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Direction of Time Sergio Albeverio (cid:2) Philippe Blanchard Editors Direction of Time Editors SergioAlbeverio PhilippeBlanchard IAMundHCM FakultätfürPhysik UniversitätBonn UniversitätBielefeld Bonn,Germany Bielefeld,Germany ISBN978-3-319-02797-5 ISBN978-3-319-02798-2(eBook) DOI10.1007/978-3-319-02798-2 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013956938 Mathematics Subject Classification: 00A06, 00A30, 03B44, 35Qxx, 35Q20, 37D35, 37N05, 37N25, 58Zxx,70Fxx,70F16,70F45,74A15,74A25,78A25,81S22,81S40,81T28,83Axx,83Cxx,83Dxx, 83Fxx,92D15,92D25,92D40 ©SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface ThepresentvolumecollectstheProceedingsoftheConference“DirectionofTime” thattookplaceattheCenterforInterdisciplinaryResearch(ZiF)ofBielefeldUni- versity,January14to19,2002.Theorganizingandscientificcommitteeincluded, inadditiontothepresenteditors,MichaelDrieschner(Bochum)andSylviePaycha (Clermond-Ferrand/Potsdam).Wearemostgratefultothemforaveryinspiringand fruitful collaboration. We are also most grateful to Annidita Balslev (Aarhus) for inspiring,highlymotivatingobservationsatanearlystageofthepreparationofthe Conference. ThesuccessoftheMeetingwasdue,firstofall,tothespeakers.Thankstotheir efforts, it was possible to take into account recent developments in various direc- tionsas well as open problemsand to make the Conference an excitingevent.We hopethat participantsandreaders willfind the articles collectedin these Proceed- ings both interesting and useful. We apologize for the delay in publishing, due to circumstancespartlyindependentofourwillandefforts.Wehopethattheperma- nentactualityofthetopicofthisbookmightattenuatethedamagescausedbythe delay. WeareverygratefultoJean-ClaudeZambriniforcompetentandinspiringadvice inthecourseofthepreparationofthisvolume. ItisapleasuretoexpressourspecialgratitudetoMarionKämperforthecritical readingofthemanuscriptandtoHanneLitschewskyforinvaluablehelpinprepar- ing the Conference and in collecting and editing the manuscripts for publication. Withouthergeneroushelpthebookwouldhaveneverhaveappeared. WegratefullyacknowledgethefinancialandlogisticsupportofZiF. Bonn,Germany SergioAlbeverio Bielefeld,Germany PhilippeBlanchard v Introduction to Time, Its Arrow and the Present Book Thecentralthemeofthemeetingwasthediscussionoffundamentalquestionslike • Whatistime? • Istimeanillusion? • Inwhichsensearepast,present,andfuturetobeunderstood? • Istimedirected? • Whatmakestimedifferentfromspace? • Wheredoesthetimearrowcomefrom? Proposinganswerstosuchfundamentalquestionsrequiresjointeffortsfromdis- parateareas,eachbringinginitsoriginalpointofview.Themeetingandthispub- licationwhichemanatesfromitwillfocusonlyonsomeaspectsoftheproblematic nature of time (see, e.g. [D, Fraser 1975] for complements). (The references are groupedintofoursections,A,B,C,D.Reference[D,Fraser1975]standsforsec- tionD,authorFraser,published1975.) Timehasbeenapreoccupationofhumanityasawholefromtheverybeginning of culture and it still remains a basic source of open problems. In fact, nothing is probablymorecloseandfamiliartousthantimebutalso,atthesametime,sofull ofmystery. Timeisdeeplyrelatedtonaturalphenomena(likeperiodicitiesinthesky,cycles in biologicalprocesses and evolution),and as such has been of concern in all cul- turesandreligions,whichhavetriedeachinitsownwaytofindanswerstothebasic questionsitraises.Alsovariousscienceshavebeenstrugglingwiththeconceptof timeanditsmultiformaspects.Buttimeis,inaddition,anexperiencewhichhasalso beensteadilyofconcernforwriters,artists,andphilosophers,andhumanityatlarge. Already from the Presocratic tradition one can observe steady oscillations be- tweenastaticviewoftime(likeParmenides)andanallpervadingdynamicalview oftime(likeinHeraclitus).Sincetheilluministicepoch,timehasratherbeenviewed ascomposedofthreecomponents,twoverylargeones,thepastandthefuture.The third one is tiny, almost non-existing. In some sense, however, we could maintain thatnoneofthesethreepartshasaveritableexistence:past,sinceitnolongerexists, future,sinceitdoesnotyetexistandpresent,sinceateachinstantitdisappears. vii viii IntroductiontoTime,ItsArrowandthePresentBook Inthemeetingandinthepresentpublicationthemainattentionisdrawntothe question of the “arrow of time”, since it is the question which better puts in evi- dencethemultiplevisions of timeboth indifferent cultures anddifferent sciences (“irreversibility”oflifeexperiencecontrastedwith“reversibility”oftimeevolution, asdescribed,e.g.,inNewton’slawsofnature). Inthepresentbookweareexaminingparticularlythreemainapproachestothis problem, namely “time and physics”, “time, philosophy, and psychology”, and fi- nally“time,mathematics,andinformationtheory”. Let us discuss these three aspects of time, starting always with some general considerationsofoursandproceedingthentoashortdiscussionofthespecificcon- tributions,inthisvolumepresented. TimeandPhysics ContemporaryphysicshasitsorigininclassicalmechanicsasformulatedbyGalilei andNewton.Itinvolvesequationsofmotiondescribingthehistoryofanidealized system. In these equations, time is nothing else but a real parameter and the basic Newton equations are invariant under time reflection. On the other hand, as was soonrealized,classicalmechanicscanalsobeformulatedinawayemphasizingthe roleofinvariantquantities:theintegralsofmotion.TheabsolutetimeofNewtonian physicswasreplacedinthedescriptionofclassicaldynamicalsystemsbythe“rel- ative time” introduced 1905 by Einstein (and Poincaré). In special relativity time andspaceformawholeanddependonthedynamics(e.g.,onehasthewell-known phenomenon of “slowing down of clocks in motion”). The notion of simultaneity is no longer absolute but past and future maintainfor each observer their absolute character. SinceEinstein’s(andHilbert’s)generalrelativity(1917),gravityisdescribedasa geometricpropertyofspace-time,andinparticulartimeisaffectedbythepresence ofmatterandenergy(althoughspace-timeitself,beforeametricisputonit,isjust conceivedasamanifold).Thematterdistributiondeterminesthespace-timegeom- etryandthesolutionsofthecorrespondingEinstein’sequations.Bothinspecialand in(local)generalrelativitytimedoesnotplayaspecialrole,exceptforbeingoneof thecomponentsofafour-dimensionalspace-time,andmakingthelocalMinkowski metricnotpositivedefinite,ofsignature,say,(+,−,−,−). Cosmologyenters,however,intothestages,whengeneralrelativityisappliedto the whole cosmos. Different cosmological models imply different histories of the universe.The presently most acceptedmodels take into accountHubble’slaw and leadtoacosmicarrowoftime,fromaninitial“bigbang”tothepresentstateofan expandinguniverse. Statistical mechanics is concerned with the description of very large systems and constitutes the theoretical foundations of thermodynamics. Although the mi- croscopic equations of motion are reversible, the emerging observed macroscopic evolution is for all practical purposes irreversible, introducing by this a “thermo- dynamic arrow of time”. The question of deriving irreversibility starting ab initio IntroductiontoTime,ItsArrowandthePresentBook ix has generated a large amount of investigations, starting on the physical side with Carnot, Clausius, Maxwell, Boltzmann, and Gibbs and on the mathematical side with the works of Poincaré, Zermelo, and others, expanding in the second half of thelastcenturyintothegeneraltheoryof(classical)dynamicalsystems. Entropyhasemergedasacentralnotionintheareaofthermodynamicsandstatis- ticalmechanics.Inphysicsentropyisameasureofdisorder.Itisalsoafundamental quantity in information theory and the relations between these two concepts and areasofresearcharecurrentlyunderintensivedevelopment.Ourperceptionofthe direction of time is connected to the fact that entropy in a closed system can only stayconstantorincrease.Theoriginofthetimearrowasweperceiveitdependson twoessentialproperties.Thefirstisalow-entropyinitialstate.Thesecondismixing, whichisnecessaryforexplainingwhyagivensystemevolvesandrearrangesfrom a low-entropy (small probability) to a higher-entropy (high-probability) state. The same properties hold true in cosmology. The cosmological arrow of time requires theuniversetohavestartedinalowentropystateandthemattertohavemixedever since.Amongthepossiblecosmologicalmodelsthebig-bangonesarestillprevail- ing,buttherearealsomodelswhichareextremelydifferentfromtheseandyetare stillbeingdiscussedtoclarifysomebasicissues(seee.g.[C,Gödel1949],[C,Segal 1976,1996],[B,Stölzner1996],[B,Yourgrau1991]). Although ordinary (non-relativistic) quantum mechanics has dramatically changedourconceptionoftheworld,timeinitisstillaparameterlikeinNewtonian physics and the fundamental dynamical equation, the Schrödinger equation, stays reversible.Butquantummechanics,ontheotherhand,isintrinsicallyprobabilistic. Decoherencecanservetorelatesomequantumprobabilitiestoclassicalprobabili- tiesbutitdoesnotmakethemintoclassicalprobabilities.Measurementsareachain ofcorrelatedconsequencesanddecoherencedoesnotexplainwhyaparticularevent isrealizedinaparticularmeasuringprocess.Measurementistheirreversibleregis- trationofamacroscopicsignal,andassuchincorporatesanintrinsicarrowoftime. The natural extension of quantum theory to include special relativity leads to quantumfieldtheory,whichisatthebasisoftheentirepresent-dayparticlephysics theory. Afurtherextensionwouldleadtoquantumgravity,aquantizationofgeneralrel- ativity.Quantumgravityisconcernedwithgainingquantitativeknowledgeofphys- ical phenomena at very high energies. The characteristic scale is the Planck scale of 10−35 m, which is very far removed from our every day experience and phys- ical intuition, and cannot be probed directly by experiment. This latter extension hasgeneratedveryintensiveactivitiesusingdifferentapproachesandpointsofview (amongthemstringtheory,loopquantumgravity,non-commutativegeometry,...), and showing intriguing connections between particle physics and cosmology. The studyof phenomenalikeblackholesneedsconsiderationsbothfromclassicaland quantumthermodynamicsleadingtonewpointsofviewontheproblematicofthe arrowoftime.Itisfairtosay,however,thatthe“theories”tryingtocopewithsuch phenomenaarestillratherjustresearchprogramsandfarfromconclusive.Inallof them,however,acausalityprincipleseemstoprevail,see,e.g.,[A,Klein2007]. Let us also mention the thermal type hypothesis of Connes and Rovelli [C,Connes,Rovelli],whichdescribestimeasanemergingstatisticalpropertysim- x IntroductiontoTime,ItsArrowandthePresentBook ilartotemperatureinstatisticalmechanics.Timeappearsinthistheoryastheresult of our missing information about reality and it is our approximate knowledge of realitythatexhibitsatimearrow. InarecentpaperC.Rovelli[A,Rovelli2000,2005]proposesawaytodescribe multiple quantum events in time by introducing a single event that can be formu- lated without reference to time. In this approach evolution in time is replaced by correlationsbetweeneventsthatcanbeobservedinspace. Thereareotherviewsofspace-timeliketheoriesbasedbothonthecausalityprin- cipleandthetheoryoftwistors,byR.Penrose[C,Penrose1994],andtheonewhich postulatesmicroscopicallydiscontinuous(e.g.p-adic)space-times[C,Khrennikov 1999]. Let us also mention in passing that an extreme position of “eliminating time from physics”has beentakenby J.Barbour [A,Barbour 1999](who continuedin atradition,e.g.E.Meyerson[A,Meyerson1908]),itremainstoseewhetherthisis reallymorethanjustaninterestingplaywithformalisms. Acalculationaltool,“causaldynamicalcalculation”[A,J.Ambjørnetal.,2008] hasbeendevelopedforcopingwith“quantumgravity”overthelastfewyears.The absenceofadhocextradimensionsanditsuseoffundamentalquantummechanical principlesmakethisapproachconceptuallysimple.Amongtheintriguingresultsof thisapproachletusmentiontheheuristicemergenceofaclassicalspace-timegeom- etryfromquantumfluctuationsandofaspace-timedimensionatveryshortdistance whichisnotequaltofour!Itremains,ofcourse,toseewhetherthisapproachcould be made into a theory which is both satisfactory in a mathematical and physical sense.Forfurtherreferencestothetopic“Timeandphysics”seeA. Let us now summarize briefly the present volume contributions to the subject “Timeandphysics”. M. Cini’s contribution is concerned with topics relating to the different di- chotomy representations of change: Parmenides versus Heraclitus, eternity versus timelessness, objectivetime versus subjective time, time’s arrow versus time’s cy- cle,reversibilityversusirreversibility.AmongtheseareadiscussionoftheEarth’s historyandofbiologicalevolution,andoftheradicaldifferenceoftheroleofchance in statistical mechanics and quantum mechanics. Topics covered include the irre- versibilityoftheSecondLaw,thetransitionfromordertodisorder,theirreversibil- ityofquantummeasurement,onticandepistemicuncertainties,andarepresentation ofQuantumMechanicsinphasespacebyusingWigner’sfunctions. A.Tetaproposesatwo-particlemodelofdecoherenceinonedimension.Itcon- sistsofaheavyandalightparticlewithcoordinatesRresp.r interactingviaapoint interactionα δ(r−R),withα intherealfieldR andδ standingforDirac’sfunc- 0 0 tion.Theinitialstateischosentobeaproductstateandtheinitialwavefunctionof theheavyparticleofmassM isasumoftwospatiallyseparatedwavepacketswith oppositemomentum.Asforthelightparticleofmassm,itislocalizedbetweenthe twowavepackets.The authordiscusses theasymptoticdynamicsof thesystemin thelimitε=m/M→0,andgivesanexplicitestimationoftheerror.Moreover,he introducesthereduceddensityfortheheavyparticleandgivesanexplicitestimation ofthedecoherenceeffect. IntroductiontoTime,ItsArrowandthePresentBook xi Th.Görnitzconfessesthatheisconvincedthattimeexistsandthatnothingisso un-influentialas the direction of time. To the question: What is time? he proposes different viewpoints namely those of classical physics (time without importance), ofspecialrelativity(anowntimefor everybody),of generalrelativity(theflowof timedependingonthesituation),ofquantumtheory(thetimedisappearanceandthe occurrenceoffactsattheborderbetweenclassicalandquantumworld)andfinallyof cosmology(timebecominguniversalandhavingabeginning).ForGörnitzphysics as any other human activity can only be an approximation of truth, in many cases averygoodone,anditreflectsmoreorlessallpossibletimeexperienceswehave beenabletomake. R. Haag starts by recalling a remark by Wolfgang Pauli,who distinguished the “realpartofphysics”(thephenomena)fromthe“imaginaryaxis”(thetheory).The- oreticalconceptsarementalconstructswhose“realityvalue”isdebatable.Haagin- troducesthenotionofeventsasindividualfacts.Hepresentstheviewthatthequan- tumstatedescribestheprobabilitiesfortheoccurrenceofvariouspossibleevents. Realityconsistsofpastfacts.Thefutureisopen.InthiswaythephysicistHaag developsanevolutionarypictureofrealitysimilartothatdescribedlongbeforeby thephilosopherA.N.Whitehead. AfruitfulinteractionbetweenPhysicsandNeuroscienceisexploredinvonder Malsburg’scontribution.Afterashortdescriptionofouroutlookontime,hestarts with the description of Cramer’s system of communication between potential ab- sorbersinthecourseofthetransmissionofaphoton.Accordingtothispicturealong thelinessketchedbyTetrodeandFokkerthedynamicsisformulatedintheEternal Universe by an action principle implying that events at all space-time points stick togetherbyatangleofadvancedandretardedsignalsfromthebeginningtotheend oftime.Nevertheless,therelationbetweenpastandfutureisnotsymmetric,energy andinformationpropagatingalwaysintothefuture.Theadoptedviewpointseemsto beinsharpcontrastwithourtraditionalintuitiveoutlookontimeaccordingtowhich thefuturehasnorealityyetandisopentothedecisionsofourfreewill.Whatour brain uses permanently is practical predictability allowing our survival. The free willintheEternalUniversedoesnotchangetherealfuturebutthepotentialfuture ofourimaginationsandperceptions.Theauthorconcludesthatthereisnocontra- dictionbetweenfreewillanddeterminism.Onthecontrary,freewillisimpossible without determinism.Reserving the status of reality only to the present is nothing elsebutanotherproofoftheextremeegocentricperspectiveofourcivilization. The contribution of R. Omnès discusses how decoherence is the most efficient causeofirreversibilityinquantumphysics,relatingtheprivilegeddirectionoftime withtheoneusuallyassociatedwiththermodynamics.Moreover,healsorelatesthis arrow of time with a “logical one” as part of the approach to quantum mechanics through “consistent histories”. Decoherence implies that the classical world is no more in opposition to the quantum one, but on the contrary is inherently required foritsproperformulation. R.Tumulka’scontributionisdevotedtothedescriptionofa“toymodel”inspired byBohmianmechanics.Introducingtwooppositearrowsoftime,athermodynamic onerelatedtoboundaryconditionsandanotheroneconnectedtoasymmetryofthe
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