dilation covariant with additional properties, we prove that it can be extended to a Möbius covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. The proof relies neither on the existence of stress-energy tensor nor any assumption on scaling dimensions. We exhibit some examples of Poincaré-dilation covariant net which cannot be extended to a Möbius covariant net, and discuss the obstructions.

math-ph↗