On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

Geometric modular action for disjoint intervals and boundary conformal field theory

In suitable states, the modular group of local algebras associated with unions of disjoint intervals in chiral conformal quantum field theory acts geometrically. We translate this result into the setting of boundary conformal QFT and interpret it as a relation between temperature and acceleration. We also discuss aspects ("mixing" and "charge splitting") of geometric modular action for unions of disjoint intervals in the vacuum state.Comment: Dedicated to John E. Roberts on the occasion of his 70th birthday; 24 pages, 3 figure

Is life a thermal horizon ?

This talk aims at questioning the vanishing of Unruh temperature for an inertial observer in Minkovski spacetime with finite lifetime, arguing that in the non eternal case the existence of a causal horizon is not linked to the non-vanishing of the acceleration. This is illustrated by a previous result, the diamonds temperature, that adapts the algebraic approach of Unruh effect to the finite case.Comment: Proceedings of the conference DICE 2006, Piombino september 200

Deformations of the Canonical Commutation Relations and Metric Structures

Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics. In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations. We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h- and q-derivatives, respectively. In these various examples, some points turn out to be at infinite distance. We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance. On the circle, this provides an explicit example of computation of the Wasserstein distance

A Prevention through Design Approach for the Environmental S&H Conditions and the Ventilation System at an Italian Underground Quarry

Even if the number of the Italian quarrying operations in underground is steadily growing, in many cases the safety criticalities are somehow underestimated, in spite of the regulations D.Lgs. 81/08 and D.Lgs. 624/96, Italian enforcements of the European Directives. Ventilation system is conceived to face very simplified requirements, whilst critical pollutants or emergency situations are not taken in due considerations. Asides, the ventilation system fault and availability analysis is seldom included in the project. The paper deals with the results of a research work started some years ago at an underground quarry exploited through drift sublevel based on drill and blast technique, to identify criteria suitable to grant effective safety and health -S&H- conditions for the workers operating in the underground in the Prevention through Design approach. Taken into account that the phases for an effective approach to the S&H problems in underground must follow a hierarchic method in which the risk management should be faced from an effective hazard reduction to a minimum at the sources, and the ventilation should be considered only as a 4th level solution, the possibilities of control at the main pollution sources, i.e. the emission of pollutants due to the rock winning and mucking operations, have been examined. The residual risk was then faced with both the original underground and airways layout definition for a new exploitation development, based on technical and efficiency considerations, and on fire emergency computer simulations. Finally, the paper summarizes the results of an availability analysis of the ventilation system for the normal operating conditions and the emergencies management, on the basis of the results of Hazard Evaluation techniques, in particular Hazard and Operability Analysis and Fault Tree Analysis

Minimal length in quantum space and integrations of the line element in Noncommutative Geometry

We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime; on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d'_L, which coincides exactly with the spectral distance d_D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator - together with their translations - d'_L and d_D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d'_L and d_D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.Comment: 29 pages, 2 figures. Minor corrections to match the published versio

An algebraic Birkhoff decomposition for the continuous renormalization group

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited

Relative entropy and the Bekenstein bound

Elaborating on a previous work by Marolf et al, we relate some exact results in quantum field theory and statistical mechanics to the Bekenstein universal bound on entropy. Specifically, we consider the relative entropy between the vacuum and another state, both reduced to a local region. We propose that, with the adequate interpretation, the positivity of the relative entropy in this case constitutes a well defined statement of the bound in flat space. We show that this version arises naturally from the original derivation of the bound from the generalized second law when quantum effects are taken into account. In this formulation the bound holds automatically, and in particular it does not suffer from the proliferation of the species problem. The results suggest that while the bound is relevant at the classical level, it does not introduce new physical constraints semiclassically.Comment: 12 pages, 1 figure, minor changes and references adde

Lung response to prone positioning in mechanically-ventilated patients with COVID-19

Background: Prone positioning improves survival in moderate-to-severe acute respiratory distress syndrome (ARDS) unrelated to the novel coronavirus disease (COVID-19). This benefit is probably mediated by a decrease in alveolar collapse and hyperinflation and a more homogeneous distribution of lung aeration, with fewer harms from mechanical ventilation. In this preliminary physiological study we aimed to verify whether prone positioning causes analogue changes in lung aeration in COVID-19. A positive result would support prone positioning even in this other population. Methods: Fifteen mechanically-ventilated patients with COVID-19 underwent a lung computed tomography in the supine and prone position with a constant positive end-expiratory pressure (PEEP) within three days of endotracheal intubation. Using quantitative analysis, we measured the volume of the non-aerated, poorly-aerated, well-aerated, and over-aerated compartments and the gas-to-tissue ratio of the ten vertical levels of the lung. In addition, we expressed the heterogeneity of lung aeration with the standardized median absolute deviation of the ten vertical gas-to-tissue ratios, with lower values indicating less heterogeneity. Results: By the time of the study, PEEP was 12 (10–14) cmH2O and the PaO2:FiO2 107 (84–173) mmHg in the supine position. With prone positioning, the volume of the non-aerated compartment decreased by 82 (26–147) ml, of the poorly-aerated compartment increased by 82 (53–174) ml, of the normally-aerated compartment did not significantly change, and of the over-aerated compartment decreased by 28 (11–186) ml. In eight (53%) patients, the volume of the over-aerated compartment decreased more than the volume of the non-aerated compartment. The gas-to-tissue ratio of the ten vertical levels of the lung decreased by 0.34 (0.25–0.49) ml/g per level in the supine position and by 0.03 (− 0.11 to 0.14) ml/g in the prone position (p < 0.001). The standardized median absolute deviation of the gas-to-tissue ratios of those ten levels decreased in all patients, from 0.55 (0.50–0.71) to 0.20 (0.14–0.27) (p < 0.001). Conclusions: In fifteen patients with COVID-19, prone positioning decreased alveolar collapse, hyperinflation, and homogenized lung aeration. A similar response has been observed in other ARDS, where prone positioning improves outcome. Therefore, our data provide a pathophysiological rationale to support prone positioning even in COVID-19