Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics PDF
Preview Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics
Progress in Mathematical Physics Volume 69 Editors-in-chief AnneBoutetdeMonvel,UniversitéParisVIIUFRdeMathematiques, ParisCX05,France GeraldKaiser,CenterforSignalsandWaves,Portland,Oregon,USA EditorialBoard C.Berenstein,UniversityofMaryland,CollegePark,USA SirM.Berry,UniversityofBristol,UK P.Blanchard,UniversityofBielefeld,Germany M.Eastwood,UniversityofAdelaide,Australia A.S.Fokas,UniversityofCambridge,UK F.W.Hehl,UniversityofCologne,Germany andUniversityofMissouri-Columbia,USA D.Sternheimer,UniversitédeBourgogne,Dijon,France C.Tracy,UniversityofCalifornia,Davis,USA Forfurthervolumes: http://www.springer.com/series/4813 Philippe Blanchard (cid:129) Erwin Brüning Mathematical Methods in Physics Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics Second Edition PhilippeBlanchard ErwinBrüning Abt.TheoretischePhysik SchoolofMathematics,Statistics, UniversitätBielefeldFak.Physik andComputerScience Bielefeld UniversityofKwaZulu-Natal Germany Durban SouthAfrica ISSN1544-9998 ISSN2197-1846(electronic) ProgressinMathematicalPhysics ISBN978-3-319-14044-5 ISBN978-3-319-14045-2(eBook) DOI10.1007/978-3-319-14045-2 LibraryofCongressControlNumber:2015931210 MathematicsSubjectClassification(MSC):46-01,46C05,46F05,46N50,47A05,47L90,49-01,60E05, 81Q10 SpringerChamHeidelbergNewYorkDordrechtLondon © SpringerInternationalPublishingSwitzerland2003,2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrors oromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Dedicatedtothememoryof YurkoVladimirGlaserandResJost, mentorsandfriends Preface to the Second Edition The first edition of this book was published in 2003. Let us first thank everyone who, over the past 11 years, has provided us with suggestions and corrections for improvingthisfirstedition.Weareextremelygratefulforthishelp. Thepastdecadehasbroughtmanychanges,buttheaimofthisbookremainsthe same.Itisintendedforgraduatestudentsinphysicsandmathematicsanditmayalso beusefulfortheoreticalphysicistsinresearchandinindustry.Astheextendedtitle ofthissecondeditionindicates, wehavefocusedourattentiontoalargeextenton topicalapplicationstoQuantumPhysics. Withthehopethatthisbookwouldbeausefulreferenceforpeopleapplyingmath- ematicsintheirwork,wehaveemphasizedtheresultsthatareimportantforvarious applications in the areas indicated above. This book is essentially self-contained. Perhapssomereaderswillusethisbookasacompendiumofresults;thiswouldbea pity,however,becauseproofsareoftenasimportantasresults.Mathematicalphysics isnotapassiveactivity,andthereforethebookcontainsmorethan220exercisesto challengereadersandtofacilitatetheirunderstanding. This second edition differs from the first through the reorganization of certain materialandtheadditionoffivenewchapterswhichhaveanewrangeofsubstantial applications. ThefirstadditionisChap.13“Sobolevspaces”whichoffersabriefintroduction tothebasictheoryofthesespacesandthuspreparestheiruseinthestudyoflinear andnonlinearpartialdifferentialoperatorsand,inparticular,inthethirdpartofthis bookdedicatedto“VariationalMethods.” While in the first edition Hilbert–Schmidt and trace class operators were only discussed briefly in Chap. 22 “Special classes of bounded operators,” this edition containsanewChap.26“Hilbert–Schmidtandtraceclassoperators.”Theremainder ofChap.22hasbeenmergedwiththeold,shorterChap.23“Self-adjointHamilton operators”toformthenewChap.23“Specialclassesoflinearoperators,”whichnow alsocontainsabriefsectiononvonNeumann’sbeautifulapplicationofthespectral theoryforunitaryoperatorsinergodictheory. Chapter 26 “Hilbert–Schmidt and trace class operators” gives a fairly compre- hensiveintroductiontothetheoryoftheseoperators. Furthermore, thedualspaces ofthespacesofcompactandoftraceclassoperatorsaredetermined,whichallows vii viii PrefacetotheSecondEdition athoroughdiscussionofseverallocallyconvextopologiesonthespaceB(H)ofall bounded linear operators on a separable Hilbert space H. These are used later in thechapter“Operatoralgebrasandpositivemappings”tocharacterizenormalstates on von Neumann algebras. This chapter also contains the definition of the partial traceoftraceclassoperatorsonthetensorproductsoftwoseparableinfinitedimen- sionalHilbertspacesandstudiesitsmainproperties.Theseresultsareofparticular importanceinthetheoryofopenquantumsystemsandthetheoryofdecoherence. ThemotivationforthenewChap.29“SpectralanalysisinriggedHilbertspaces” comesfromthefactthat,ononeside,Dirac’sbraandketformalismiswidelyand successfullyemployedintheoreticalphysics,butitsmathematicalfoundationisnot easilyaccessible.Thischapteroffersanearlyself-containedmathematicalbasisfor thisformalismandprovesinparticularthecompleteness?thereisawordmissingin thissentencecompletenessofthesetofgeneralizedeigenfunctions. InChap.30“Operatoralgebrasandpositivemappings,”westudyindetailpos- itive and completely positive mappings on the algebra B(H) of all bounded linear operatorsonaHilbertspaceHrespectivelyonitssubalgebras.WeexplaintheGNS construction for positive linear functionals in detail and characterize states, i.e., normalizedpositivelinearfunctionals,intermsoftheirequivalentcontinuityproper- ties(normal,completelyadditive,tracial).Next,Stinespring’sfactorizationtheorem characterizescompletelypositivemapsintermsofrepresentations.Sinceallrepre- sentationsofB(H)aredeterminedtoo,wecangiveaself-containedcharacterization ofallcompletelypositivemappingsonB(H). Thelastnewchapter,Chap.31“Positivemappingsinquantumphysics,”presents severalresultswhichareveryimportantforthefoundationsofquantumphysicsand quantum information theory. We start with a detailed discussion of Gleason’s the- orem on the general form of countable additive probability measures on the set of projectionsofaseparableHilbertspace. Usingsomeoftheresultsoftheprevious chapter,wethengiveaself-containedcharacterizationofquantumoperationsspecif- ically,quantumchannelmaps(Krausform)andconcludewithabriefdiscussionof thestrongerformoftheseresultsiftheunderlyingHilbertspaceisfinitedimensional (Choi’scharacterizationofquantumoperations). Onthebasisofthemathematicalresultsobtainedinthisandearlierchapters, it isstraightforwardtointroducesomequiteprominentconceptsinquantumphysics, namelyopenquantumsystems,reduceddynamics,anddecoherence.Thisisdonein thelastsectionofthischapter. BielefeldandDurban Ph.Blanchard May2014 E.Brüning Preface Courses in modern theoretical physics have to assume some basic knowledge of the theory of generalized functions (in particular distributions) and of the theory of linear operators in Hilbert spaces. Accordingly, the faculty of physics of the UniversityofBielefeldofferedacompulsorycourseMathematischeMethodender Physikforstudentsinthesecondsemesterofthesecondyear,whichnowhasbeen given for many years. This course has been offered by the authors over a period of about 10 years. The main goal of this course is to provide basic mathematical knowledgeandskillsastheyareneededformoderncoursesinquantummechanics, relativistic quantum field theory, and related areas. The regular repetitions of the course allowed, on the one hand, testing of a number of variations of the material and, on the other hand, the form of the presentation. From this course, the book Distributionen und Hilbertraumoperatoren. Mathematische Methoden der Physik. Springer-VerlagWien,1993emerged.Thepresentbookisatranslated,considerably revised,andextendedversionofthisbook.Itcontainsmuchmorethanthiscourse sinceweaddedmanydetailedproofs,manyexamples,andexercisesaswellashints linking the mathematical concepts or results to the relevant physical concepts or theories. This book addresses students of physics who are interested in a conceptually and mathematically clear and precise understanding of physical problems, and it addressesstudentsofmathematicswhowanttolearnaboutphysicsasasourceand asanareaofapplicationofmathematicaltheories,i.e.,allthosestudentswithinterest inthefascinatinginteractionbetweenphysicsandmathematics. Itisassumedthatthereaderhasasolidbackgroundinanalysisandlinearalgebra (inBielefeldthismeansthreesemestersofanalysisandtwooflinearalgebra). On this basis the book starts in PartA with an introduction to basic linear functional analysisasneededfortheSchwartztheoryofdistributionsandcontinuesinPartB withtheparticularitiesofHilbertspacesandthecoreaspectsofthetheoryoflinear operatorsinHilbertspaces.PartCdevelopsthebasicmathematicalfoundationsfor modern computations of the ground state energies and charge densities in atoms andmolecules,i.e.,basicaspectsofthedirectmethodsofthecalculusofvariations includingconstrainedminimization.Apowerfulstrategyforsolvinglinearandnon- linearboundaryandeigenvalueproblems, whichcoverstheDirichletproblemand ix x Preface itsnonlineargeneralizations,ispresentedaswell.Anappendixgivesdetailedproofs ofthefundamentalprinciplesandresultsoffunctionalanalysistotheextenttheyare neededinourcontext. Withgreatpleasurewewouldliketothankallthosecolleaguesandfriendswho have contributed to this book through their advice and comments, in particular G. Bolz, J. Loviscach, G. Roepstorff, and J. Stubbe. Last but not least we thank the editorialteamofBirkhäuser—Bostonfortheirprofessionalwork. BielefeldandDurban Ph.Blanchard June2002 E.Brüning
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