The Bisognano-Wichmann property on nets of standard subspaces, some sufficient conditions

We discuss the Bisognano-Wichmann property for local Poincar\'e covariant nets of standard subspaces. We give a sufficient algebraic condition on the covariant representation ensuring the Bisognano-Wichmann and Duality properties without further assumptions on the net called modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property. Furthermore, we give an outlook in the standard subspace setting on the relation between the Bisognano-Wichmann property and the Split property.Comment: Final version. To appear in Annales Henri Poincar\'

An algebraic condition for the Bisognano-Wichmann Property

The Bisognano-Wichmann property for local, Poincar\'e covariant nets of standard subspaces is discussed. We present a sufficient algebraic condition on the covariant representation ensuring Bisognano-Wichmann and Duality properties without further assumptions on the net. Our modularity condition holds for direct integrals of scalar massive and massless representations. We conclude that in these cases the Bisognano-Wichmann property is much weaker than the Split property. Furthermore, we present a class of massive modular covariant nets not satisfying the Bisognano-Wichmann property.Comment: Invited contribution to the Proceedings of the 14th Marcel Grossmann Meeting - MG14 (Rome, 2015

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.Comment: v2: 18p, largely extended and more results added. Title adjuste

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, M\"obius 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.Comment: 34 pages, 3 tikz figure

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.Comment: 30 page

Scale and M\"obius covariance in two-dimensional Haag-Kastler net

Given a two-dimensional Haag-Kastler net which is Poincar\'e-dilation covariant with additional properties, we prove that it can be extended to a M\"obius 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 Poincar\'e-dilation covariant net which cannot be extended to a M\"obius covariant net, and discuss the obstructions.Comment: 35 pages, 9 Tikz figures. See http://www.mat.uniroma2.it/~tanimoto/smc18.pdf for figures with better fadin

Split property for free massless finite helicity fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincar\'e 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 M\"obius 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.Comment: v2: Minor corrections, comments and references added, as to appear in Ann. H. Poin

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

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