New Spaces in Physics; Formal and Conceptual Reflections

437 Pages·6.007 MB·English

by  Mathieu Anel, Gabriel Catren

NewSpacesinPhysics
After the development of manifolds and algebraic varieties in the previous century,
mathematicians and physicists have continued to advance concepts of space. This
bookanditscompanionexplorevariousnewnotionsofspace,includingbothformal
and conceptual points of view, as presented by leading experts at the New Spaces in
MathematicsandPhysicsworkshopheldattheInstitutHenriPoincaréin2015.
Thisvolumecoversabroadrangeoftopicsinmathematicalphysics,includingnon-
commutativegeometry,supergeometry,derivedsymplecticgeometry,highergeometric
quantization, intuitionistic quantum logic, problems with the continuum description
of spacetime, twistor theory, loop quantum gravity, and geometry in string theory.
It is addressed primarily to mathematical physicists and mathematicians, but also to
historiansandphilosophersofthesedisciplines.
mathieu anel isaVisitingAssistantProfessoratCarnegieMellonUniversity.
His research interests include higher category theory, algebraic topology, and topos
theory.
gabriel catrenisPermanentResearcherinphilosophyofphysicsattheFrench
National Centre for Scientific Research (CNRS). His research interests include the
foundationsofclassicalandquantummechanics,andthefoundationsofgaugetheories.

New Spaces in Physics
Formal and Conceptual Reflections
Editedby
MATHIEU ANEL
CarnegieMellonUniversity
GABRIEL CATREN
CNRS-UniversitédeParis
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DOI:10.1017/9781108854399
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Title:Newspacesinphysics:formalandconceptualreflections/
editedbyMathieuAnel,CarnegieMellonUniversity,Pennsylvania,
GabrielCatren,CentreNationaldelaRechercheScientifique(CNRS),Paris.
Description:NewYork:CambridgeUniversityPress,2020.|Includes
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Identifiers:LCCN2020006655(print)|LCCN2020006656(ebook)|ISBN
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ISBN-VolumeII978-1-108-49062-7Hardback
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Contents
ContentsforNewSpacesinMathematics pagevii
Introduction 1
MathieuAnelandGabrielCatren
PARTI NONCOMMUTATIVEAND
SUPERCOMMUTATIVEGEOMETRIES 21
1 NoncommutativeGeometry,theSpectralStandpoint 23
AlainConnes
2 TheLogicofQuantumMechanics(Revisited) 85
KlaasLandsman
3 SupergeometryinMathematicsandPhysics 114
MikhailKapranov
PARTII SYMPLECTICGEOMETRY 153
4 DerivedStacksinSymplecticGeometry 155
DamienCalaque
5 HigherPrequantumGeometry 202
UrsSchreiber
PARTIII SPACETIME 279
6 StruggleswiththeContinuum 281
JohnC.Baez
v
vi Contents
7 Twistor Theory: A Geometric Perspective for
DescribingthePhysicalWorld 327
RogerPenrose
8 QuantumGeometryofSpace 373
MuxinHan
9 StringyGeometryandEmergentSpace 407
MarcosMariño
Contents for New Spaces in Mathematics
ContentsforNewSpacesinPhysics pagevii
Introduction 1
MathieuAnelandGabrielCatren
PARTI DIFFERENTIALGEOMETRY 29
1 AnIntroductiontoDiffeology 31
PatrickIglesias-Zemmour
2 NewMethodsforOldSpaces:SyntheticDifferentialGeometry 83
AndersKock
3 MicrolocalAnalysisandBeyond 117
PierreSchapira
PARTII TOPOLOGYANDALGEBRAICTOPOLOGY 153
4 Topo-logie 155
MathieuAnelandAndréJoyal
5 SpacesasInfinity-Groupoids 258
TimothyPorter
6 HomotopyTypeTheory:TheLogicofSpace 322
MichaelShulman
PARTIII ALGEBRAICGEOMETRY 405
7 SheavesandFunctorsofPoints 407
MichelVaquié
8 Stacks 462
NicoleMestranoandCarlosSimpson
vii
viii ContentsforNewSpacesinMathematics
9 TheGeometryofAmbiguity:AnIntroductiontotheIdeas
ofDerivedGeometry 505
MathieuAnel
10 Geometryindg-Categories 554
MaximKontsevich
Introduction
MathieuAnelandGabrielCatren
Contents
NewSpacesinPhysics 1
1 SummariesoftheChapters 8
2 Acknowledgments 18
References 18
NewSpacesinPhysics
Two fundamental scientific revolutions took place in physics during the
first decades of the 20th century: Einstein’s geometric description of the
gravitational interaction by means of the general theory of relativity and the
developmentofquantummechanics.Thesetworevolutionsradicallymodified
ourunderstandingofthelawsthatrulethephysicalphenomenatakingplaceat
opposite(astrophysicalandmicroscopic)spatiotemporalscales.
Ononehand,generalrelativityintroducedintophysicsessentialgeometric
ideasandtoolsmainlydevelopedduringthe19thcenturyinpuremathematics,
notably differential geometry, Riemannian geometry, and tensor calculus.
Moreover, general relativity provided the motivating example of the general
program – launched by H. Weyl around 1918 – intended to provide similar
geometricdescriptionsoftheotherfundamental(electromagneticandnuclear)
interactions.This“geometrizationprogram”wasfinallyachievedinthe1950s
intheframeworkoftheYang–Millstheoriesandacquiredasolidmathematical
foundationandgeometricinterpretationwiththetheoryofCartan(forgeneral
relativity)andEhresmann(forYang–Millstheories)connectionsonprincipal
1