Foundations of quantum theory : from classical concepts to commutative operator algebras

881 Pages·2017·5.08 MB·English

by  Landsman

Fundamental Theories of Physics 188
Klaas Landsman
Foundations
of Quantum
Theory
From Classical Concepts to Operator
Algebras
Fundamental Theories of Physics
Volume 188
Series editors
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Nicolaas P. Landsman, Nijmegen, The Netherlands
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Klaas Landsman
Foundations of Quantum
Theory
From Classical Concepts to Operator Algebras
KlaasLandsman
IMAPP
Radboud University
Nijmegen
TheNetherlands
ISSN 0168-1222 ISSN 2365-6425 (electronic)
FundamentalTheories of Physics
ISBN978-3-319-51776-6 ISBN978-3-319-51777-3 (eBook)
DOI 10.1007/978-3-319-51777-3
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ToJeremyButterfield
Preface
‘DerKopf,sogesehen,hatmitdemKopf,sogesehen,auchnichtdieleisesteA¨hnlichkeit
(...) Der Aspektwechsel. “Du wu¨rdest doch sagen, dass sich das Bild jetzt ga¨nzlich
gea¨ndert hat!” Aber was ist anders: mein Eindruck? meine Stellungnahme? (...) Ich
beschreibedieA¨nderungwieeineWahrnehmung,ganz,alsha¨ttesichderGegenstandvor
meinenAugengea¨ndert.’(Wittgenstein,PhilosophischeUntersuchungenII,§§127,129).1
Asthewell-knownpictureaboveismeanttoallegorize,somephysicalsystems
admitadualdescriptionineitherclassicalorquantum-mechanicalterms.According
to Bohr’s “doctrine of classical concepts”, measurement apparatuses are examples
ofsuchsystems.Moregenerally—ashammereddownbydecoherencetheorists—
theclassicalworldaroundusisacaseinpoint.Aswillbearguedinthisbook,the
measurementproblemofquantummechanics(highlightedbySchro¨dinger’sCat)is
causedbythisduality(ratherthanresolvedbyit,asBohrissaidtohavethought).
1‘Theheadseeninthiswayhasn’teventheslightestsimilaritytotheheadseeninthatway(...)
Thechangeofaspect.“Butsurelyyou’dsaythatthepicturehaschangedaltogethernow!Butwhat
isdifferent:myimpression?myattitude?(...)Idescribethechangelikeaperception;justasifthe
objecthaschangedbeforemyeyes.’Translation:G.E.M.Anscombe,P.M.S.Hacker,&J.Schulte
(Wittgenstein,2009/1953,pp.205–206).
vii
viii Preface
The aim of this book is to analyze the foundations of quantum theory from the
point of view of classical-quantum duality, using the mathematical formalism of
operatoralgebrasonHilbertspace(and,moregenerally,C*-algebras)thatwasorig-
inallycreatedbyvonNeumann(followedbyGelfandandNaimark).Insupportof
thisanalysis,butalsoasamatterofindependentinterest,thebookcoversmanyof
the traditional topics one might expect to find in a treatise on the foundations of
quantummechanics,likepureandmixedstates,observables,theBornruleandits
relationtobothsingle-caseprobabilitiesandlong-runfrequencies,Gleason’sTheo-
rem,thetheoryofsymmetry(includingWigner’sTheoremanditsrelatives,culmi-
natinginarecenttheoremofHamhalter’s),Bell’sTheorem(s)andthelike,quantiza-
tiontheory,indistinguishableparticle,largesystems,spontaneoussymmetrybreak-
ing, the measurement problem, and (intuitionistic) quantum logic. One also finds
a few idiosyncratic themes, such as the Kadison–Singer Conjecture, topos theory
(whichnaturallyinjectsintuitionismintoquantumlogic),andanunusualemphasis
onbothconceptualandmathematicalaspectsoflimitsinphysicaltheories.
AllofthisisheldtogetherbywhatwecallBohrification,i.e.,themathematical
interpretation of Bohr’s classical concepts by commutative C*-algebras, which in
turnarestudiedintheirquantumhabitatofnoncommutativeC*-algebras.
Thusthebookismostlywritteninmathematicalphysicsstyle,butitsrealsubject
isnaturalphilosophy.Henceitsintendedreadershipconsistsnotonlyofmathemati-
calphysicists,butalsoofphilosophersofphysics,aswellasoftheoreticalphysicists
whowishtodomorethan‘shutupandcalculate’,andfinallyofmathematicianswho
areinterestedinthemathematicalandconceptualstructureofquantumtheory.
Toserveallthesegroups,thenativemathematicallanguage(i.e.ofC*-algebras)
isintroducedslowly,startingwithfinitesets(asclassicalphasespaces)andfinite-
dimensionalHilbertspaces.Inaddition,alladvancedmathematicalbackgroundthat
is necessary but may distract from the main development is laid out in extensive
appendices on Hilbert spaces, functional analysis, operator algebras, lattices and
logic, and category theory and topos theory, so that the prerequisites for this book
are limited to basic analysis and linear algebra (as well as some physics). These
appendices not only provide a direct route to material that otherwise most readers
wouldhaveneededtoextractfromthousandsofpagesofdiversetextbooks,butthey
alsocontainsomeoriginalmaterial,andmaybeofinteresteventomathematicians.
Insummary,theaimsofthisbookaresimilartothoseofitspeerlessparadigm:
‘DerGegenstanddiesesBuchesistdieeinheitliche,und,soweitalsmo¨glichundangebracht,
mathematischeinwandfreieDarstellungderneuenQuantenmechanik(...).Dabeisolldas
HauptgewichtaufdieallgemeinenundprinzipiellenFragen,dieimZusammenhangemit
dieser Theorie entstanden sind, gelegt werden. Insbesondere sollen die schwierigen und
vielfachnochimmernichtrestlosgekla¨rtenInterpretationsfragenna¨heruntersuchtwerden.’
(vonNeumann,MathematischeGrundlagenderQuantenmechanik,1932,p.1).2
2‘Theobjectofthisbookistopresentthenewquantummechanicsinaunifiedpresentationwhich,
sofarasitispossibleanduseful,ismathematicallyrigorous.(...)Thereforetheprincipalemphasis
shallbeplacedonthegeneralandfundamentalquestionswhichhaveariseninconnectionwiththis
theory.Inparticular,thedifficultproblemswithinterpretation,manyofwhichareevennownot
fullyresolved,willbeinvestigatedindetail.’Translation:R.T.Beyer(vonNeumann,1955,p.vii).
Preface ix
Twootherquotationstheauthoroftenhadinmindwhilewritingthisbookare:
‘Andalthoughthewholeofphilosophyisnotimmediatelyevident,stillitisbettertoadd
somethingtoourknowledgedaybydaythantofillupmen’smindsinadvancewiththe
preconceptionsofhypotheses.’(Newton,draftprefacetoPrincipia,1686).3
‘JuisthetfeitdateengeniealsDESCARTESvolkomennaastdelijnvanontwikkelingisbli-
jvenstaan,dievanGALILEInaarNEWTONvoert(...)[is]eenphasevandenindehistorie
zoovaakherhaaldenstrijdtusschendebescheidenheiddermathematisch-physischemeth-
ode,dienanauwkeurigonderzoekdeverschijnselendernatuurinsteedsmeeromvattende
schematametbehulpvandeexactetaaldermathesiswilbeschrijvenendenhoogmoedvan
hetphilosophischedenken,datine´e´ngenialengreepdeheelewereldwilomvatten(...).’
(Dijksterhuis,ValenWorp,1924,p.343).4
Acknowledgements
1. Researchunderlyingthisbookhasbeengenerouslysupportedby:
• RadboudUniversityNijmegen,partlythroughasabbaticalin2014.
• The Netherlands Organization for Scientific Research (NWO), initially by
funding various projects eventually contributing to this book, and most re-
centlybypayingtheOpenAccessfee,makingthebookwidelyavailable.
• TheTempletonWorldCharityFoundation(TWCF),byfundingtheOxford–
Princeton–NijmegencollaborationExperimentalTestsofQuantumReality.
• Trinity College (Cambridge), by appointing the author as a Visiting Fellow
CommonerduringtheEasterTerm2016,whenthebookwaslargelyfinished.
2. Theauthorwasfortunateinhavingbeensurroundedbyoutstandingstudentsand
postdocs,whomadeessentialcontributionstotheinsightsdescribedinthisbook.
InalphabeticalorderthesewereChristianBudde,MartijnCaspers,RonnieHer-
mens, Jasper van Heugten, Chris Heunen, Bert Lindenhovius, Robin Reuvers,
BasSpitters,MarcoStevens,andSanderWolters.Thosewerethedays!
3. The author is indebted to Jeremy Butterfield, Peter Bongaarts, Harvey Brown,
DennisDieks,SiegfriedEchterhoff,AernoutvanEnter,JanHamhalter,Jaapvan
Oosten,andBasTerwijnforcommentsonthemanuscript.Inaddition,through
critical feedback on a Masterclass at Trinity, Owen Maroney and Fred Muller
indirectly(butconsiderably)improvedChapter11onthemeasurementproblem.
4. AngelaLaheefromSpringerthoughtfullyguidedthepublicationprocessofthis
bookfromthebeginningtotheend.ThanksalsotohercolleagueAldoRampioni.
Finally,itisapleasuretodedicatethisbooktoJeremyButterfield,inrecognitionof
hisideas,aswellasofhisunrelentingsupportandfriendshipoverthelast25years.
3Newton(1999),p.61.
4‘TheveryfactthatageniuslikeDescarteswascompletelysidelinedinthedevelopmentleading
fromGalileitoNewton(...)representsaphaseinthestruggle—thathassooftenbeenrepeated
throughout history—between the modesty of the approach of mathematical physics, which af-
terpreciseinvestigationsattemptstodescribenaturalphenomenainincreasinglycomprehensive
schemesusingtheexactlanguageofmathematics,andthehaughtinessofphilosophicalthought,
whichwantstocomprehendtheentireworldinonedazzlinggrasp.’Translationbytheauthor.
Contents
Introduction ....................................................... 1
PartI C (X)andB(H)
0
1 Classicalphysicsonafinitephasespace .......................... 23
1.1 Basicconstructionsofprobabilitytheory ....................... 24
1.2 Classicalobservablesandstates............................... 26
1.3 Purestatesandtransitionprobabilities ......................... 31
1.4 Thelogicofclassicalmechanics .............................. 34
1.5 TheGNS-constructionforC(X)............................... 36
Notes ......................................................... 38
2 Quantummechanicsonafinite-dimensionalHilbertspace .......... 39
2.1 QuantumprobabilitytheoryandtheBornrule................... 40
2.2 Quantumobservablesandstates .............................. 43
2.3 Purestatesinquantummechanics............................. 46
2.4 TheGNS-constructionformatrices ............................ 50
2.5 TheBornrulefromBohrification ............................. 54
2.6 TheKadison–SingerProblem ................................ 57
2.7 Gleason’sTheorem ......................................... 59
2.8 ProofofGleason’sTheorem ................................. 62
2.9 EffectsandBusch’sTheorem................................. 71
2.10 ThequantumlogicofBirkhoffandvonNeumann ............... 75
Notes ......................................................... 80
3 Classicalphysicsonageneralphasespace ........................ 83
3.1 Vectorfieldsandtheirflows.................................. 85
3.2 PoissonbracketsandHamiltonianvectorfields.................. 88
3.3 SymmetriesofPoissonmanifolds ............................. 90
3.4 Themomentummap........................................ 94
Notes .........................................................101
xi