Discrete or Continuous? The Quest for Fundamental Length in Modern Physics

282 Pages·2014·1.45 MB·English

by  Amit Hagar

DISCRETE OR CONTINUOUS?
TheQuestforFundamentalLengthinModernPhysics
The idea of infinity plays a crucial role in our understanding of the universe,
with the infinite spacetime continuum perhaps the best-known example – but is
spacetime really continuous? Throughout the history of science, many have felt
thatthecontinuummodelisanunphysicalidealization,andthatspacetimeshould
bethoughtofas“quantized”atthesmallestofscales.
Combining novel conceptual analysis, a fresh historical perspective, and con-
crete physical examples, this unique book tells the story of the search for the
fundamentalunitoflengthinmodernphysics,fromearlyclassicalelectrodynam-
ics to current approaches to quantum gravity. Original philosophical theses, with
directimplicationsfortheoreticalphysicsresearch,arepresentedanddefendedin
an accessible format that avoids complex mathematics. Blending history, philos-
ophy, and theoretical physics, this refreshing outlook on the nature of spacetime
shedslightononeofthemostthought-provokingtopicsinmodernphysics.
amit hagar is an Associate Professor in the Department of History and Phi-
losophy of Science, Indiana University Bloomington, where he specializes in the
foundationsofmodernphysics.

DISCRETE OR CONTINUOUS?
The Quest for Fundamental Length in Modern Physics
AMIT HAGAR
IndianaUniversity
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
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Informationonthistitle:www.cambridge.org/9781107062801
©A.Hagar2014
Thispublicationisincopyright.Subjecttostatutoryexception
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Firstpublished2014
PrintedintheUnitedKingdombyClays,StIvesplc
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ISBN978-1-107-06280-1Hardback
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Tomyparents
Timeisnature’swayofkeepingeverythingfromhappeningatonce;
spaceiswhatpreventseverythingfromhappeningtome.
AttributedtoJohnWheeler

Contents
Preface pageix
1 Introduction 1
2 Argumentsfrommath 7
2.1 Outline 7
2.2 Zeno’sparadoxofextension 9
2.3 Topologyandtheargumentagainstcollision 13
2.4 Geometryandthetileargument 20
2.5 Richeralternativestothereallinemodel 27
2.6 ThephysicalChurch–Turingthesis 31
2.7 Pureversusappliedmathematics 35
3 Argumentsfromphilosophy 37
3.1 Outline 37
3.2 Metaphysicalmotivations 39
3.3 Epistemologyandtheprimacyof“length” 44
3.4 Isdiscretenesstranscendental? 49
3.5 Enoughwiththe“isms” 52
4 Electrodynamics,QED,andearlyQFT 54
4.1 Outline 54
4.2 Classicalelectrodynamics 56
4.3 FromQMtoQED 63
4.4 Theriseofrenormalization 83
4.5 Philosophicalramifications 88
vii
viii Contents
5 Quantumgravity:prehistory 97
5.1 Outline 97
5.2 Earlysteps 98
5.3 Gravitons,measurability,andthePlanckscale 107
5.4 Non-commutativegeometry 112
5.5 (Re)entersgravity 118
5.6 Quantizinggravity–thephilosophicaldebate 126
6 Einsteinonthenotionoflength 141
6.1 Outline 141
6.2 Constructingtheprinciples 142
6.3 TheSwann–Einsteincorrespondence 149
6.4 ReadingEinstein 156
6.5 EinsteinandtheconstructiveapproachtoSTR 160
6.6 Geometryanddynamics,again 170
7 Quantumgravity:currentapproaches 171
7.1 Outline 171
7.2 Stringtheory 174
7.3 Backgroundindependentstrategies 181
7.4 Emergentgravity 193
7.5 Onthe“disappearance”ofspacetime 194
8 Theproofisinthepudding 201
8.1 Outline 201
8.2 Thequestforquantumgravityphenomenology 203
8.3 Consistencyproofs 208
8.4 Theperilsofinnovation 213
8.5 Thereandbackagain 228
9 Coda 231
References 239
Index 264