Physics and Number theory

275 Pages·2006·2.35 MB·english

by  Vladimir G. Turaev

titelei10_nyssen.qxd 19.9.2006 8:45 Uhr Seite 1
IRMA Lectures in Mathematics and Theoretical Physics 10
Edited by Vladimir G. Turaev
Institut de Recherche Mathématique Avancée
Université Louis Pasteur et CNRS
7 rue René Descartes
67084 Strasbourg Cedex
France
titelei10_nyssen.qxd 19.9.2006 8:45 Uhr Seite 2
IRMA Lectures in Mathematics and Theoretical Physics
Edited by Vladimir G. Turaev
This series is devoted to the publication of research monographs, lecture notes, and other
materials arising from programs of the Institut de Recherche Mathématique Avancée
(Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical
physics and to make them accessible to wide circles of mathematicians, physicists, and
students of these disciplines.
Previously published in this series:
1 Deformation Quantization, Gilles Halbout(Ed.)
2 Locally Compact Quantum Groups and Groupoids, Leonid Vainerman(Ed.)
3 From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi(Eds.)
4 Three courses on Partial Differential Equations, Eric Sonnendrücker(Ed.)
5 Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher(Ed.)
6 Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature
7 Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier,
Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.)
8 AdS-CFT Correspondence: Einstein Metrics and Their Conformal Boundaries,
Oliver Biquard(Ed.)
9 Differential Equations and Quantum Groups,
D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schaefke(Eds.)
10 Physics and Number Theory, Louise Nyssen(Ed.)
Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)
titelei10_nyssen.qxd 19.9.2006 8:45 Uhr Seite 3
Physics and
Number Theory
Louise Nyssen
Editor
titelei10_nyssen.qxd 19.9.2006 8:45 Uhr Seite 4
Editor:
Louise Nyssen
Institut de Recherche Mathématique Avancée
Université Louis Pasteur et CNRS
7 Rue René Descartes
67084 Strasbourg Cedex
France
2000 Mathematics Subject Classification : 11F; 11L, 11M, 81T, 52C, 68R15
ISBN 978-3-03719-028-9
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie;
detailed bibliographic data are available in the Internet at http://dnb.ddb.de.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting,
reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission
of the copyright owner must be obtained.
©2006 European Mathematical Society
Contact address:
European Mathematical Society Publishing House
Seminar for Applied Mathematics
ETH-Zentrum FLI C4
CH-8092 Zürich
Switzerland
Phone: [phone]
Email: [email protected]
Homepage: www.ems-ph.org
Typeset using the author’s TEX files: I. Zimmermann, Freiburg
Printed in Germany
9 8 7 6 5 4 3 2 1
Preface
Asinglebookiscertainlynotenoughtodescribetherichandhistoricalrelationships
between physics and number theory. This volume presents a selection of problems
whicharecurrentlyinfulldevelopmentandinspiretheresearchofmanypeople. All
thepapersbeginwithasurveywhichwillmakeitpossibleevenfornon-specialiststo
understandthemandwillgiveanideaofthegreatvarietyofsubjectsandtechniques
inthisfrontierarea.
Thefirstpaper,“Thephaseofoscillationsandprimenumbers: classicalandquan-
tum”,byMichelPlanat,isanexampleofthestrongconnectionbetweenphysicsand
mathematics. Itstartsfromaconcreteproblemandbringsintoplayanimpressiveva-
rietyofmathematicaltechniques,especiallyinnumbertheory. Thepaperprovidesan
accessibleintroductiontotheproblemofphase-lockinginoscillatingsystems,bothat
aclassicallevelandataquantumlevel. Themathematicalformulationofthedifferent
aspects of this problem requires numerous tools: first, you see how prime numbers
appear, together with continuous fractions and the Mangoldt function. Then, come
somehyperbolicgeometryandtheRiemannζ-function. Onthequantumside,roots
ofunityandRamanujansumsarerelatedtonoiseinoscillations,andwhendiscussing
phaseinquantuminformation,theauthorusesBostandConnesKMSstates,Galois
ringsandfieldsalongwithsomefiniteprojectivegeometry.
Next there are two papers about crystallography. From a physical point of view,
a crystal is a solid having an essentially discrete diffraction diagram. It can be pe-
riodic or not. From a mathematical point of view, lattices in Rn are good tools to
describeperiodiccrystals,butnotaperiodicones. Verylittleisknownaboutaperiodic
crystals, apart from the so called quasicrystals, whose diffraction diagrams present
someregularity: theyareinvariantunderdilatationbyafactorthatmaybeirrational.
Theyarewelldescribedbysomediscretesetscalledcut-and-projectsets,whicharea
generalisationoflattices. Inhispaper“OnSelf-SimilarFinitelyGeneratedUniformly
Discrete(SFU-)SetsandSpherePackings”,Jean-LouisVerger-Gaugryisinterestedin
cut-and-projectsetsinRn. Thefirstpartofthepaperisasurveyofthelinkbetweenthe
geometryofnumbersandaperiodiccrystalsinphysics,fromthemathematicalpoint
of view. In the second part, the author proves some new results about the distances
betweenthepointsofcut-and-projectsets. Byconsideringeachpointasthecentreof
asphere,onegetsaspherepackingproblemwhichishopefullyagoodmodelforatom
packing. In “Nested quasicrystalline discretisation of the line”, Jean-Pierre Gazeau,
ZuzanaMasáková,andEditaPelantováfocusoncut-and-projectsetsobtainedfroma
squarelatticeinR2,withtheideaofconstructingaperiodicwavelets. Theyreviewthe
geometrical properties of such sets, their combinatorial properties from the point of
viewoflanguagetheoryandtheirrelationtononstandardnumerationsystemsbased
vi
on θ when the cut-and-project set is self-similar for an irrational scaling factor θ.
Finally,theyprovideanalgorithmwhichgeneratescut-and-projectsets.
In“Hopfalgebrasinrenormalizationtheory: localityandDyson–Schwingerequa-
tions from Hochschild cohomology”, Christoph Bergbauer and Dirk Kreimer refor-
mulateaproblemfromquantumfieldtheoryinalgebraicterms. Inthefirstpart,the
authorsexplainhow,byconstructinganappropriateHopfalgebrastructureonrooted
trees, onegetsanalgebraicformulationoftherenormalizationprocessforFeynman
graphs. After an introductory overview of those Hopf algebras, they give numerous
examples of their generalizations. In the second part, they recall the definition of
theHochschildcohomologyassociatedtosuchHopfalgebras. Aphysicalinterpreta-
tionofthecocyclesleadstotheDyson–Schwingerequations,whichdescribetheloop
expansionofGreenfunctioninarecursiveway. Thismethodisastraightforwardalter-
nativetotheusualone. ThepaperendsupwithadescriptionoftheDyson–Schwinger
equationfromdifferentpointsofview: itprovidestranscendentalnumbers,itisatool
todefineageneratingfunctionforthepolylogarithmanditistheequationofmotion
forarenormalizablequantumfieldtheory.
“Fonctionζ etmatricesaléatoires”byEmmanuelRoyer,isanextensivesurveyon
thezeroesofL-functions. TheanalyticpropertiesoftheRiemannζ-functiondescribe
the behaviour of prime numbers. More generally, one can attach an L-function to
certainarithmeticorgeometricobjects,andthusobtainsomeinformationonthemby
analyticmethods. ThefunctionsdescribedinthispaperaretheRiemannζ-function
and L-functions attached to Dirichlet characters, to automorphic representations, or
tomodularforms. TheGeneralisedRiemannHypothesispredictsthatallthezeroes
ofthoseL-functionslieonthelineRe(z)= 1,andotherimportantconjecturesrelate
2
theirstatisticalbehaviouralongthislinetothestatisticalbehaviourofthespectrumof
largeunitarymatrices.
The paper by Philippe Michel, “Some recent applications of Kloostermania”, is
devotedtoKloostermansumswhichareaspecialkindofalgebraicexponentialsums.
Theywherefirstdevelopedtoboundthenumberofrepresentationsofalargeintegern
byadiagonalquaternarydefinitequadraticform
n=ax2+bx2+cx2+dx2.
1 2 3 4
Theyeventuallyturnedouttobeoneofthosefascinatingobjectswithtwofaces: the
Petersson–Kuznetsov trace formula relates sums of Kloostermans sums, data of an
arithmetico-geometric nature, to the Fourier coefficients of modular forms, data of
a spectral nature. First written by Kuznetsov for SL (Z) the formula was extended
2
by Deshouillers and Iwaniec to arbitrary congruence subgroups. The use of this
connectioninbothdirectionsisapowerfultoolinanalyticnumbertheory. Thereare
manyapplications. Forexample, onecangetgoodestimatesoflinearcombinations
ofFouriercoefficents,orgetanupperboundforthedimensionofthespaceofweight
one modular forms, or solve many instances of the subconvexity problem. The last
applicationpresentedinthepaperisthat,usingthePetersson–Kuznetzovtraceformula,
vii
onecanrefinetheerrorterminWeyl’slawandthattheserefinementscanbeinterpreted
intermsofQuantumChaos.
Finally,ArianeMézard’s“IntroductionàlacorrespondancedeLanglandslocale”,
is a survey of the local Langlands correspondence. When K is a non archimedian
localfield,andWK itsWeilgroup,thelocalclassfieldisomorphism
Wab (cid:2)K∗
K
can be interpreted as a bijection between continuous irreducible representations of
K∗ = GL1(K) and one dimensional continuous representation of WK. The Lang-
lands conjecture is a generalisation of this isomorphism in higher dimension: for
any n ≥ 1 there should be an isomorphism between isomorphism classes of contin-
uous irreducible admissible representations of GLn(K) and isomorphism classes of
n-dimensional(cid:4)-semi-simplecontinuousrepresentationsoftheWeil–Delignegroup
W(cid:5) . Inherpaper,ArianeMézardstudiesthebijectionbetweenisomorphismclassesof
K
irreducibleadmissiblerepresentationsofGL2(Qp)overQ(cid:5),andisomorphismclasses
ofsomerepresentationsofWQp inGL2(Q(cid:5)). Sheexplainsthemainstepsoftheproof
inthe(cid:5)-adiccase(thismeans(cid:5)(cid:6)=p,inwhichcasetheconjectureisprovedforanyn).
She also points out where and why problems arise in the p-adic case ((cid:5) = p). She
describesthenewobjectsandexplainsthestrategyforn=2. Thep-adicconjecture
isstillopen.
Iwouldliketotaketheopportunitytothanktherefereesfortheirselflessworkand
numerousconstructiveremarks,MarcusSlupinskiforhelpwithEnglish,andtoexpress
specialgratitudetoVladimirTuraevwhoseconstanthelpduringthepreparationofthe
volumewasagreatsupport.
Strasbourg,August2006 LouiseNyssen

Table of Contents
Preface.....................................................................v
MichelPlanat
Thephaseofoscillationsandprimenumbers: classicalandquantum.............1
Jean-LouisVerger-Gaugry
Onself-similarfinitelygenerateduniformlydiscrete(SFU-)sets
andspherepackings........................................................39
Jean-PierreGazeau,ZuzanaMasáková,andEditaPelantová
Nestedquasicrystallinediscretisationsoftheline .............................79
ChristophBergbauerandDirkKreimer
Hopfalgebrasinrenormalizationtheory: localityandDyson–Schwinger
equationsfromHochschildcohomology.....................................133
EmmanuelRoyer
Fonctionζ etmatricesaléatoires ...........................................165
PhilippeMichel
SomerecentapplicationsofKloostermania..................................225
ArianeMézard
IntroductionàlacorrespondancedeLanglandslocale ........................ 253