Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation

752 Pages·2013·4.758 MB·English

by  Valter Moretti

Dedicatedtothoseyoungandbrilliantcolleagues,mathematicians
andphysicists,forcedtofleeItalyforothercountriesinorder
togivetheircontribution,bigorsmall,toscientificresearch.
UNITEXT – La Matematica per il 3+2
Volume 64
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Valter Moretti
Spectral Theory and
Quantum Mechanics
With an Introduction to the Algebraic
Formulation
ValterMoretti
DepartmentofMathematics
UniversityofTrento
Translatedby:SimonG.Chiossi,DepartmentofMathematics,PolitecnicodiTorino
Translated and extended version of the original Italian edition: V. Moretti, Teoria
SpettraleeMeccanicaQuantistica,©Springer-VerlagItalia2010
UNITEXT–LaMatematicaperil3+2
ISSN2038-5722 ISSN2038-5757(electronic)
ISBN978-88-470-2834-0 ISBN978-88-470-2835-7(eBook)
DOI10.1007/978-88-470-2835-7
LibraryofCongressControlNumber:2012945983
SpringerMilanHeidelbergNewYorkDordrechtLondon
©Springer-VerlagItalia2013
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Preface
Imusthavebeen8or9whenmyfather,amanoflettersbutwell-readineverydis-
ciplineandwithacuriousmind,toldmethisstory:“AgreatscientistnamedAlbert
Einsteindiscoveredthatanyobjectwithamasscan’ttravelfasterthanthespeedof
light”.TomybewildermentIreplied,boldly:“Thiscan’tbetrue,ifIrunalmostat
thatspeedandthenacceleratealittle,surelyIwillrunfasterthanlight,right?”.My
father wasadamant: “No,it’simpossible todowhatyousay, it’saknownphysics
fact”.AfterawhileIadded:“Thatbloke,Einstein,must’vecheckedthisthingmany
times...howdoyousay,hedidmanyexperiments?”.TheanswerIgotwasutterly
unexpected:“No,notevenoneIthink,heusedmaths!”.
What did numbers and geometrical figures have to do with the existence of a
limitspeed?Howcouldonestandbehindsuchanapparentlynonsensicalstatement
astheexistence ofamaximumspeed,althoughcertainlytrue(Itrustedmyfather),
just based on maths? How could mathematics have such big a control on the real
world?Andphysics?Whatonearthwasit,andwhatdidithavetodowithmaths?
ThiswasoneofthemostbeguilingandirresistiblethingsIhadeverheardtillthat
moment...Ihadtofindoutmoreaboutit.
Thisisanextendedandenhanced versionofanexistingtextbookwritteninItalian
(andpublishedbySpringer-Verlag).Thateditionandthisonearebasedonacommon
partthatoriginated,inpreliminaryform,whenIwasaPhysicsundergraduateatthe
University of Genova. The third-year compulsory lecture course called Institutions
ofTheoreticalPhysicswasthesecondexamthathaduspupilsseriouslyclimbingthe
walls(thefirstbeingthefamousPhysicsII,coveringthermodynamicsandclassical
electrodynamics).
QuantumMechanics,taughtinthatcourse,elicitedanovelandinvolvedwayof
thinking,atruechallengeforcravingstudents:formonthswehesitantlyfalteredona
hazyanduncertainterrain,notunderstandingwhatwasreallykeyamongthenotions
weweretrying–struggling,Ishouldsay–tolearn,togetherwithacompletelynew
formalism:linearoperatorsonHilbertspaces.Atthattime,actually,wedidnotreal-
isewewereusingthismathematicaltheory,andformanymatesofminethematter
VI Preface
wouldhavebeen,rightlyperhaps,completelyfutile;Dirac’sbravectorswerewhat
theywere,andthat’sit!Theywerecertainlynotelementsinthetopologicaldualof
the Hilbert space. The notions of Hilbert space and dual topological space had no
right of abode in the mathematical toolbox of the majority of my fellows, even if
theywouldsooncomebackinthroughtthebackdoor,withthecourseMathematical
MethodsofPhysicstaughtbyprof.G.Cassinelli.Mathematics,andthemathematical
formalisation of physics, had always been my flagship to overcome the difficulties
thatstudyingphysicspresentedmewith,tothepointthateventually(afteraPh.D.in
theoreticalphysics)Iofficiallybecameamathematician.Armedwithamathsback-
ground–learntinanextracurricularcourseofstudythatIcultivatedovertheyears,
inparalleltoacademicphysics–andeagertobroadenmyknowledge,Itriedtoform-
aliseeverynotionImetinthatnewandrivetinglecturecourse.AtthesametimeI
was carrying along a similar project for the mathematical formalisation of General
Relativity,unawarethattheworkputintoQuantumMechanicswouldhavebeenin-
commensurablybigger.
The formulation of the spectral theorem as it is discussed in §8, 9 is the same
IlearntwhentakingtheTheoreticalPhysicsexam,whichforthisreasonwasadia-
logueofthedeaf.Latermyinterestturnedtoquantumfieldtheory,atopicIstillwork
ontoday,thoughintheslightlymoregeneralframeworkofquantumfieldtheoryin
curvedspacetime.Notwithstanding,myfascinationwiththeelementaryformulation
ofQuantumMechanicsneverfadedovertheyears,andtimeandagainchunkswere
addedtotheopusIbegunwritingasastudent.
Teaching Master’s and doctoral students in mathematics and physics this ma-
terial,therebyinflictingonthemtheresultofmyeffortstosimplifythematter,has
provedtobecrucialforimprovingthetext;itforcedmetotypesetinLATEXthepile
ofloosenotesandcorrectseveralsections,incorporatingmanypeople’sremarks.
ConcerningthisIwouldliketothankmycolleagues,thefriendsfromthenews-
groupsit.scienza.fisica,it.scienza.matematicaandfree.it.scienza.fisica,andthemany
students–someofwhicharenowfellowsofmine–whocontributedtoimprovethe
preparatorymaterialofthetreatise,whetherdirectlyofnot,inthecourseoftime:S.
Albeverio,P.Armani,G.Bramanti,S.Bonaccorsi,A.Cassa,B.Cocciaro,G.Collini,
M.DallaBrida,S.Doplicher,L.DiPersio,E.Fabri,C.Fontanari,A.Franceschetti,
R.Ghiloni,A.Giacomini,V.Marini,S.Mazzucchi,E.Pagani,E.Pelizzari,G.Tes-
saro,M.Toller,L.Tubaro,D.Pastorello,A.Pugliese,F.SerraCassano,G.Ziglio,
S.Zerbini.Iamindebted,forvariousreasonsalsounrelatedtothebook,tomylate
colleague Alberto Tognoli. My greatest appreciation goes to R. Aramini, D. Cada-
muroandC.Dappiaggi,whoreadvariousversionsofthemanuscriptandpointedout
anumberofmistakes.
I am grateful to my friends and collaborators R. Brunetti, C. Dappiaggi and N.
Pinamontiforlastingtechnicaldiscussions,suggestionsonmanytopicscoveredand
forpointingoutprimaryreferences.
LastlyIwouldliketothankE.Gregoriofortheinvaluableandon-the-spottech-
nicalhelpwiththeLATEXpackage.
Preface VII
InthetransitionfromtheoriginalItaliantotheexpandedEnglishversionamas-
sive number of (uncountably many!) typos and errors of various kind have been
amended. I owe to E. Annigoni, M. Caffini, G. Collini, R. Ghiloni, A. Iacopetti,
M. Oppio and D. Pastorello in this respect. Fresh material was added, both math-
ematical and physical, including a chapter, at the end, on the so-called algebraic
formulation.
In particular, Chapter 4 contains the proof of Mercer’s theorem for positive
Hilbert–Schmidtoperators.Thenow-deeperstudyofthefirsttwoaxiomsofQuantum
Mechanics,inChapter7,comprisesthealgebraiccharacterisationofquantumstates
∗
intermsofpositivefunctionalswithunitnormontheC -algebraofcompactoperat-
∗ ∗
ors.GeneralpropertiesofC -algebrasand -morphismsareintroducedinChapter8.
Asaconsequence,thestatementsofthespectraltheoremandseveralresultsonfunc-
tional calculus underwent a minor but necessary reshaping in Chapters 8 and 9.
I incorporated in Chapter 10 (Chapter 9 in the first edition) a brief discussion on
abstract differential equations in Hilbert spaces. An important example concerning
Bargmann’s theorem was added in Chapter 12 (formerly Chapter 11). In the same
chapter,afterintroducingtheHaarmeasure,thePeter–Weyltheoremonunitaryrep-
resentationsofcompactgroupsisstated,andpartiallyproved.Thisisthenappliedto
thetheoryoftheangularmomentum.Ialsothoroughlyexaminedthesuperselection
rulefortheangularmomentum.ThediscussiononPOVMsinChapter13(formerly
Chapter12)isenrichedwithfurthermaterial,andIincludedaprimeronthefunda-
mentalideasofnon-relativisticscatteringtheory.Bell’sinequalities(Wigner’sver-
sion)aregivenconsiderablymorespace.Attheendofthefirstchapterbasicpoint-set
topologyisrecalledtogetherwithabstractmeasuretheory.Theoverallefforthasbeen
tocreateatextasself-containedaspossible.Iamawarethatthematerialpresented
has clear limitations and gaps. Ironically – my own research activity is devoted to
relativistic theories – the entire treatise unfolds at a non-relativistic level, and the
quantumapproachtoPoincaré’ssymmetryisleftbehind.
I thank my colleagues F. Serra Cassano, R. Ghiloni, G. Greco, A. Perotti and
L. Vanzo for useful technical conversations on this second version. For the same
reason,andalsofortranslatingthiselaborateopustoEnglish,Iwouldliketothank
mycolleagueS.G.Chiossi.
Trento,September2012 ValterMoretti
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Contents
1 Introductionandmathematicalbackgrounds...................... 1
1.1 Onthebook ............................................... 1
1.1.1 Scopeandstructure .................................. 1
1.1.2 Prerequisites ........................................ 4
1.1.3 Generalconventions.................................. 4
1.2 OnQuantumMechanics..................................... 5
1.2.1 QuantumMechanicsasamathematicaltheory............ 5
1.2.2 QMinthepanoramaofcontemporaryPhysics............ 7
1.3 Backgroundsongeneraltopology............................. 10
1.3.1 Open/closedsetsandbasicpoint-settopology ............ 10
1.3.2 Convergenceandcontinuity ........................... 12
1.3.3 Compactness........................................ 14
1.3.4 Connectedness ...................................... 15
1.4 Round-uponmeasuretheory................................. 16
1.4.1 Measurespaces...................................... 16
1.4.2 Positiveσ-additivemeasures .......................... 19
1.4.3 Integrationofmeasurablefunctions..................... 22
1.4.4 Riesz’stheoremforpositiveBorelmeasures ............. 25
1.4.5 Differentiatingmeasures .............................. 27
1.4.6 Lebesgue’smeasureonRn ............................ 27
1.4.7 Theproductmeasure ................................. 31
1.4.8 Complex(andsigned)measures........................ 32
1.4.9 Exchangingderivativesandintegrals.................... 33
2 NormedandBanachspaces,examplesandapplications ............ 35
2.1 NormedandBanachspacesandalgebras....................... 36
2.1.1 Normedspacesandessentialtopologicalproperties........ 36
2.1.2 Banachspaces....................................... 40
2.1.3 Example:theBanachspaceC(K;Kn),thetheoremsofDini
andArzelà–Ascoli ................................... 42