An algebraic formulation of causality for noncommutative geometry

We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.Comment: 24 pages, minor changes from v2, to appear in Classical and Quantum Gravit

Causality in noncommutative two-sheeted space-times

We investigate the causal structure of two-sheeted space-times using the tools of Lorentzian spectral triples. We show that the noncommutative geometry of these spaces allows for causal relations between the two sheets. The computation is given in details when the sheet is a 2- or 4-dimensional globally hyperbolic spin manifold. The conclusions are then generalised to a point-dependent distance between the two sheets resulting from the fluctuations of the Dirac operator.Comment: 26 pages, 2 figure

Noncommutative geometry, Lorentzian structures and causality

The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However, noncommutative geometry has mainly been developed using the Euclidean signature, and the typical Lorentzian aspects of space-time, the causal structure in particular, are not taken into account. We present an extension of noncommutative geometry \`a la Connes suitable the for accommodation of Lorentzian structures. In this context, we show that it is possible to recover the notion of causality from purely algebraic data. We explore the causal structure of a simple toy model based on an almost commutative geometry and we show that the coupling between the space-time and an internal noncommutative space establishes a new `speed of light constraint'.Comment: 24 pages, review article. in `Mathematical Structures of the Universe', eds. M. Eckstein, M. Heller, S.J. Szybka, CCPress 201

Acquire Driving Scenarios Efficiently: A Framework for Prospective Assessment of Cost-Optimal Scenario Acquisition

Scenario-based testing is becoming increasingly important in safety assurance for automated driving. However, comprehensive and sufficiently complete coverage of the scenario space requires significant effort and resources if using only real-world data. To address this issue, driving scenario generation methods are developed and used more frequently, but the benefit of substituting generated data for real-world data has not yet been quantified. Additionally, the coverage of a set of concrete scenarios within a given logical scenario space has not been predicted yet. This paper proposes a methodology to quantify the cost-optimal usage of scenario generation approaches to reach a certainly complete scenario space coverage under given quality constraints and parametrization. Therefore, individual process steps for scenario generation and usage are investigated and evaluated using a meta model for the abstraction of knowledge-based and data-driven methods. Furthermore, a methodology is proposed to fit the meta model including the prediction of reachable complete coverage, quality criteria, and costs. Finally, the paper exemplary examines the suitability of a hybrid generation model under technical, economical, and quality constraints in comparison to different real-world scenario mining methods.Comment: Accepted to be published as part of the 26th IEEE International Conference on Intelligent Transportation Systems (ITSC) 2023, Bilbao, Spain, September 24-28, 202

Direct generation of a multi-transverse mode non-classical state of light

Quantum computation and communication protocols require quantum resources which are in the continuous variable regime squeezed and/or quadrature entangled optical modes. To perform more and more complex and robust protocols, one needs sources that can produce in a controlled way highly multimode quantum states of light. One possibility is to mix different single mode quantum resources. Another is to directly use a multimode device, either in the spatial or in the frequency domain. We present here the first experimental demonstration of a device capable of producing simultanuously several squeezed transverse modes of the same frequency and which is potentially scalable. We show that this device, which is an Optical Parametric Oscillator using a self-imaging cavity, produces a multimode quantum resource made of three squeezed transverse modes

Noncommutative geometry of Zitterbewegung

Based on the mathematics of noncommutative geometry, we model a 'classical' Dirac fermion propagating in a curved spacetime. We demonstrate that the inherent causal structure of the model encodes the possibility of Zitterbewegung - the 'trembling motion' of the fermion. We recover the well-known frequency of Zitterbewegung as the highest possible speed of change in the fermion's 'internal space'. Furthermore, we show that the latter does not change in the presence of an external electromagnetic field and derive its explicit analogue when the mass parameter is promoted to a Higgs-like field. We discuss a table-top experiment in the domain of quantum simulation to test the predictions of the model and outline the consequences of our model for quantum gauge theories.Comment: 15 page

The Lorentzian distance formula in noncommutative geometry

For almost twenty years, a search for a Lorentzian version of the well-known Connes' distance formula has been undertaken. Several authors have contributed to this search, providing important milestones, and the time has now come to put those elements together in order to get a valid and functional formula. This paper presents a historical review of the construction and the proof of a Lorentzian distance formula suitable for noncommutative geometry.Comment: 16 pages, final form, few references adde

Inequality of opportunity in the land of opportunities : 1968-2001

We measure inequality of opportunity for earnings acquisition in the U.S. between 1968 and 2001. Following recent theories of social justice, earnings determinants are divided into two parts: Circumstances, which are characteristics outside individual control and effort representing factors impacting earnings but under individuals’ responsibility. Equality of opportunity requires that inequality of circumstances must be corrected while differences of effort must remain unaltered. Circumstances are represented by parental education and occupation, ethnic origin, place of birth and age. Effort is modeled with schooling choices and labour supply decisions. Using the PSID from 1968 to 2001, we provide two alternative assessments of inequality of opportunity using counterfactual distributions. The statistical framework is semi-parametric and builds on duration models. Finally, we conclude that inequality of opportunity represents between 20 and 43% of earnings inequality, but decreases all over the period reaching around 18% in 2001