Remarks on time-energy uncertainty relations

Using a recent construction of observables characterizing the time of occurence of an effect in quantum theory, we present a rigorous derivation of the standard time-energy uncertainty relation. In addition, we prove an uncertainty relation for time measurements only.Comment: 9 pages, to be pubblished in Rev. Math. Phys. issue in honor of H. Arak

The very steep spectrum radio halo in Abell 697

In this paper we present a detailed study of the giant radio halo in the galaxy cluster Abell 697, with the aim to constrain its origin and connection with the cluster dynamics. We performed high sensitivity GMRT observations at 325 MHz, which showed that the radio halo is much brighter and larger at this frequency, compared to previous 610 MHz observations. In order to derive the integrated spectrum in the frequency range 325 MHz--1.4 GHz, we re--analysed archival VLA data at 1.4 GHz and made use of proprietary GMRT data at 610 MHz. {Our multifrequency analysis shows that the total radio spectrum of the giant radio halo in A\,697 is very steep, with $\alpha_{\rm~325 MHz}^{\rm~1.4 GHz} \approx 1.7-1.8$. %\pm0.1$. Due to energy arguments, a hadronic origin of the halo is disfavoured by such steep spectrum. Very steep spectrum halos in merging clusters are predicted in the case that the emitting electrons are accelerated by turbulence, observations with the upcoming low frequency arrays will be able to test these expectations.}Comment: 10 pages, 8 figures, A&A in pres

Modular dynamics in diamonds

We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.Comment: 48 page