Elementary Introduction to Quantum Geometry

293 Pages·2022·51.374 MB·English

by  Jan Ambjorn

Elementary Introduction to
Quantum Geometry
This graduate textbook provides an introduction to quantum gravity, when spacetime
is two-dimensional. The quantization of gravity is the main missing piece of theoretical
physics, but in two dimensions it can be done explicitly with elementary mathematical
tools, but it still has most of the conceptional riddles present in higher dimensional (not
yet known) quantum gravity.
It provides an introduction to a very interdisciplinary field, uniting physics (quantum
geometry) and mathematics (combinatorics) in a non-technical way, requiring no prior
knowledge of quantum field theory or general relativity.
Using the path integral, the chapters provide self-contained descriptions of random walks,
random trees and random surfaces as statistical systems where the free relativistic par-
ticle, the relativistic bosonic string and two-dimensional quantum gravity are obtained as
scaling limits at phase transition points of these statistical systems. The geometric nature
of the theories allows one to perform the path integral by counting geometries. In this
way the quantization of geometry becomes closely linked to the mathematical fields of
combinatorics and probability theory. By counting the geometries, it is shown that the
two-dimensional quantum world is fractal at all scales unless one imposes restrictions on
the geometries. It is also discussed in simple terms how quantum geometry and quantum
matter can interact strongly and change the properties both of the geometries and of the
matter systems.
It requires only basic undergraduate knowledge of classical mechanics, statistical mechan-
ics and quantum mechanics, as well as some basic knowledge of mathematics at under-
graduate level. It will be an ideal textbook for graduate students in theoretical and statisti-
cal physics and mathematics studying quantum gravity and quantum geometry.
Key features:
• Presents the first elementary introduction to quantum geometry
•  Explores how to understand quantum geometry without prior knowledge beyond
bachelor level physics and mathematics
• Contains exercises, problems and solutions to supplement and enhance learning

Elementary Introduction to
Quantum Geometry
Jan Ambjørn
First edition published 2022
by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
and by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
© 2023 Jan Ambjørn
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Names: Animasaun, Isaac Lare, author. | Shah, Nehad Ali, author. | Wakif,
Abderrahim, author. | Mahanthesh, Basavarajappa, author. | Sivaraj,
Ramachandran, author. | Koríko, Olubode Kolade, author.
Title: Ratio of momentum diffusivity to thermal diffusivity : introduction,
meta-analysis, and scrutinization / Isaac Laare Animasaun, Nehad Ali
Shah, Abderrahim Wakif, Basavarajappa Mahanthesh, Ramachandran Sivaraj,
Olubode Kolade Koriko.
Description: First edition. | Boca Raton : Chapman & Hall/CRC Press, 2023.
| Includes bibliographical references and index. |
Identifiers: LCCN 2022005066 (print) | LCCN 2022005067 (ebook) | ISBN
9781032108520 (hbk) | ISBN 9781032310893 (pbk) | ISBN 9781003217374 (ebk)
Subjects: LCSH: Materials--Thermal properties. | Thermal diffusivity.
Classification: LCC TA418.52 .A55 2023 (print) | LCC TA418.52 (ebook) |
DDC 620.1/1296--dc23/eng/20220401
LC record available at https://lccn.loc.gov/2022005066
LC ebook record available at https://lccn.loc.gov/2022005067
ISBN: 978-1-032-33555-1 (hbk)
ISBN: 978-1-032-34100-2 (pbk)
ISBN: 978-1-003-32056-2 (ebk)
DOI: 10.1201/9781003320562
Typeset in Nimbus
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface.......................................................................................................................ix
Author.......................................................................................................................xi
Chapter1 PreliminaryMaterialPart1:ThePathIntegral................................1
1.1 TheClassicalAction...............................................................1
1.2 StatisticalMechanics..............................................................2
1.3 ClassicaltoQuantum..............................................................5
1.4 TheFeynmanPathIntegralinQuantumMechanics..............7
1.5 TheFeynman-KacPathIntegralandImaginaryTime..........12
1.6 ProblemSetsandFurtherReading.......................................14
Chapter2 TheFreeRelativisticParticle........................................................15
2.1 ThePropagator......................................................................15
2.2 ThePathIntegral...................................................................17
2.3 RandomWalksandUniversality..........................................21
2.4 ProblemSetsandFurtherReading........................................23
Chapter3 One-DimensionalQuantumGravity..............................................25
3.1 ScalarFieldsinOneDimension...........................................25
3.2 HausdorffDimensionandScalingRelations........................30
3.3 ProblemSetsandFurtherReading.......................................34
Chapter4 BranchedPolymers........................................................................35
4.1 DefinitionsandGeneralities.................................................35
4.2 RootedBranchedPolymersandUniversality.......................37
4.3 TheTwo-PointFunction.......................................................39
4.4 IntrinsicPropertiesofBranchedPolymers...........................42
4.5 MulticriticalBranchedPolymers..........................................44
4.6 GlobalandLocalHausdorffDimensions.............................46
4.7 ProblemSetsandFurtherReading.......................................47
Chapter5 RandomSurfacesandBosonicStrings..........................................49
5.1 TheAction,GreenFunctionsandCriticalExponents..........49
5.2 RegularizingtheIntegrationoverGeometries......................55
5.3 Digression:SummationoverTopologies..............................63
v
vi Contents
5.4 ScalingoftheMass...............................................................68
5.5 ScalingoftheStringTension................................................76
5.6 ProblemSetsandFurtherReading.......................................79
Chapter6 Two-DimensionalQuantumGravity.............................................81
6.1 Solving2DQuantumGravitybyCountingGeometries.......81
6.2 CountingTriangulationsoftheDisk....................................83
6.3 MultiloopsandtheLoop-InsertionOperator........................92
6.4 ExplicitSolutionforBipartiteGraphs..................................93
6.5 TheNumberofLargeTriangulations...................................96
6.6 TheContinuumLimit...........................................................99
6.7 OtherUniversalityClasses..................................................102
6.8 Appendix.............................................................................104
6.9 ProblemSetsandFurtherReading.....................................105
Chapter7 TheFractalStructureof2DGravity............................................107
7.1 UniversalityandtheMissingCorrelationLength...............107
7.2 TheTwo-LoopPropagator..................................................107
7.3 TheTwo-PointFunction.....................................................114
7.4 TheLocalHausdorffDimensionin2DGravity.................117
7.5 ProblemSetsandFurtherReading.....................................120
Chapter8 TheCausalDynamicalTriangulationmodel...............................121
8.1 LorentzianVersusEuclideanSetUp..................................121
8.2 DefiningandSolvingtheCDTModel................................122
8.3 GCDT:ShowcasingQuantumGeometry...........................131
8.4 GCDTDefinedasaScalingLimitofGraphs.....................137
8.5 TheClassicalContinuumTheoryRelatedto2DCDT.......140
8.6 ProblemSetsandFurtherReading.....................................142
AppendixA PreliminaryMaterialPart2:GreenFunctions.............................145
AppendixB ProblemSets1–13.......................................................................161
B.1 ProblemSet1......................................................................161
B.2 ProblemSet2......................................................................164
B.3 ProblemSet3......................................................................167
B.4 ProblemSet4......................................................................170
B.5 ProblemSet5......................................................................175
B.6 ProblemSet6......................................................................178
B.7 ProblemSet7......................................................................185
B.8 ProblemSet8......................................................................191
B.9 ProblemSet9......................................................................195
B.10 ProblemSet10....................................................................204
Contents vii
B.11 ProblemSet11....................................................................210
B.12 ProblemSet12....................................................................217
B.13 ProblemSet13....................................................................219
AppendixC SolutionstoProblemSets1–13...................................................225
C.1 SolutionstoProblemSet1.................................................225
C.2 SolutionstoProblemSet2.................................................229
C.3 SolutionstoProblemSet3.................................................232
C.4 SolutionstoProblemSet4.................................................235
C.5 Solutionstoproblemset5..................................................238
C.6 SolutionstoProblemSet6.................................................243
C.7 SolutionstoProblemSet7.................................................247
C.8 SolutionstoProblemSet8.................................................250
C.9 SolutionstoProblemSet9.................................................252
C.10 SolutionstoProblemSet10...............................................257
C.11 SolutionstoProblemSet11...............................................260
C.12 SolutionstoProblemSet12...............................................263
C.13 SolutionstoProblemSet13...............................................267
References.............................................................................................................273
Index......................................................................................................................275