α.ItssolutionscanbeimmediatelyobtainedfromitsinverseFouriertransform.IntermsofA(q,p)(theinverseFouriertransformofa(k,y))andofPα(q,p)eq.(5)showstobenolongeranintegralequationbutasimplealgebraicequation:A(q,p)Pα(q,p)=αPα(q,p)
whichhasthesolutions
(7)
α=A(q,p)(8)
Pα(q,p)=fα(Π(q,p))δ(A(q,p)−α)(9)
withf(Π)anarbitraryfunctionofthevariableΠconjugatedtoA.InfactanyotherdependenceoffαonqandpcouldbeexpressedasadependenceonA(q,p)whichwouldbeeliminatedbyreplacingA(q,p)withα.Thisarbi-trarinessreflectsthefactthattheremaybeaninfinityofclassicalensemblesinwhichthevariableAhasthevalueα.Eq.(7)impliesthat,givenacouple
4
ofvaluesq,pofthevariablesq,p,thevariableAhasnecessarilythevalue(8).Thisseemsatrivialstatementbutitwillturnouttobeessentiallater.Itshouldbenotedthat(9)holdsforanydynamicalvariable,functionofq,p.Thecaseoftheenergyisnoexception,inspiteofthequestionableroleofthetimeasitsconjugatevariable,sincetheHamiltonianH=H(q,p)canbeexpressed,bymeansofasuitablecanonicaltransformationQ=Q(q,p),P=P(q,p),asafunctionE(P)ofthenewmomentumnotdependingonthenewcoordinateQ.ThereforeagivenvaluePofPyieldsauniquelydeterminedvalueE(P)oftheenergy.WecantakethereforeΠ=Q.ForclosedsystemsPistheactionvariableJ=J(q,p),andE(J)isindependentoftheconjugatedanglevariableΘ=QofJ.
Thelimitingcasef=constantisthemostinterestingforthegeneralizationwehaveinmind,becauseinthisparticularclassicalensemblethevariableΠiscompletelyundetermined.
Inthiscaseandonlyinthiscasetheensembleacquiresaveryimportantproperty.Infact,byindicatingwith{.,.}PBthePoissonBracketofAwithanarbitraryvariableB,onehas
<{A,B}PB>α=−
=−
namely
dpdqPα(p,q)[(∂A/∂q)(∂B/∂p)−(∂A/∂p)(∂B/∂q)]=dpdqB(p,q)[(∂A/∂q)(∂Pα/∂p)−(∂A/∂p)(∂Pα/∂q)]
(10)
5
<{A,B}PB>α=0
becausewhenPαdependsonlyonAwehave(∂Pα/∂Π)=0.
(11)
Eq.(11)impliesthatbotheqs.(3)and(11)areinvariantunderarbitraryinfinitesimalcanonicaltransformations
A′=A+ε{A,B}PB
(12)
Fromeq.(11)itfollowsthereforethat,forthedispersionfreeensembleinwhichAhasthevalueαandΠiscompletelyundetermined,thecharacteristicfunctionsatisfies,inadditionto(5),alsotheequation
dxdha(h−k,x−y)(1/h¯2)(kx−hy)α=0.(13)
Eq.(13)representsa“classicaluncertaintyprinciple”expressingthecon-ditiontobefulfilledbyclassicalensembleshavingthepropertythatwhenagivenvariableAhasthevalueαtheconjugatevariableΠisundetermined.Conversely,ifweimposethatthecharacteristicfunctionofanensemblesat-isfieseqs.(5)and(13)weselectonlytheensemblesinwhichthe“uncertaintyprinciple”issatisfied.
3Quantumpostulate
Ourreformulationofquantumtheorywillbebasedontheassumptionthateqs.(3)and(11)shouldholdforanyvariableA.Thiswillimposeautomat-6
icallyforallthepossibleensemblesthevalidityoftheuncertaintyprinciple.Howevertheexplicitformoftheseequationsintermsofαgivenby(5)and(13)willhavetobemodified,becauseeqs.(8)and(9)arenolongervalidinquantumtheory.Inadditiontothisfirstassumption,therefore,wewillimposethefulfil-mentofanextrapostulate,basedontheconvinctionthat,insteadofpos-tulatingtheconventionalrepresentationofphysicalquantitiesbymeansofoperatorsinHilbertspace,itismoresatisfactorytoassumeasafoundingstoneofquantumtheorytheexperimentalfactthatphysicalquantitiesexist(e.g.angularmomentum)whosepossiblevaluesformadiscreteset,invariantundercanonicaltransformations,characteristicofeachvariableinquestion.Anequallycompellingphysicalstartingpointfortheadoptionofthispostulatemightbethestabilityofmatter.InfactthisrequirementimpliesthenecessaryexistenceofaminimumvalueE0belowwhichnolowervaluecanbeassumedbytheenergyofanelectron-nucleusboundstate.
Inanycaseweneedonlyassume(QuantumPostulate)thatatleastonevariableLexistswhichhasfinitegapsinthecontinuousrangeL(q,p)impliedbyitsfunctionaldependenceL(q,p)onqandp(whichcanbothassumeanyvalueinthecontinuousrange−∞,+∞)inwhichitcannotassumevaluesexceptforoneormorediscretevaluesλi.Thisinfactmeansthat,sinceLcannothavevaluesintherangebetweenλiandλi−ε,and/orbetweenλiandλi+η,(withε,η,finite)eqs.(8)(9)donotholdintheseranges.
7
AsaconsequenceweconcludethatL(q,p)cannotbeexpressedintheform(2),namelythatthequantumcharacteristicvariablesC(k,y)cannotsatisfythecrucialproperty(6).
Therefore,sincebydefinitionallvariablesshouldbeexpressedintermsofauniquesetofcharacteristicvariables,weconcludethatforallthevariables2ˆ,eq.(2)shouldbereplacedbyA
ˆ=Aˆ(k,y)dydka(k,y)C
(14)
ˆ(k,y)obeyinganewruleofmultipli-withasetofcharacteristicvariablesCcationreplacingeq.(6).
Inordertofindtherequiredmodificationofeq.(6)westartbyaskinghowtheeigenvalueequation(5)shouldbemodifiedinordertoallow,besides(insteadorinadditionto)acontinuousrangeofpossiblevalues,alsoforˆ.ThisamountstosaythatitstheexistenceofdiscreteeigenvaluesλiofL
Fouriertransformshouldnolongerreducetothealgebraicrelation(7)butshouldbecomeatrueFredholmhomogeneousintegralequation,which,asiswellknown,hastheproperty,undersuitableconditions,ofallowingfortheexistenceofdiscreteeigenvalues.
ˆwillactuallyhaveeigenvaluesbelongingtoaWhetheragivenvariableA
discreteoracontinuous(orevenboth)spectrumwilldependonitsfunctionaldependenceonpˆandqˆ.Thereareinanycasesomestringentrequirements
thatthemodifiedkernelshouldsatisfytoattainthisgoal,namely:ˆonpa)thebasicinformationonthefunctionaldependenceofAˆandqˆcontainedinthekernela(h-k,x-y)shouldremainunaltered;
b)thecorrelationbetweenthecoupleofvariablesh,xandk,ywhichisnecessaryinordertotransformeq.(5)intoatrueFredholmintegralequationshouldbeuniversal,namelyindependentofthevariablechosenandofthestateconsidered;
c)theclassicalkernelshouldberecoveredwhenk=y=0becauseeq.(5)ˆ(0,0)>α=1shouldgiveeq.(4)whichmuststillbevalid;ford)theclassicalkernelshouldberecoveredalsoforh=x=0becausetherelation(6)shouldstillbevalidwhenk=handx=y.Thesimplest(andfromthispointofviewunique)waytosatisfya)andb)istomultiplytheclassicalkernela(h−k,x−y)byafactorg(kx−hy)whoseargumentisunambiguouslyfixedbytherequirementthat,fordimensionalreasons,xshouldbecorrelatedtokandhtoy.Furthermoreinorderthatc)andd)arefulfilled,itmustbeg(0)=1.Themodifiedintegralequationreplacingeq.(5)shouldthereforereadˆ(k,y)>i=αiˆ(h,x)>idxdha(h−k,x−y)g(kx−hy)(15)Eq.(15)hasafirstimportantconsequence.Infactthecondition(3),ˆ(k,y)intheformwhichmayberewrittenintermsofthenewvariablesC
9
dydka(k,y)
ˆ(k,y)Cˆ(k′,y′)>idy′dk′a(k′,y′)=αiˆ(k,y)>idydka(k,y)(16)leadstoeq.(15)onlyifeq.(6)isreplacedby
ˆ(k,y)Cˆ(k′,y′)+Cˆ(k′,y′)Cˆ(k,y)]=g(ky′−k′y)Cˆ[(k+k′),(y+y′)](1/2)[C
(17)
Thisequation,however,cannotbesatisfiedbyordinaryc-numbers.Thismeansthat,ifwewanttoallowfortheexistenceofdiscretevaluesofatleastˆweareforcedtorepresentallonevariableL
ˆshouldbeInordertofulfilltheconditionthattheeigenvaluesαiofA
invariantunderthecanonicaltransformations(12)onemustinfactimposethateq.(11)shouldhold.However,ifweuseforthenewcharacteristicˆ(k,y)theclassicalPBsoftheoldvariablesvariablesC
h)(kq+yp)(i/¯h)[(k+k)q+(y+y)p]
{e(i/¯,eh)(kq+yp)}PB=[(k′y−ky′)/h¯2]e(i/¯
′
′
′
′
(18)
weimmediatelyseethateq.(17)isnolongerinvariantunder(12)whichhasthereforetobereplacedby
ˆ′=Aˆ+ε{Aˆ,Bˆ}QPBA
(19)
WehavethereforetoderivethecorrespondingquantumPoissonBracketsˆ(k,y),Cˆ(k′,y′)fromtheconditionofinvariance(QPB)ofthetwovariablesC
of(17)under(19).HereagainweneednotintroduceexplicitlythestandarddefinitionofthePB’softheseq-numbersintermsofoperators.Onthecontrary,theirformwillbeobtainedasaconsequenceofourformalism.WewillonlyneedtodefineQPB’s,forconsistencywitheq.(17),bymeansofthefollowinggeneralizationoftheclassicalPBs
ˆ(k,y),Cˆ(k′,y′)}QPB=f(ky′−k′y)Cˆ[(k+k′),(y+y′)]{C
(20)
wheref(λ)isanoddfunctionofitsargumentsatisfying,forconsistencywitheq.(18),theconditionlimλ→0f(λ)=−λ/h¯2
From(17)and(20)wenowobtainimmediately
ˆ(h,x)>i=0dxdha(h−k,x−y)f(kx−hy)11(21)
Thisistherequiredgeneralizationofeq.(13).
Thefurthersteprequiredtocompleteourformalismisthedeterminationofthefunctionsf(.)andg(.).Theknowledgeofthesefunctionswillthenal-ˆ(k,y)>iandthelowtheexplicitderivationofthecharacteristicfunctionˆforanystate<>ibysolvingeqs.(15)andcorrespondingeigenvalueαiofA(21).Thisgoaliseasilyattainedbyimposingtheconditionthatbothrela-tions(16)and(20)shouldbeinvariantunderthecanonicaltransformations(19).Thisconditionleadsinfacttothetwoequationsf(λ)f(µ−ν)+f(µ)f(ν−λ)+f(ν)f(λ−µ)=0
(Jacobiidentity)
(22)
g(λ)f(µ+ν)=g(λ+µ)f(ν)+g(λ−ν)f(µ)
Theseequationshavethefollowingsolutionsg(ky′−k′y)=cos[b(ky′−k′y)/h¯];
(23)
f(ky′−k′y)=(1/bh¯)sin[b(k′y−ky′)/h¯]
(24)
withbaparameterwhichisstillundetermined.Itshouldbestressedthattheclassicalstatisticaltheoryisnotrecuperatedbymakingh¯→0,butbylettingtheadimensionalparameterb→0(absenceofcorrelations).However,althoughb→0isavalidmathematicallimitfortheexpressions(25),b=0andb=0yieldtworadicallydifferenttheories,becauseinthefirstcasethevariablesarec-numberswhileinthesecondonetheyareq-numbers.
12
ˆ(k,y)>iofeqs.(15)(21)willnowyieldeasilythecorre-ThesolutionspondingexpressionforPi(q,p)bymeansofPi(q,p)=(2πh¯)
−2
h)(kq+yp)
dydke(−i/¯i(25)
Beforediscussingthepropertiesofthis(pseudo)probabilitydensitywewillhoweverworkouttheresultsofourtheoryinsomesimplecases.
4Simpleexamples
Wewillfirstsolvethetwoequations(15)(21)forthevariablesqˆ,pˆandsuc-ˆ=(1/2)pcessivelyfortheenergyHˆ2+(1/2)ω2qˆ2oftheharmonicoscillator.Thiswillshowexplicitlyhowtheformalismleadsbothtotheexistenceofvariableswhoseeigenvaluesbelongtoacontinuousrangeaswellasofotheroneswithadiscretespectrum.
1.Variableqˆ.Fromtheclassicalexpression(2)oneobtains
aq(k,y)=
dqdpqexp[−i(py+qk)/h¯]=ih¯δ(y)[∂δ(k)/∂k](26)
Theeigenvalueequation(15)readsˆ(k,y)>qo=ih¯qoˆ(k,y)>qo=dxdhδ(x−y)[∂δ(h−k)/∂h]g(kx−hy)(27)ˆ(k,y)>qo/∂k]=−ih¯[∂becauseg(0)=1and[∂g(λ)/∂λ]λ=0=0,whereq0isthevalueofqˆwhichlabelsthestate.Thesolutionof(27)isˆ(k,y)>qo=exp[ikqo/h¯]φ(y)13(28)
withφ(y)anarbitraryfunction.Ontheotherhandeq.(21)reads0=ih¯
ˆ(k,y)>qoˆ(h,x)>qo=y(29)becausef(0)=0and[∂f(λ)/∂λ]λ=0=1/h¯2.Eq.(29)givesimmediately
φ(y)=δ(y)
Byintroducing(28)(30)ineq.(25)oneobtains
¯)−1δ(q−qo)Pqo(q,p)=(2πh
(31)(30)
whichcoincideswiththeclassicalprobabilitydensityoftheensembleinwhichthevariableqˆhasthevalueqo.
Thisshowsthatthepossiblevaluesofthequantumvariableqˆspanthesamecontinuousrangefrom−∞to+∞oftheclassicalvariableqˆ.Thisisbecausethesolutionof(27)and(29)involvesonlytheclassicallimitsofg(.)andf(.)anddoesnotdependontheactualvalueofb.InthiscasetheQPB’scoincidewiththeclassicalPB’s.
2.Variablepˆ.Thecompletesymmetrybetweenqˆandqˆallowsustowrite
ˆ(k,y)>po=exp[iypo/h¯]δ(k)namely¯)−1δ(p−po)Ppo(q,p)=(2πh
(33)(32)
14
ˆ=(1/2)p3.VariableHˆ2+(1/2)ω2qˆ2.Fromtheclassicalexpressionweobtain
h(k,y)=
dqdp(1/2)[p2+ω2q2]exp[−i(py+qk)/h¯]=
(34)
=−(¯h2/2)[δ(k)∂2δ(y)/∂y2+ω2δ(y)∂2δ(k)/∂k2]
Eq.(15)readsˆ(k,y)>Eo=Eo2
ˆ(h,x)>Eo]dxdhδ(x−y)δ(h−k)[∂2/∂x2+ω2∂2/∂h2][g(kx−hy)(35)Since,from(24)wehave
[∂2g(hy−kx)/∂x2]x=y;h=k=−k2b2/h¯2[∂2g(hy−kx)/∂h2]x=y;h=k=−y2b2/h¯
weobtain
ˆ(k,y)>Eoˆ(k,y)>Eo=(1/2)[k2b2−h¯2ω2∂2/∂k2+ω2y2b2−h¯2∂2/∂k2](37)Fromeq.(21)weobtain
ˆ(k,y)>Eo=0[k∂/∂y−ω2y∂/∂k]Eq.(38)canbesolvedbysettingˆ(k,y)>Eo=F(k)G(y)leadingtoF(k)=exp[µk2/ω2]
15
G(y)=exp[µy2]
(40)(39)(38)
2
(36)
ˆ(0,0)>=1)Introducing(39)and(40)in(37)weeasilyfind(sinceEo=bh¯ω(41)
ˆ(k,y)>Eo=exp[−bk2/2¯hω]exp[−by2ω/2¯h]Byintroducing(42)in(25)weobtain¯b)exp[−p2/2¯hωb]exp[−q2ω/2¯hb]PEo(q,p)=(1/2πh
(42)
(43)
Fortheexcitedstatestheseparabilitycondition(39)doesnothold.Eqs.(37)and(38)arehoweversufficienttodeterminecompletelythecorrespond-ingcharacteristicfunctionsandeigenvalues(10).
5Theuncertaintyprinciple
Wewillfinallydiscussthepropertiesofthe(pseudo)probabilitydensitiesPi(q,p)givenby(25).Thiswillalsoallowustodeterminetheparameterb.ˆ(k,y)>iandjforWestartbywritingeq.(21)forbothˆ(−k,−y)>jandthesecondonebyi=j;wemultiplythefirstonebyˆ(−k,−y)>iandfinallyintegrateoverk,y.Bysubtractingthesecondequationfromthefirstoneweobtain0=(αi−αj)
ˆ(−k,−y)>ijdydk(44)16
Thisamountstowriting3
dydkˆ(−k,−y)>ij=Nδij(45)whereNisanormalizationconstant,havingthedimensionsofanaction,independentofthevariableAˆandofthestate<>i.From(25)and(45)weobtain
dqdp[(2πh¯)2Pi(q,p)/N]Pj(q,p)=δij
(46)
Atthisstagewehavetofixourunitofaction2πh¯.Tothispurposewecompare(46)withitssemiclassicallimitgivenbytheoldtheoryofquantaofPlanckandBohrwherethevolumeoftheregionofphasespaceinwhichtheclassicalA(q,p)hasthevalueαiandallthepointsq,phaveequalconstantprobabilityKiinsideitandzeroprobabilityoutside,isassumedtobeequaltoPlanck’sconstant(2πh¯).Theninthissemiclassicaltheory,wehave,forthenormalizationofprobability
2πh¯Ki=1
(48)
andfor(46)
(2πh¯)3Ki2/N=1
(49)
fromwhichwegetN=2πh¯.Eq.(46)becomestherefore
dqdpPi(q,p)Pi(q,p)=(2πh¯)−1≡Pav
(50)
Thelaststepofourworkisnowthedeterminationoftheparameterb.Infactitisimmediatetoseethat,introducingintoeq.(50)theexpression(43)forPEo(q,p)oftheharmonicoscillatorgroundstate,oneobtainsb=1/2.Sincethisvalueisindependentofthevariableandofthestatechosen,thisresultiswhollygeneralandconsequentlyourreformulationofquantumtheoryiscomplete.
Eq.(50)expressesanewformoftheuncertaintyprincipleforpositionandmomentum.Infact,byintroducinginthenormalizationconditionthemeanvaluePavdefinedbythisequation,weobtain
dqdpP(q,p)=Pavδqδp=1(51)
whereδqδpisthevolumeofphasespaceinwhichP(q,p)isreplacedbyPavandiszerooutside.Wethenimmediatelyobtain
δqδp=2πh¯.
(52)
Itisimportanttostressthat(52)doesnothavetheformoftheconven-tionalHeisenberginequality,whichgivesnoupperlimittothepossiblevalueoftheuncertaintyproduct∆q∆pofthemeansquarevaluesofqandp,butinvolvesonlyitsminimumvalue.Wewillreturnontheimplicationsofthisdifferenceinthediscussion.
18
6Thedynamicalevolution
WefinallyindicatehowthedynamicalevolutionofthepseudoprobabilitydistributionP(q,p)inanygivenstategivenby(25)canbeworkedout.ItisˆsufficienttousetheHamiltonianH
ˆ=H
ˆ(k,y)dydkh(k,y)C
(53)
asthegeneratoroftheinfinitesimaldisplacementintimeˆ(k,y)/dt={Cˆ(k,y),Hˆ}QPB=dC
ˆ(j,x)dxdjh(j−k,z−y)f(jy−kx)C
()
Eq.()yieldsaChapman-KolmogorovequationforthetimedependenceofthepseudoprobabilitydensityP(q,p;t):
(d/dt)P(q,p;t)=
dq′dp′K(q,p;q′,p′)P(q′,p′;t)
(55)
Intheclassicallimiteq.(55)reducestotheLiouvilleequation.
Wehavethereforeattainedourgoal,namelytheconstructionofaformalprobabilistictheory(withthegeneralizationofprobabilitiestonegativevaluesaccordingtoFeynman’sinterpretation)ofthequantumworldinphasespacebymeansofastraightforwardgeneralizationofclassicalstatisticalmechanics.
7
Comparisonwiththeconventionalformu-lationofQuantumMechanics
Thepresentformulationofquantumtheoryisclearlyidenticaltothecon-ventionalformalismofQuantumMechanics.InfactifweconsidertheWeyl
19
operator
C
andq
,p
(k,y)
(57)
y+kq
Therefore,if|ψ>isthestatevectorcorrespondingtoourstate<.>wehaveˆ(k,y)Cˆ(k′,y′)+Cˆ(k′,y′)Cˆ(k,y)>=2Re<ψ|Cˆ(k,y),Cˆ(k′,y′)}QPB>=(2/h<{C¯)Im<ψ|CFrom(57)(58)itfollowsalsothatP(q,p)=W(q,p)
(60)
(k′,y′)|ψ>(58)(k′,y′)|ψ>
(59)
whereW(q,p)istheWignerfunction(4)ofthestate|ψ>.Eq.(55)coincidesthereforewiththestandardequationforthetimeevolutionoftheWignerfunction.Thisresultshowsthatthisfunctionhasaprivilegedstatusamongotherfunctions
(6)
usedintheliteraturetodescribeQuantumMechanicsin
phasespace,becauseitcanbederiveddirectlyfromourquantumpostulate.
8Discussion
ThephysicalmeaningofnegativeprobabilitiesiswellclarifiedbyFeynman’sownwords:“Itisthatasituationforwhichanegativeprobabilityiscalculated
20
isimpossible,notinthesensethatthechanceforitshappeningiszero,butratherinthesensethattheassumedconditionsofpreparationorverificationareexperimentallyunattainable.”Admittedly,asherecognizes,a“strongmentalblock”againstthisextentionoftheprobabilityconceptiswidespread.Oncethishasbeenovercome,however,thepresentformulationofquantumtheoryhasseveraladvantages.
Firstofall,asalreadyanticipatedintheintroduction,manyparadoxestypicalofthewave-particledualitydisappear.Ontheonehandinfact,asalreadyshownbyFeynman,itbecomespossibletoexpressthecorrelationsbetweentwodistantparticlesintermsoftheproductoftwoprobabilitiesindependentfromeachother(3)(7).Allthespeculationsonthenatureofanhypotheticalsuperluminalsignalbetweenthembecomesthereforemeaning-less.Ontheotherhandthelongtimedebatedquestionaboutthemeaningofthesuperpositionofstatevectorsformacroscopicobjects(Schr¨odinger’scats)mayalsobesetasideasequallybaseless,togetherwiththemanyproposalsofdetectionof“emptywaves”.ItisnotthepracticaluseoftheformalismofQuantumMecanics,ofcourse,whichisputinquestion.However,fromaconceptualpointofview,theeliminationofthewavesfromquantumtheoryisinlinewiththeprocedureinauguratedbyEinsteinwiththeeliminationoftheaetherinthetheoryofelectromagnetism.
Secondly,thisapproacheliminatestheconventionalhybridprocedureofdescribingthedynamicalevolutionofasystem,whichconsistsofafirststage
21
inwhichthetheoryprovidesadeterministicevolutionofthewavefunction,followedbyahandmadeconstructionofthephysicallymeaningfulprobabil-itydistributions.Iftheprobabilisticnatureofthemicroscopicphenomenaisfundamental,andnotsimplyduetoourignoranceasinclassicalstatisti-calmechanics,whyshoulditbeimpossibletodescribetheminprobabilistictermsfromtheverybeginning?
ThethirdadvantageisconnectedwiththepossibilityofdissipatingtheambiguityoftheconventionaltheoryabouttwophysicallydifferentaspectsofthequantumuncertaintiesinherenttotheHeisenberginequality.Ithasbeenrecognizedinfactthatthisinequalitycontainstwocontributionsofdifferentorigin(8).Itsminimumvalueisinfactanontologicaluncertainty,ofquantumnature,whilethecontributionexceedingthisminimumisofepistemicnature,namelyexpressesastatisticaleffectduetoimperfectknowledgeofreality.Infact,whiletheirreduciblequantumcontributionrequiresthatareductionof∆xshouldnecessarilyimplyasimultaneousincreaseof∆p(orviceversa),forthestatisticalcontributionbothuncertaintiescanbereducedatthesametimebymoreaccuratemeasurementsuntiltheminimumvalueisreached.Inthepresentformulationofquantumtheory,however,onlythequantumontologicaluncertaintiesarepresent,withoutanyspuriousstatisticalcon-tribution.Thisisbecausetheuncertaintyprincipleinourtheoryisgivenbytheequality(52),involvingonlytheminimumvalueoftheHeisenberginequality.
22
Thelast,butnotleast,appealofthisapproachisthatitmaybecosideredasaconceptual“Gestaltswitch”ofthetypesuggestedbyThomasKuhn(9)concerningthestatusofthe“LawsofNature”.Aswitchfromthe“au-tocratic”rulethattheLawsprescribeeverythingthatmusthappentothe“democratic”principlethatanythingwhichisnotforbiddenbytheLawsmayhappen.Ifchancehasanirreducibleoriginthefundamentallawscannotpre-scribeeverything:theycanonlyexpressconstraintsfollowingfromstabilityrequirementsofmatter,orprohibitionsderivingfromsymmetrypropertiesoftheUniverse,orgeneralprincipleswarrantingtheexistenceofpatternsoforder.Inotherwordstheyshouldallowfortheoccurrenceofdifferenteventsunderequalconditions.Ifthisistrue,itbecomesmeaninglesstoask:howcanthiseventhappen?Theanswercanonlybe:ithappensbecauseitisnotforbidden.Thelanguageofprobability,suitablyadaptedtotakeintoaccountalltherelevantconstraints,seemsthereforetobetheonlylanguagecapableofexpressingthisfundamentalroleofchance.
Acknowledgement
ThecontributionofmyfriendandcolleagueGianniJona-Lasiniohasbeencrucialinclarifyingthemathematicalnatureofthecharacteristicvariablesandingivingtotheformulationofthetheoryatightlogicalstructure.Iwishthereforetoexpressmygratitudeforhisconstructivecriticism,withoutwhichthispaperwouldnothavebeenaccomplished.Illuminatingdiscussions
23
withFrancescoGuerraandMaurizioServaarealsogratefullyacknowledged.
24
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25
