Lectures on Gravity and Entanglement

The AdS/CFT correspondence provides quantum theories of gravity in which spacetime and gravitational physics emerge from ordinary non-gravitational quantum systems with many degrees of freedom. Recent work in this context has uncovered fascinating connections between quantum information theory and quantum gravity, suggesting that spacetime geometry is directly related to the entanglement structure of the underlying quantum mechanical degrees of freedom and that aspects of spacetime dynamics (gravitation) can be understood from basic quantum information theoretic constraints. In these notes, we provide an elementary introduction to these developments, suitable for readers with some background in general relativity and quantum field theory. The notes are based on lectures given at the CERN Spring School 2014, the Jerusalem Winter School 2014, the TASI Summer School 2015, and the Trieste Spring School 2015.Comment: 75 pages, beta version: please e-mail correction

Extension of information geometry for modelling non-statistical systems

In this dissertation, an abstract formalism extending information geometry is introduced. This framework encompasses a broad range of modelling problems, including possible applications in machine learning and in the information theoretical foundations of quantum theory. Its purely geometrical foundations make no use of probability theory and very little assumptions about the data or the models are made. Starting only from a divergence function, a Riemannian geometrical structure consisting of a metric tensor and an affine connection is constructed and its properties are investigated. Also the relation to information geometry and in particular the geometry of exponential families of probability distributions is elucidated. It turns out this geometrical framework offers a straightforward way to determine whether or not a parametrised family of distributions can be written in exponential form. Apart from the main theoretical chapter, the dissertation also contains a chapter of examples illustrating the application of the formalism and its geometric properties, a brief introduction to differential geometry and a historical overview of the development of information geometry.Comment: PhD thesis, University of Antwerp, Advisors: Prof. dr. Jan Naudts and Prof. dr. Jacques Tempere, December 2014, 108 page

Development of a novel 3D simulation modelling system for distributed manufacturing

This paper describes a novel 3D simulation modelling system for supporting our distributed machine design and control paradigm with respect to simulating and emulating machine behaviour on the Internet. The system has been designed and implemented using Java2D and Java3D. An easy assembly concept of drag-and-drop assembly has been realised and implemented by the introduction of new connection features (unified interface assembly features) between two assembly components (modules). The system comprises a hierarchical geometric modeller, a behavioural editor, and two assemblers. During modelling, designers can combine basic modelling primitives with general extrusions and integrate CAD geometric models into simulation models. Each simulation component (module) model can be visualised and animated in VRML browsers. It is reusable. This makes machine design re-configurable and flexible. A case study example is given to support our conclusions

Discrete space-time geometry and skeleton conception of particle dynamics

It is shown that properties of a discrete space-time geometry distinguish from properties of the Riemannian space-time geometry. The discrete geometry is a physical geometry, which is described completely by the world function. The discrete geometry is nonaxiomatizable and multivariant. The equivalence relation is intransitive in the discrete geometry. The particles are described by world chains (broken lines with finite length of links), because in the discrete space-time geometry there are no infinitesimal lengths. Motion of particles is stochastic, and statistical description of them leads to the Schr\"{o}dinger equation, if the elementary length of the discrete geometry depends on the quantum constant in a proper way.Comment: 22 pages, 0 figure

Reconstructing Quantum Geometry from Quantum Information: Spin Networks as Harmonic Oscillators

Loop Quantum Gravity defines the quantum states of space geometry as spin networks and describes their evolution in time. We reformulate spin networks in terms of harmonic oscillators and show how the holographic degrees of freedom of the theory are described as matrix models. This allow us to make a link with non-commutative geometry and to look at the issue of the semi-classical limit of LQG from a new perspective. This work is thought as part of a bigger project of describing quantum geometry in quantum information terms.Comment: 16 pages, revtex, 3 figure

The existence of thick triangulations -- an "elementary" proof

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also emphasized.Comment: 7 pages Short not