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  • The unreasonable effectiveness of quantum theory: Logical inference

    approach

    Hans De Raedt1, Mikhail Katsnelson2,

    Hylke Donker2 and Kristel Michielsen3

    1 32

  • TwowaysofthinkingI. Reductionism(microscopicapproach)Everythingisfromwater/fire/earth/gauge

    fields/quantumspacetimefoam/strings...andtherestisyourproblem

    II. Phenomenology:operatingwithblackboxes

  • TwowaysofthinkingIIKnowledgebegins,sotospeak,inthemiddle,andleadsintothe

    unknown bothwhenmovingupward,andwhenthereisadownwardmovement.Ourgoalistograduallydissipatethe

    darknessinbothdirections,andtheabsolutefoundation thishugeelephantcarryingonhismightybackthetoweroftruth it

    existsonlyinafairytales(HermannWeyl)

    Weneverknowthefoundations!Howcanwehaveareliableknowledgewithoutthe

    base?

  • Isfundamentalphysicsfundamental?Classical thermodynamics is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability

    of its basic concepts (A. Einstein)

    The laws describing our level of reality are essentially independent on the background laws. I wish our colleagues from true theory (strings, quantum gravity, etc....) all kind of success

    but either they will modify electrodynamics and quantum mechanics at atomic scale (and then they will be wrong) or they

    will not (and then I do not care). Our way is down

    But how can we be sure that we are right?!

  • Unreasonableeffectiveness Quantumtheorydescribesavastnumberofdifferentexperimentsverywell

    WHY?

    NielsBohr*:Itiswrongtothinkthatthetaskofphysicsistofindouthownatureis.Physicsconcernswhatwecansay aboutnature.

    *A. Petersen, The philosophy of Niels Bohr, Bulletin of the Atomic Scientists 19, 8 14 (1963).

  • Mainmessageofthistalk

    Logicalinferenceappliedtoexperimentsforwhich1. Thereisuncertaintyabouteachindividualevent2. Thefrequenciesofobservedeventsarerobustwith

    respecttosmallchangesintheconditionsBasicequationsofquantumtheory

    Notaninterpretationofquantumtheory

    Derivationbasedonelementaryprinciplesofhumanreasoningandperception

  • SternGerlach experiment Neutralatoms(orneutrons)

    passthroughaninhomogeneousmagnetic

    field

    Inferencefromthedata:directionalquantization

    Idealization

    SourceS emitsparticleswithmagneticmoment

    MagnetM sendsparticletooneoftwodetectors

    Detectorscounteveryparticle

  • IdealizedSternGerlach experiment

    Event=clickofdetectorD+ or(exclusive)D

    Thereisuncertaintyabouteachevent Wedonotknowhowtopredictaneventwithcertainty

  • Somereasonableassumptions(1)

    ForfixedaandfixedsourceS,thefrequenciesof+and eventsarereproducible

    IfwerotatethesourceS andthemagnetM bythesameamount,thesefrequenciesdonotchange

  • Somereasonableassumptions(2)

    Thesefrequenciesarerobustwithrespecttosmallchangesina

    Basedonallotherevents,itisimpossibletosaywithsomecertaintywhattheparticulareventwillbe(logicalindependence)

  • Logicalinference

    Shorthandforpropositions: x=+1 D+clicks x=1 D clicks MthevalueofMisM athevalueofaisa Zeverythingelsewhichisknowntoberelevanttotheexperimentbutisconsideredtofixed

    WeassignarealnumberP(x|M,a,Z)between0and1toexpressourexpectationthatdetectorD+ or(exclusive)Dwillclickandwanttoderive,notpostulate,P(x|M,a,Z)fromgeneralprinciplesofrationalreasoning

    Whatarethesegeneralprinciples?

  • Plausible,rationalreasoninginductivelogic,logicalinference

    G.Plya,R.T.Cox,E.T.Jaynes, Fromgeneralconsiderationsaboutrationalreasoningitfollows

    thattheplausibilitythatapropositionA (B)istruegiventhatpropositionZ istruemaybeencodedinrealnumberswhichsatisfy

    ExtensionofBooleanlogic,applicabletosituationsinwhichthereisuncertaintyaboutsomebutnotallaspects

    Kolmogorovsprobabilitytheoryisanexamplewhichcomplieswiththerulesofrationalreasoning

    Isquantumtheoryanotherexample?

    0 ( | ) 1( | ) ( | ) 1 ; NOT ( | ) ( | ) ( | ) ; AND

    P A ZP A Z P A Z A AP AB Z P A BZ P B Z AB A B

  • Plausible,rationalreasoninglogicalinference

    Plausibility Isanintermediatementalconstructtocarryoutinductivelogic,rationalreasoning,logicalinference

    Mayexpressadegreeofbelieve(subjective) Maybeusedtodescribephenomenaindependentofindividualsubjectivejudgmentplausibility iprob (inferenceprobability)

  • ApplicationtotheSternGerlach experiment

    WerepeattheexperimentN times.ThenumberoftimesthatD+(D) clicksisn+(n)iprob fortheindividualeventis

    Dependenton RotationalinvarianceDifferenteventsarelogicallyindependent:

    Theiprob toobserven+andn eventsis

  • Howtoexpressrobustness?

    HypothesisH0:given we observen+andn HypothesisH1:given+ weobserven+andn TheevidenceEv(H1/H0)isgivenby

    Frequenciesshouldberobustwithrespecttosmallchangesin weshouldminimize,inabsolutevalue, thecoefficientsof, 2,

  • Removedependenceon (1)

    Choose

    Removesthe1st and3rd termRecovertheintuitiveprocedureofassigningtotheiprob oftheindividualevent,thefrequencywhichmaximizestheiprob toobservethewholedataset

    ( | , ) xnP x ZN

  • Removedependenceon (2)

    Minimizingthe2nd term(Fisherinformation)forallpossible(small) and

    InagreementwithquantumtheoryoftheidealizedSternGerlach experiment

    2

    1

    ( | , )( | ,

    1)F x

    P x ZP x Z

    I

  • Derivationofbasicresultsofquantumtheorybylogicalinference

    Genericapproach1. Listthefeaturesoftheexperimentthataredeemed

    toberelevant2. Introducetheiprob ofindividualevents3. Imposeconditionofrobustness4. Minimizefunctional equationofquantumtheory

    whenappliedtoexperimentsinwhichi. Thereisuncertaintyabouteacheventii. Theconditionsareuncertainiii. Frequencieswithwhicheventsareobservedare

    reproducibleandrobust againstsmallchangesintheconditions

    Weneedtoaddsomedynamicalinformationonthesystem

  • Schrdingersfirstderivationofhisequation*

    StartfromtimeindependentHamiltonJacobiequation

    Writefortheaction toobtain

    Incomprehensiblestep:insteadofsolving(X),minimize

    tofind

    21 ( ) ( ) 02

    S x V x Em x

    222( ) [ ( ) ] ( ) 0

    2K x V x E xm x

    222( ) [ ( ) ] ( )

    2K xQ dx V x E xm x

    (X)

    2

    22

    ( ) ( ) ( ) 0xK V x E xx

    *Ann. Physik 384, 361 (1926) (Band 79 Folge 4 Seite 361)

  • Schrdingerssecondjustificationofhisequation*

    Callshisfirstderivationincomprehensibleandthengivesaphysicaljustificationofhisequationbyanalogywithoptics

    Fromtheviewpointofrationalreasoning,thefirstderivationisallbutincomprehensible

    *Ann.der.Physik 384, 489 (1926) (Band 79 Folge 4 Seite 489)

  • LogicalinferenceSchrdingerequation

    Genericprocedure: Experiment Thetrueposition oftheparticleisuncertainandremainsunknown

    iprob thattheparticleatunknownpositionactivatesthedetector

    atposition :

    Pulsed light source

    Particle moving on this line

    Photon emitted by particle

    Detector jx

    ( | , )P x Z

  • Robustness

    Assumethatitdoesnotmatterifwerepeattheexperimentsomewhereelse

    Conditionforrobustfrequencydistributionminimizethefunctional(Fisherinformation)

    withrespectto

    ( | , ) ( | , ) ; arbitraryP x Z P x Z

    21 ( | , )( )( | , )F

    P x ZI dxP x Z x

  • Imposeclassicalmechanics(laSchrdinger)

    Ifthereisnouncertaintyatallclassicalmechanics HamiltonJacobiequation

    Ifthereisknownuncertainty

    Reducesto(X)if

    21 ( ) ( ) 02

    S V Em

    2( ) 2 [ ( ) ] ( | , ) 0S xdx m V x E P x Zx

    (X)

    (XX)

  • Robustness+classicalmechanics

    canbefoundbyminimizingwiththeconstraintthat(XX)shouldholdWeshouldminimizethefunctional

    =Lagrangemultiplier Nonlinearequationsfor and

    2 21 ( | , ) ( )( ) 2 [ ( ) ] ( | , )( | , )

    P x Z S xF dx m V x E P x ZP x Z x x

  • Robustness+classicalmechanics

    Nonlinearequationsfor and anbeturnedintolinearequationsbysubstituting*

    Minimizingwithrespecttoyields

    Schrdingerequation

    ( ) /2( | , ) ( | , ) iS xx Z P x Z e

    ( | , ) ( | , )( ) 4 2 [ ( ) ] ( | , ) ( | , )x Z x ZF dx m V x E x Z x Zx x

    *E.Madelung,QuantentheorieinhydrodynamischerForm,Z.Phys.40,322 326(1927)

    ( | , )x Z

    2

    2

    ( | , ) ( ) ( | , ) 02

    x Z m V x E x Zx

  • Timedependent,multidimensionalcaseThespaceisfilledbydetectorswhicharefired(ornotfired)atsomediscrete(integer)time

    Attheveryendwehaveasetofdatapresentedas0(noparticleinagivenboxatagiveninstantor1

    Logicalindependenceofevents:

  • TimedependentcaseIIHomogeneityofthespace:

    Evidence:

  • TimedependentcaseIIIMinimizingFisherinformation:

    Takingintoaccounthomogeneityofspace;continuumlimit:

    Hamilton Jacobiequations:

  • TimedependentcaseIVMinimizingfunctional:

    Substitution

    Equivalentfunctionalforminimization:

  • TimedependentcaseVTimedependentSchrdingerequation

    Itislinear (superpositionprinciple)whichfollowsfromclassicalHamiltonian(kineticenergyismv2/2)and,inportantly,frombuildingonecomplexfunctionfromtworeal(SandS+2areequivalent).

    Averynontrivialoperationdictatedjustbydesiretosimplifytheproblemasmuchaspossible(topassfromnonlineartolinearequation).

    Requiresfurthercarefulthinking!

  • PauliequationWhatisspin?Justduality(e.g.,color blueorred).Nothingis

    rotating(yet!)

    IsospininnuclearphysicsSublatticeindexingraphene

    Justaddcolor(k=1,2)

  • PauliequationIIFisherinformationpartjustcopiesthepreviousderivation

    fortheevidence

    Expansion

  • PauliequationIII

    ha