Table Of ContentLecture Notes in Mathematics 2051
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E. stands for Centro Internazionale Matematico Estivo, that is, International
MathematicalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,
Italy,andwelcomedbytheworldmathematical community:itcontinues successfully,year
foryear,tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
50139Florence 50134Florence
Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’Universita´edellaRicerca)
ThiscourseispartiallysupportedbyGDR-GDREonCONTROLEDESEQUATIONSAUX
DERIVEESPARTIELLES(CONEDP).
Vincent Rivasseau Robert Seiringer
(cid:2)
Jan Philip Solovej Thomas Spencer
(cid:2)
Quantum Many Body Systems
Cetraro, Italy 2010
Editors:
Alessandro Giuliani
Vieri Mastropietro
Jakob Yngvason
123
VincentRivasseau JanPhilipSolovej
Universite´Paris-Sud DepartmentofMathematics
LaboratoiredePhysiqueThe´orique UniversityofCopenhagen
Orsay,France Copenhagen,Denmark
RobertSeiringer ThomasSpencer
McGillUniversity InstituteforAdvancedStudy
DepartmentofMathematicsandStatistics SchoolofMathematics
Montreal,QC,Canada Princeton,NJ,USA
ISBN978-3-642-29510-2 ISBN978-3-642-29511-9(eBook)
DOI10.1007/978-3-642-29511-9
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012941621
MathematicsSubjectClassification(2010):82B10,81V70,82B28,82B44
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Preface
The idea that matter is composed by a huge number of particles (atoms) obeying
simple laws of motion and that the macroscopic properties of bodies emerge as
collectivephenomenastartingfromthissimpledescriptiondatesbacktoDemocritus
(460–370BC). Since then, physicists of all times have struggled with this fasci-
nating idea in order to really understand the variety of our world starting from a
simplemicroscopicdescription.Quantummany-bodytheoryisnowadaysthearea
of physics that provides the framework for understanding the collective behavior
ofvastassembliesofinteractingquantumparticles.Indeed,weknowthatmatteris
composedofatomsandmolecules,interactinginaccordwiththelawsofquantum
mechanics.Theequationdescribingtheevolutionandbehaviorofsuchassemblies,
the Schro¨dinger equation with Coulomb forces, is well known: it can be written
downandstudied,andthe propertiesofthesystem caninprinciplebe understood
ataqualitativeandquantitativelevel,startingfromthisfundamentallaw.However,
the number of variables required for describing the behavior of the microscopic
constituentsofanymacroscopicbodyisenormous(itisoftheorderofthenumber
of particles, which is (cid:3)1023 fora mole of substance).Therefore,exceptfor a few
specialbutveryimportantcases(e.g.,idealgases),thedeductionofthemacroscopic
propertiesfromthefundamentalmicroscopicequationisaformidablemathematical
challenge.Thisis,afterall,notsurprising:weknowfromexperimentsthatcomplex
andunexpectedphenomenalikesuperconductivity,superfluidity,andBose–Einstein
condensation,together with more familiar phenomenalike magnetism or metallic
behavior, emerge from the collective behavior of a huge number of quantum
particles.Theseremarkablephenomenaareamacroscopicmanifestationofthelaw
ofquantummechanicsandcannotbeunderstoodbyusingNewtonianphysics.
Fromthepointofviewofmathematicalphysicists,many-bodytheoryprovides
analmostidealfield;thebasicequationisknownandone“just”needstosolveitin
ordertodeduceandunderstandanumberofinterestingphenomena.Indeed,ifthe
solutionsoftheSchro¨dingerequationwithalargenumberofvariableswereknown
thiswouldclearlybeofextraordinaryimportanceforphysicalapplicationsaswell
asfortechnologicaldevelopments.However,themathematicaldifficultyoffinding
suchsolutionsisenormous,butinpastyearsanumberofincreasinglypowerfuland
v
vi Preface
sophisticatedtechniqueshavebeendevelopedtoextractrelevantinformationwith-
outhavingtosolvetheequationitself.Thesemethodsincludemultiscaleanalysis,
functional inequalities, localization estimates, cluster expansions, supersymmetry
andstochasticandconformalgeometry.Thebeautifulmathematicsrelatedtothese
developmentshasalsoprovedtobeusefulasabridgebetweendifferentdisciplines,
ranging from algebraic geometry to the study of the shape of bird flocks. The
interactionofmathematicalandphysicalcomplexityhasprovedtobeveryfruitful
forbothfields.
The aim of the CIME school on Quantum Many-Body Systems, which took
place in Cetraro (Italy) from August 30 to September 4, 2010, was to provide
an introduction to the beautiful and powerful mathematical techniques developed
in this field. The school was attended by 30 participants from several different
countries, including Austria, Denmark, France, Germany, Italy, the UK, Ukraine
andtheUSA.Althoughtheschoolwasprimarilyintendedforgraduatestudents,the
interestingtopicsandthehighreputationofthelecturersalsoattractedseveralmore
seniorresearchers.
The school consisted of four series of lectures, presented by V. Rivasseau,
R. Seiringer,J.P. Solovej,and T.Spencer,andeach serieswas organizedintofour
lessonsof2heach(twointhemorningandoneintheafternoon).Inaddition,one
afternoonwasdevotedtoshortpresentationsoftheresearchactivityoftheyounger
participantsandone eveningto a shortdescriptionof the activitiesof someof the
seniorparticipants.
The lectures of Prof. Rivasseau gave an introduction to some results in solid-
statephysics,obtainedviaconstructiveRenormalizationGroupmethods.Whilethe
focuswasontheproofofFermiliquidbehaviorforasystemofnonrelativistictwo-
dimensionalfermionsabovetheBCStransitiontemperatureonelecturewasdevoted
to the exciting perspectives opened by the use of the same methods to quantum
gravity.
ThelecturesofProf.SeiringerweredevotedtothemathematicalphysicsofBose
gases and Bose–Einstein condensation and included a rigorous proof of the latter
fordilute,interactinggasesintheso-calledGross–Pitaevskilimit.Startingfromthe
basic notionshe also discussed advancedtopics like the analysis of rotating traps
andtheemergenceoflatticesofquantizedvortices.
Prof. Solovej provided a comprehensive introduction to quantum Coulomb
systems. He gave a self-containedpresentationof the functionalanalytic methods
used to prove thermodynamic stability of coulombic matter, following a recently
developedapproachthatallows to treat, on the same footing,translation andnon-
translationinvariantsystemsofchargedfermionsandbosons.
Finally, Prof. Spencer described the rigorous and powerful methods of super-
symmetry and their application to the problem of the localization–delocalization
transition in the Anderson model and in random matrices. Moreover, he gave a
tutorial review of some classical results and techniques, such as the use of Ward
IdentitiesintheXYmodel.
The atmosphere at the school was very lively, many questions and comments
arose during and after each lecture, and scientific discussions took place; the
Preface vii
students profited very much from the possibility of close interactions with the
lecturers.
As Editors of these Lectures Notes we would like to thank the people and
institutions who contributed to the success of the course. In particular, it is our
pleasure to thank the Scientific Committee of CIME for supporting our project;
the Director,Prof. Pietro Zecca, and the Secretary,Prof. Elvira Mascolo, for their
supportduring the organization.Special thanks go to the lecturers, who offered a
uniqueoccasiontotheparticipantstoenterthisbeautifulfield.
Roma,Italy AlessandroGiuliani
Roma,Italy VieriMastropietro
Vienna,Austria JakobYngvason
(cid:127)
Acknowledgements
We gratefully acknowledge financial support from the ERC Starting Grant
CoMBoS-239694, from the CIME foundation and from the International Associ-
ationinMathematicalPhysics.
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