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Colored Discrete Spaces - Higher Dimensional Combinatorial Maps and Quantum Gravity

English · Hardback

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Description

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This book provides a number of combinatorial tools that allow a systematic study of very general discrete spaces involved in the context of discrete quantum gravity. In any dimension D, we can discretize Euclidean gravity in the absence of matter over random discrete spaces obtained by gluing families of polytopes together in all possible ways. These spaces are then classified according to their curvature. In D=2, it results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of D=2 quantum gravity is recovered.
Previous results in higher dimension regarded triangulations, converging towards  a continuum random tree, or gluings of simple building blocks of small sizes, for which multi-trace matrix model results are recovered in any even dimension. In this book, the author develops a bijection with stacked two-dimensional discrete surfaces for the most general colored building blocks, and details how it can be used to classify colored discrete spaces according to their curvature.  The way in which this combinatorial problem arrises in discrete quantum gravity and  random tensor models is discussed in detail.

List of contents


Colored Simplices and Edge-Colored Graphs.- Bijective Methods.- Properties of Stacked Maps.- Summary and Outlook.

Summary

This book provides a number of combinatorial tools that allow a systematic study of very general discrete spaces involved in the context of discrete quantum gravity. In any dimension D, we can discretize Euclidean gravity in the absence of matter over random discrete spaces obtained by gluing families of polytopes together in all possible ways. These spaces are then classified according to their curvature. In D=2, it results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of D=2 quantum gravity is recovered.
Previous results in higher dimension regarded triangulations, converging towards  a continuum random tree, or gluings of simple building blocks of small sizes, for which multi-trace matrix model results are recovered in any even dimension. In this book, the author develops a bijection with stacked two-dimensional discrete surfaces for the most general colored building blocks, and details how it can be used to classify colored discrete spaces according to their curvature.  The way in which this combinatorial problem arrises in discrete quantum gravity and  random tensor models is discussed in detail.

Product details

Authors Luca Lionni
Publisher Springer, Berlin
 
Languages English
Product format Hardback
Released 01.01.2018
 
EAN 9783319960227
ISBN 978-3-31-996022-7
No. of pages 218
Dimensions 156 mm x 242 mm x 20 mm
Weight 506 g
Illustrations XVIII, 218 p. 107 illus., 98 illus. in color.
Series Springer Theses
Springer Theses
Subjects Natural sciences, medicine, IT, technology > Physics, astronomy > Theoretical physics

Gravitation, Geometrie, B, Gravity, geometry, Physics, Physics and Astronomy, Relativity physics, Mathematical Methods in Physics, Classical and Quantum Gravity, Classical and Quantum Gravitation, Relativity Theory

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