Magnifying Spacetime: How Physics Changes with Scale: How Physics Changes with Scale

186 Pages·2019·5.036 MB·English

by  Daniel Nathan Coumbe

PHYSICS RESEARCH AND TECHNOLOGY

M S
AGNIFYING PACETIME

H P C S
OW HYSICS HANGES WITH CALE

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or
by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no
expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No
liability is assumed for incidental or consequential damages in connection with or arising out of information
contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in
rendering legal, medical or any other professional services.
P R T
HYSICS ESEARCH AND ECHNOLOGY

Additional books and e-books in this series can be found on
Nova’s website under the Series tab.
PHYSICS RESEARCH AND TECHNOLOGY

M S
AGNIFYING PACETIME

H P C S
OW HYSICS HANGES WITH CALE

DANIEL COUMBE
Copyright © 2019 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted
in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying,
recording or otherwise without the written permission of the Publisher.

We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to
reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and
locate the “Get Permission” button below the title description. This button is linked directly to the title’s
permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN,
or ISSN.

For further questions about using the service on copyright.com, please contact:
Copyright Clearance Center
Phone: [phone] Fax: [phone] E-mail: [email protected].

NOTICE TO THE READER
The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or
implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is
assumed for incidental or consequential damages in connection with or arising out of information
contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary
damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts
of this book based on government reports are so indicated and copyright is claimed for those parts to the
extent applicable to compilations of such works.

Independent verification should be sought for any data, advice or recommendations contained in this
book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons
or property arising from any methods, products, instructions, ideas or otherwise contained in this
publication.

This publication is designed to provide accurate and authoritative information with regard to the subject
matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in
rendering legal or any other professional services. If legal or any other expert assistance is required, the
services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS
JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A
COMMITTEE OF PUBLISHERS.

Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data

Names: Coumbe, Daniel, author.
Title: Magnifying spacetime: how physics changes with scale / Daniel Coumbe
(Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark).
Other titles: Physics research and technology.
Description: Hauppauge, New York: Nova Science Publishers, Inc., [2019] |
Series: Physics research and technology
Identifiers: LCCN 2019013905| ISBN 9781536153194 (softcover) | ISBN
1536153192 (softcover) | ISBN 9781536153200 (eBook) | ISBN 1536153206 (eBook)
Subjects: LCSH: Space and time. | Scaling laws (Statistical physics) |
Quantum gravity. | Fractals. | Renormalization group.

Published by Nova Science Publishers, Inc. † New York
Contents
ListofFigures xi
ListofTables xiii
Preface xv
Units,ConventionsandCommonAbbreviations xviii
Acknowledgments xxi
Introduction xxiii
1 ScaleTransformations 1
1.1. GlobalScaleTransformations 1
1.2. LocalScaleTransformations 3
2 Fractals 7
2.1. TheCoastlineParadox 7
2.2. FractalDimensions 8
2.2.1. TheTopologicalDimension 8
2.2.2. TheHausdorffDimension 9
2.2.3. TheSpectralDimension 10
2.2.4. TheWalkDimension 13
2.2.5. Myrheim-Meyer Dimension 13
2.2.6. CorrelationDimension 14
vi Contents
2.3. FractalsAboveUsandBelowUs 15
2.3.1. FractalsinCosmology 15
2.3.2. FractalsinQuantumMechanics 16
3 AMinimumScale? 19
3.1. AtomsofSpacetime 19
3.2. EvidenceforaMinimalLength 22
3.2.1. ALowerBoundonDistanceMeasurements 22
3.2.2. BlackHoleLimitations 23
3.2.3. Heisenberg’sMicroscope 25
3.2.4. High-EnergyConvergence 28
3.2.5. FluctuationsoftheConformalFactor 29
3.2.6. ModifiedFeynmanPropagator 32
3.2.7. LatticeQuantumGravity 33
3.3. SpecialRelativityandaMinimalLength 34
3.4. PhenomenologicalQuantumGravity 36
4 TheRenormalisationGroup 39
4.1. Overview 39
4.2. Kadanoff’sBlock-SpinModel 40
4.3. TheBetaFunction 41
4.4. RenormalisationGroupOperators 43
5 TheAsymptoticSafetyScenario 45
5.1. Weinberg’sGreatIdea 45
5.2. APotentialProblem 49
6 QuantumGravityontheLattice 53
6.1. LatticeRegularisation 53
6.2. GeometricObservables 54
6.3. EuclideanDynamicalTriangulations 59
6.3.1. ConformalInstability 63
6.4. CausalDynamicalTriangulations 65
Contents vii
7 IstheDimensionofSpacetimeScaleDependent? 69
7.1. Why4Dimensions? 69
7.2. TheEvidenceforDimensionalReduction 71
7.2.1. StringTheory 71
7.2.2. CausalDynamicalTriangulations 71
7.2.3. EuclideanDynamicalTriangulations 75
7.2.4. Horava-LifshitzGravity 78
7.2.5. AsymptoticSafety 79
7.2.6. LoopQuantumGravity 81
7.2.7. TheWheeler-DeWittEquation 82
7.2.8. CausalSetTheory 83
7.2.9. Non-CommutativeGeometry 83
7.3. APossibleSolutiontoanOldProblem 84
7.4. DimensionalReductionintheSky 85
7.5. ExperimentalTests 88
7.5.1. Cosmology 88
7.5.2. GeVScales 90
7.5.3. TeVScales 91
7.6. WhatisDimensionalReductionReallyTellingUs? 91
7.6.1. Overview 91
7.6.2. ScaleDependentLength 92
7.6.3. ADualDescription? 97
8 ScaleDependentSpacetime 101
8.1. EinsteinandWeyl 101
8.2. RenormalisingSpacetime 106
8.2.1. Motivation 106
8.2.2. EstimatingΩ(k) 109
8.2.3. ImmediateImplications 110
8.2.4. ImplicationsforQuantumGravity 111
FinalThoughts 117
References 119
AbouttheAuthor 143
viii Contents
Index 145
RelatedNovaPublications 149