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Collapse. What else? Nicolas Gisin (Dated: January 31, 2017) We present the quantum measurement problem as a serious physics problem. Serious because withoutaresolution,quantumtheoryisnotcomplete,asitdoesnottellhowoneshould-inprinciple - perform measurements. It is physical in the sense that the solution will bring new physics, i.e. new testable predictions; hence it is not merely a matter of interpretation of a frozen formalism. I arguethatthetwopopularwaysaroundthemeasurementproblem,many-worldsandBohmian-like mechanics, do, de facto, introduce effective collapses when “I” interact with the quantum system. Hence,surprisingly, in many-worldsand Bohmian mechanics, the“I”plays amore active role than incollapsemodels. Finally,Iarguethateitherthereareseveralkindsofstuffsoutthere,i.e. physical 7 dualism, some stuff that respects the superposition principle and some that doesn’t, or there are 1 specialconfigurationsofatomsandphotonswherethesuperpositionprinciplebreaksdown. Or,and 0 thisIargueisthemost promising, thedynamicshastobemodified,i.e. intheform ofastochastic 2 Schrödingerequation. n a J I. THE QUANTUM MEASUREMENT pointer itself should be measured by coupling it to yet 8 PROBLEM another measurement system, and so on. 2 If one insists, the theory remains silent. But the de- Quantum theory is undoubtedly an extraordinarily fender of the theory get virulent: “If you don’t know ] h successfulphysicstheory. Itisalsoincrediblyfascinating: how to perform measurements”, they claim, “then you p somehow, by ”brute mental force”, one can understand are not a good physicist!”. Ok, physicists do know how - t the strange and marvelous world of atoms and photons! to perform measurements, indeed, especially experimen- n Furthermore,itisamazinglyconsistent,inthesensethat talphysicists. Butshouldn’t allphysicstheoriestell how a it is amazingly difficult to modify the formalism: appar- to perform measurements, at least in principle? u q ently,anychangehereorthereactivatesnon-locality,i.e. Somehow, quantum theory is incomplete. I belong to [ allowsonetoexploitquantumentanglementforarbitrary the generation that learned that one should never write 1 fastcommunication[1–3]. Howcouldthefathersdevelop such a claim in a paper, at least if one wants to pub- such a consistent theory based on the very sparse exper- v lish it in a respectable journal Admittedly, one has to imental evidence they had? In the landscape oftheories, 0 be careful with such incompleteness claims. The idea is 0 whateverthatmeans,quantumtheorymustbequiteiso- clearlynottogobacktoclassicalphysics,i.e. toamech- 3 lated, so that if one looks for a theory in the neighbor- anistic theory in which cogwheels and billiardballs push 8 hood, onehas to meetit. However,the quantumformal- othercogwheelsandbilliardballs. Theideaisalsonotto 0 ismisnotconsistentifonedemandsthataphysicstheory complementquantum theory with localelements ofreal- . 1 tells how one should, in principle, make measurements, ity, using EPR’s terminology [7], nor with local beables 0 as we develop in this article; it is also not consistent if in Bell’s terminology [8]. The idea is simply to comple- 7 one treats the observer as a quantum system [4]. ment quantum theory is such a way that it tells how, in 1 : However, quantum theory is a physics theory and all principle, one performs measurements. v physics theory should tell what is measurable and how i to perform measurements. About the first of these two To illustrate the kind of complements I am looking X for, one could, for example, postulate that the world is points, quantum theory tells that all self-adjoint opera- r madeoutoftwosortsofstuff, onetowhichthequantum a tor correspond to a measurable quantity. More precisely mechanical superposition principle1 applies and one to andprobablymorecorrect(itdependsonthetextbooks), which it doesn’t apply. Quantum theory would describe quantum theory claims that every physical quantity is only the first kind of stuff and measurements happen representedby aself-adjointoperatorandeveryphysical whenonecouples(somehow)the twosortsofstuffs. The quantitycanbemeasured(almostbydefinitionofaphys- readerswill haverecognizedstandardCopenhageninter- ical quantity). Note that often these measurable physi- pretation of quantum theory: the theory applies only calquantitiesarecalledobservables. Sofarsogood. But to the “small stuff”, while the superposition principle let’sturntothesecondpointabove: howtoperformmea- doesn’t apply to the “large stuff”. Measurements hap- surements. Here quantum theory is surprisingly silent. penwhenonecouplesasmallquantumsystemtoalarge Often it is said that one should couple the system un- derinvestigationtoameasurementapparatus,frequently called a pointer, and then measure the latter [5] (for a recent application of this, related to the measurement 1 The superposition principle states, in words, that if some stuff problem, see [6]). Hence, to measure a physical quantity can be either in one state or in another, it can also be in the of interest of your quantum system, you should measure firststateandinthesecondone,i.e. insuperpositionofthetwo another system. This is the infamous shifty split, as the states. 2 measurementapparatus. As sketchedhere, this dualistic idea, i.e. that there are two kinds of stuffs, is not yet Physics is all about extracting information a complete theory. First, because it doesn’t tell how to about How Nature Does It. recognize the two sorts of stuffs (besides that the superposition principle applies to one and not to the other). And for physicists extracting information means per- Second, because it doesn’t tell how to couple these two formingmeasurements. Hencethemeasurementproblem kinds of stuffs. Moreover, one may argue that a com- has to be taken seriously. plete theory should also describe the other - non quan- Let me stress that I consider the quantum measure- tum - kind of stuff. Nevertheless, I believe that this line ment problem as a serious and real physics problem. It ofthoughtdeservestobe investigatedmoreindepth, see is serious because without a solution quantum theory is section VI. incomplete, as discussed above. It is real in the sense What are the alternatives to some sort of dualism? that it’s solutionwill providenew physics,with new and Assume there is only one sort of stuff, but certain testable predictions. Hence it is not merely a matter of arrangements of this stuff make it special. For example, interpretation of a given formalism: to solve it, one has assume everything is made out of elementary particles, to go beyond today’s physics. but certain arrangements of atoms and photons make them actas measuring apparatuses. Hence the question: To conclude this section and to be transparent, I should state that I am a naive realist (as most physi- Which configurations of atoms and photons charac- cists): there is a world out there and the grand enter- terize measurement setups? priseofPhysicsaims atunderstanding it, see footnote 2. Additionally, following Schrödinger [10], I consider that Looking for such special configurations is an interest- “I” am not part of it: my aim is to understand the out- ing line of research. Thanks to the world-wide develop- side world,butI amnotincluding myselfin thatoutside ment in quantum technologies, we should soon be able world. Ofcourse,Iammadeoutofatomsandotherstuff to investigate highly complex configurations of (natural that can and should be studied by physics. But physics or artificial) atoms and (optical or micro-wave)photons. is not about explaining my presence. As much as possi- Shallthesedevelopmentsleadtoabreakthroughinquan- ble (and I believe it is entirely possible),physics theories tumphysics? Possibly,thoughmostphysicistsbetonthe should not postulate that “I” have to exist for the world contrary, i.e. bet that arbitrarily complex quantum pro- to function. This may seem too philosophical, but we cessorswillbedeveloped,showingnosignof“collapses”, shall see that it has consequences for possible solutions no sign of any breakdown of the superposition principle. to the measurement problem. Let me stress that this is notdualisminaphysicssense: theworldouttherecould well be made out of a single kind of stuff. II. WHAT IS PHYSICS In summary, I believe that the scientific method will never explain why there is something rather than nothing, nor will it explain why “I” am here. Physics must Quantumtheoryexplainsverywellwhy itismoreand assume both that “I” exist and that there is a worldout more difficult to keep coherence when the complexity there,sothat“I”cangainbetterandbetterunderstand- increases: because of the so-called decoherence phenom- ing of the outside world, i.e. of How Nature Does It. ena. So, are we facing the end of “clean physics”? Shall we have to stay with the fact that, apparently, quantum Let’sreturntothequantummeasurementproblemand theoryholdsatallscales,i.e. the superpositionprinciple look for alternatives to what we already discussed, i.e. is truly universal, but for all practical purposes (FAPP, physical dualism (the assumption that there are more as Bell would have said [9]) there is a sort of complexity than one sort of stuffs out there), to the existence of law of Nature - a sort of 2nd law - that states that “special”configurationsofatomsandphotonsthatmake it is ultimately harder and harder to demonstrate the them act like measurement devices, and to the end of superposition principle experimentally for larger and clean physics3. more complex systems? Who knows. But for sure, we - the physics community - should not give up the grand enterprise that easily2. Recall that piddlinglaboratoryoperations isto betraythe greatenterprise. Aseriousformulationwillnotexcludethebigworldoutsidethe laboratory.” 3 Inordertoavoidreceivingakm-longe-mailfromChrisFuchs,let 2 In[9]JohnS.Bellwrote: “Inthebeginningnaturalphilosophers mesayafewwordsaboutqubism[11]. Qubismchangesthegoal tried to understand the world around them. Trying to do that ofphysics. ItisnolongeraboutfindingoutHowNaturedoesIt, theyhituponthe greatideaofcontrivingartificiallysimplesit- qubism restricts physics to what “I” can say about the future. uations inwhich the number of factors involved isreduced to a Moreprecisely, about how “I” shouldbet on futureevents. For minimum. Divideandconquer. Experimental sciencewasborn. methisisnotonlyabetrayofthegreatenterprise,itisalmosta But experiment is a tool. The aim remain: to understand the sortofsolipsismwhereeverythingisabout“me”andmybelieves. world. To restrict quantum mechanics to be exclusively about Well,attheend,IamnotsureI’llavoidthekm-longe-mail. 3 III. MANY-WORLDS i.e. determinism applies to everything in the entire universe. Indeed, sincethere can’tbe anyinfluences coming from outside and since the Schrödinger equation - the Why not simply assume that quantum theory is com- only dynamical equationof the theory - is deterministic, plete and the superposition principle universal? This everything that happens today, e.g. what I am writ- leads straight to some many-worlds interpretations of ing, the way each reader reacts, the details of all solar quantumtheory[12,13]. Indeed,sincequantumtheoryis eruptions, etc, was all encoded in some “quantum fluc- amazingly successful and since quantum theory without tuations”4 ofthe initialstate ofthe universe5. Giventhe any addition (i.e. without any vague collapse postulate) complexity of the (many-) worlds, it had to be encoded leadstothemany-worlds,whynotmerelyadoptamany- in some infinitesimal digits of some quantum state, pos- worlds view? sibly in the billionths of billionths decimal place. I am In the many-worlds view, the measurement problem always astonished that some people seriously believe in is circumvented by the claim that everything that has a that. Mathematical real numbers are undoubtedly very chance to happen, whatever tiny chance, does actually useful when doing our theory. But are they physically happen. Hence, it is a sort of huge catalog of everything real[14]6? Dotheseinfinitesimaldigitshavearealimpact that could happen. More precisely, it is the catalog of on the real world? Is this still proper physics? For sure, everything that has happened and of everything that is such assumptions can’t be tested. Hence, for me, hyper- happeningandofeverythingthatwilleverhappen. Sim- determinism is a non-sense [14], though it is the domi- ply, we are not aware of the entire catalog, only of that nanttrendintoday’shigh-energyphysicsandcosmology part of the catalog corresponding to the world in which (though see [16, 17]). Apparently, the many followers of “we”happentolivein. Butisn’tphysicspreciselyabout, today’s trend elevate (unconsciously) the linearity of the and only about, that part of the catalog? What is the Schrödinger equation and the superposition principle to explanatory power of claiming that everything happens, some sort of ultimate quasi-religious truth, some truth but “we” are not awareof everything? And what is that in which they believes even more than in their own free “we”? will. Note that it is not the first time in science history It is a fact that “I” exist. Actually, it is the fact that that some equations get elevated above reason: follow- I know best. Should “I” be satisfied with a theory that ers of Laplace did also elevate the deterministic Newton tells that I exist in a hugely enormous number of copies equations to some sort of ultimate truth. We now what andthatallthetheoryprovidesisacatalogofeverything was the destiny of that belief. that“I”oracopyofmyselfexperiences? Nottomention Insummary,inordertomakepredictionsinthemany- the vast majority of worlds in which the atoms of my worlds,one introduces some effective collapses that hap- bodydon’tmakeupahuman,probablynotevenathing. penwhenthesystemiscoupledto“I”.Hence,thetheory Actually, the theory says a bit more, it also tells about is not complete, but relies - somehow - on “I”, i.e. on correlations. If“I”seethisnow,thenthereareonlysome some concept foreign to the theory. events“I”mayseeinthe future. Andvice-versa,astime doesn’t properly exist in the many-worlds. Note that to achieve this, one conditions the catalog on what “I see now”, i.e. one uses an effective collapse: one limits the IV. BOHMIAN QUANTUM MECHANICS analysis to that part of the catalog in which “I” see this now. There is yet another way to avoid the quantum mea- In summary, in many-worlds theories, it is “I” that surement problem. Assume that at all times there is continuously collapse the state-vector, at least for the one and only one “event” that is singled-out7. As time purposeofallowingthetheorytomakepredictionsabout passes, the list of singled-out events must be consistent, what “I” am observing. In other worlds,in many-worlds as in consistent histories [18]. A nice example assumes the “I” is not merely a passive observer, but plays an that,atthe endofthe day,everythingwe everobserveis active role. the position of some stuff. Hence, let’s assume that the Admittedly,many-worldsisalogicallyconsistentinter- physicalquantity“position”isalwayswelldeterminedby pretation, at least as long as one doesn’t insist that “I” someadditionalvariable(additionalwithrespecttostan- exists. Moreover, it is the most natural one if one sticks dard quantum theory). Interestingly, this can be made to standard Hilbert-space quantum theory (i.e. without consistent [19, 20], though at the cost of some counter- measurements). But logical consistency is only a neces- sary condition for a physics theory. Solipsism is another example ofa logicallyconsistenttheory,somehowonthe other extreme to many-worlds: in solipsism only “I” ex- 4 Don’taskmewhatthatmeans. ists. But, as I stated in the previous section, I am a 5 or,equivalently, inthefinalstateoftheUniverse. realist: I just don’t see how one can do physics without 6 recallthattheassumptionthatrealnumbersarephysicallyreal impliesthattherecouldbeaninfiniteamountsofinformationin assuming the “I” and the “world”. afinitevolumeofspace[14,15]. Letmeaddressanotherissuewithmany-worlds. Itisa 7 or one collection of events that are singled-out. We may name deterministic theory, even a hyper-deterministic theory, thiscollectionasone“big”or“composed” event. 4 intuitive phenomena [21, 22] and assuming it applies to one. But is it fundamentally different? the entire universe (as soon as one cuts out some piece Let’sreturntoBohmianquantummechanics. As said, of the universe, one may encounter paradoxes [4]). itisaremarkableexistenceproofofnon-localhiddenvari- Note that one may also apply similar ideas to other ables. But does it answerthe deep questionof the quan- physical quantities than position, leading to various tummeasurementproblem? Idon’tthinkso. Aswiththe modal interpretations of quantum theory [23, 24]. With many-worlds,itassumeshyper-determinismandrelieson position as the special physical quantity, the reader has infinitesimaldigitsforits predictions. Hence,despitethe recognized Bohmian quantum mechanics [19, 20]. It is deterministic equations, it is not a deterministic theory a nice existence proof of non-local hidden variables that [14], as I elaborate in the next section. Moreover and deserves to be more widely known [25]. It is non-local disappointingly, it doesn’t make any new prediction. despite the fact that the additional variables are points in space,i.e. highly localised. But the dynamics ofthese point-particles is non-local: be acting here one can in- V. NEWTONIAN DETERMINISM stantaneouslyinfluencethetrajectoriesofpoint-particles there,ata distance. Atfirst,this mightbe consideredas Some readers may wonder whether I would also have quiteodd. Butquantumphysicsisnon-local,inthesense argued against classical Newtonian mechanics. After of violating Bell inequalities. Hence, the non-locality of all, it also relies on deterministic equations and New- Bohmian mechanics is quite acceptable. Actually, there ton’s universal gravitation theory is also non-local; and is just no choice: in order to recover the predictions of what about the “I”. Let me start with the second as- quantum theory and the experimental data, all theories pect,non-locality. Ihavenoproblemwithquantumnon- must incorporate the possibility of Bell inequality viola- locality (the possibility to violate Bell inequalities), be- tions, i.e. some non-locality. cause quantum randomness precisely prevents the possi- One ugly aspect, in my opinion, of Bohmian mechan- bilitytousequantumnon-localitytosendclassicalinfor- ics is that the additional variables must remain hidden 10 mation[27–30]. However,Newton’snon-locality canbe forever. Ifnot,ifonecouldsomehowcollectinformation used,inprinciple,tosendinformationwithoutanyphys- about their locations beyond the statistical predictions ical support carrying this information: move a rock on of quantum theory, then one could activate non-locality, the moon (with a small rocket) and measure the grav- i.e. one could use entanglementnot only to violate some itational field on earth. According to Newton’s theory Bell inequality, but to send classical information at an this allows one to communicate in a non-physical way - arbitrarilylargespeed[26]. Butcanoneaddvariablesto i.e. without any physical stuff carrying the information a physics theory while claiming that they are ultimately - and at an arbitrarily high speed [31]. This is deeply not accessible? Bohmians answer that the hidden posi- disturbing and did already disturb Newton himself [32]: tions determine the results of measurements, hence are That Gravity should be innate, inherent and essential not entirely hidden. Indeed, when one observes a result, to Matter, so that one Body may act upon another at a onecanapplyaneffectivecollapseasoneknowsthatthe Distance thro a Vacuum, without the mediation of any hidden positions are now distributed within the reduced thing else, by and through which their Action and Force wave-function corresponding to quantum statistics. But maybeconveyedfromonetoanother,istomesogreatan there is no way to know more about the location of that particle8. This raises the question when should one ap- Absurdity, that I believe noMan who has in philosophical Matters a competent Faculty of thinking, can ever fall ply such an effective collapse? The answer presumably into it. is: “when “I” register a measurement result”, a bit like Admittedly, when I first learned about Newton’s uni- in many-worlds. versalgravitationtheoryathighschoolIfounditbeauti- IfindittemptingtocompareBohmianmechanicswith ful,notnoticinghowabsurditis. MayIsuggestthatone a toy-theory in which one has added as additional vari- should always teach Newton together with the comment ablesalltheresultsofallthemeasurementsthatwillever that it is efficient but absurd? I believe this would be be performed in the future, though with the restriction great pedagogy. that none of these additional variables can be access be- TodayweknowthatrelativitysolvedtheissueofNew- fore the corresponding measurements take place. Note ton’s non-locality and that experiments have confirmed that in such a way one can turn any theory into a deterministic one9. But, for sure, no physicists would take quantumnon-localitybeyondanyreasonabledoubts[33– 35]. such a toy-theory seriously. Admittedly, Bohmian me- Let’s now turn to the other similarity between chanics is much more elegant than the above sketched Bohmianmechanics and classicalphysics,that is the deterministic nature of Newton’s equations. For clocks, 8 Moreover,therearesituationsinwhichthehiddenparticleleaves atracewhereitwasnot[21,22]. 9 Notethough,thatwiththesketchedconstruction,timeisneces- 10 which, by the way, predicts the possibility to violate Bell’s in- sarilybuildintothetoy-theory. equality. 5 harmonic oscillators and generally integrable dynamical Clearly, dividing the world out there into “small” and systems,the stability is suchthatthe infinitesimal digits “large” is not good enough. But couldn’t there be stuff of the initial condition do play no role. For chaotic sys- towhichthesuperpositionprincipledoesn’tapply? Some tems, on the contrary, these infinitesimal digits quickly have argued that the hypothetical non-quantum stuff is dominate the dynamics. Hence, since these mathemati- space-time and/or gravity [37–41]. This is certainly a calinfinitesimal digitdonotphysicallyexist, chaoticdy- possibility. ButIamreluctanttoputmybetsonthis,be- namical system are not deterministic. This fact doesn’t causeeverythingis connected to space-timeandto grav- change anything in practice (FAPP, as Bell would have ity. Hence, if it is the coupling between the “quantum shouted [9]), but it demonstrates that classical Newto- stuff” and the hypothetical “non-quantum stuff” that nianmechanicsissimplynotadeterministicphysicsthe- determines when a measurement happens, then, contin- ory: despite the use of deterministic equations, it does uously, everything always undergoes measurements. In not describe deterministic physics [14]. such a case, either the superposition principle is contin- Thereis,however,ahugedifferencebetweenNewton’s uously broken and one should never have seen superpo- determinism (of the equations) and Bohmian or many- sitions, or the non-quantum stuff undergoes a bit of su- worlds. In the former there is no entanglement. Hence, perposition. one can separate the world into systems, hence “I” can Moreformally,denoting|QS0itheinitialstateofsome act on each of them individually. In an enormously en- quantum stuff that interacts with some non-quantum tangled world, on the contrary, there is no way to sepa- stuff |NQS0i, then, after an arbitrary short time the rate sub-systems, there is no way to acton just one sub- quantum and non-quantum stuff get entangled: system. Determinism plus entanglement make things in- t=ǫ tractable[36]. Accordingly,either“I”cannotact,or“I” |QS0i|NQS0i → X|QSji|NQSji (1) do induce effective collapsesthat disentangle the subsys- j tems. Butthen,why notinclude theseeffective collapses But if the non-quantum stuff can’t at all be in superpo- in the theory? sition, then state (1) can’t exist, not even for a split of a second. Hence, there would be instantaneously collapse VI. DUALISM also for the quantum stuff. People have speculated that this bit of superposition getsquickly,thoughnotinstantaneously,washedout[37– In summary, so far we saw 4 sorts of attempts to cir- 41]. Why not. But then, why introduce such a non- cumvent the quantum measurement problem: quantum stuff in the first place? Why not merely as- 1. dualism as in orthodox Copenhagen quantum me- sume that all stuff undergo superpositions, but only in chanics, some (precisely) limited way? Readers recognize here spontaneous collapse theories; more on this section VII. 2. some configurations of atoms and photons make Before closing this section, let’s see whether there is them act as measurement setups that break super- not anotherplausible wayto divide the stuff into several positions, sorts, i.e. dualism12. There is obviously one that goes backallthewaytoDescartes: “materialstuff”and“non- 3. all possible results co-exist in some many-worlds, materialstuff”. Thesuperpositionprinciplewouldapply only to the material stuff. This is admittedly extremely 4. all results were already encode in some additional crude, certainly not yet a theory, not even a valid sketch non-localvariables,hidden for ever,as in Bohmian of a theory, because essentially nothing is saidabout the mechanics. “non-material stuff”. Moreover, one should not make Let me recall that since I consider the quantum mea- the situation more confused by thinking that the “non- surementproblemasarealphysicsproblem,it’ssolution material stuff” is our “mind”, as this would imply that will necessarily lead to new physics, including new and the first measurement that ever happened had to wait testable predictions. It is a fact that so far attempts 2, for us. However,I like to argue that one should also not 3 and 4 did not bring up as much good and new physics reject dualism too quickly. After all, it might well be as attempt 1 did. But I should add that this argument that there is stuff out there to which the superposition might be a bit unfair, because attempt 1 came first and principle does not apply. had thus a significant advantage. Anyway, let’s consider attempt number 1, i.e. dualism. Itisprobablyfairtosaythatmostphysicistswouldre- jectdualism11. Butcoulditbethattheygotoofasthere? is dualist. The sames wouldsimultaneously claim withjoy how proud they aretoworkinafield whererational thinking domi- nates. Ok,Ileavethatlineofthoughts tosociologists. 12 Physics divides “me” and the outside world. I do not consider this as fundamental dualism, but only as the scientific method. 11 Bytheway,manywouldevendosovirulently,whileatthesame HereIamaskingwhetherarealphysicaldualismisaviablepath timeclaimingtoadheretotheCopenhageninterpretation,which towardsaresolutionofthequantum measurementproblem. 6 Let’s return to the quantum measurement problem. + 2hL†i L −L†L −hL†i hL i |ψ idt X(cid:16) j ψt j j j j ψt j ψt(cid:17) t AlthoughIamsortofadualistfromaphilosophicalpoint j of view13, I don’t think that dualism is the right solu- + (L −hL i )|ψ idξ (2) tion for the measurement problem. It might be that in X j j ψt t j somedecades,ifthe problemremainswithoutsignificant j progress,onemayhavetorevisitadualisticsolution,but whereH istheusualHamitonian,L ’sare(Lindbladlin- j atpresentwebettersticktotheassumptionthatthereis one and only one sort of stuff out there in the realworld ear) operators, hLjiψt = hψhψt|tL|jψ|ψtiti are the expectation and that the superposition principle applies to it. values of the operatorsL and the dξ ’s are independent j j complex Wiener processes satisfying: VII. MODIFIED SCHRO¨DINGER EQUATION M[dξj] = 0 (3) M[dξ dξ ] = 0 (4) j k Recallthatthesuperpositionprinciplestates,inwords, M[dξ dξ∗] = δ dt (5) j k jk thatifsomestuffcanbeeitherinonestateorinanother, it can also be in the first state and in the second one, where M[...] denotes the mean value. Note that eq. (2) i.e. in superposition of the two states. The linearity of preserves the norm of |ψ i. t the Schrödinger equation implies then that such super- Equation (2) describes a sort of Brownian motion in position last for ever. Consequently, in a theory without Hilbert space of the state-vector |ψ i. It is the analog of t the measurement problem and in which everything (ex- a stochastic description of Browian motion at the indi- cept “I”) satisfy the superposition principle (and with- vidual particle level. It is assumed that it is not merely out hyper-determinism) it must be the case that it is an approximation, but the foundamental dynamical law the Schrödinger equation that has to be modified. First describing how isolated quantum systems evolve. Hence attemptstomodifytheSchrödingerequationtriedtoex- it predicts deviations from the standard Schrödinger dy- tendittosomenon-linearbutstilldeterministicequation namics, i.e. it predicts new physics. Consequently, at [42–44]. But this turned out to be hopeless, as could be least, such modified dynamical laws could be wrong! expected from the discussions in the previous sections. Note that one unpleasant characteristic of such a mod- A quite convincing argument came from the observation ified dynamics is that the very same equation (2) can that any such deterministic nonlinear generalization of also be derived with standard quantum theory by as- the Schrodinger equation activates non-locality, i.e. pre- suming some coupling between the quantum system and dictsthepossibilityofarbitrarilyfastcommunication[1– its environment when one conditions the system’s state 3]. on some continuous measurement outcomes carried out Hence, one has to go for a non-deterministic gener- on the environment. This makes it highly non trivial alization14. Non-deterministic merely means not deter- to demonstrate an evolution satisfying equation (2) as ministic, that is it does not say how the equation should a fundamental evolution, as one would have to convinc- be, it only says how the equation should not be. How- inglyshowthatthesystemdoesnotinteractsignificantly ever, assuming that the evolution is Markovian and the with its environment. solution continuous in time, then - for those who know To illustrate eq. (2) and for simplicity, let’s consider stochastic differential equations - possibilities are quite the case with a single operator L, furthermore assume easy to find [1, 47–49]. Essentially there is only one [50]. it is self-adjoint and commutes with the Hamiltonian H. This solution depends on some operators, a bit like the Then, interestingly, the solutions to (2) follow a sort of Schrödinger equation depends on the Hamiltonian. At Brownian motion and eventually tends to an eigenstate this point, all that remains is to fix this operator of the |li of L. Moreover,the probability to tend to a given |li new,non-linearandstochastictermofthehypothetically equals the quantum probability |hl|ψ0i|2, see Fig. 1. fundamental dynamical equation of the complemented When averaging over all solutions of (2), i.e. averag- quantum theory and look for the new predictions. ing over all Wiener processes dξ , one obtains a density j Let’sbeabitmoreexplicit. ConsiderthefollowingItoˆ matrix ρ(t) that satisfies the linear evolution equation: stochastic differential equation, see e.g. [47, 51]: dρ(t) = −i[H,ρ(t)] (6) |dψ i = −iH|ψ idt dt t t − L†L ρ(t)+ρ(t)L†L −2L ρ(t)L† X(cid:16) j j j j j j(cid:17) j 13 Idon’tbelievethateverythingismerelymatterandenergy,not Equation (6) is the analog of a Fokker-Planck equation evenstuffdescribedbyanyphysicstheoryatanygivenpointin describing the probability distribution of an ensemble of time. 14 Note that sticking to a linear equation is also hopeless, as the Browian particles. Letus emphasizethatsincethe densitymatricesatall Schrodinger equation is the only linear equation that preserves thenormofthestatevector, seealso[45,46] times follow a closed form equation, this modification of 7 modified Schrödingerequation predicts a quasiinstanta- neous collapse: it suffices that a single particle gets lo- calized by the stochastic nonlinear terms of equation (2) 9 fortheentirepointertolocalize,i.e. thepointerlocalizes 8 about 1020 times faster than individual particles. 7 I remain convinced that collapse models of the form sketched above, see [55] and references there in, is the 6 mber 5 best option we have today to solve the unacceptable Photon nu 4 quantum measurement problem. 3 2 VIII. CONCLUSION 1 I want to understand Nature. For me this requires 0 1 2 3 4 5 Time that “I” exist and that there is something out there to be understood, in particular that there is a world out FIG. 1: Example of some solutions to eq. (2) in case of a there. Physics is all about extracting information about photon-number measurements, i.e. H = L = a†a. The con- How Nature Does It. For physicists extracting informa- vergence to the eigenstates can be clearly seen. Taken from tion means performing measurements. Hence the mea- [51]. surement problem has to be taken seriously. It is a real physicsproblemanditssolutionwillprovidenewphysics and new and testable predictions. the Schrödingerequationdoes notleadto the possibility TakingstandardHilbert-spacequantumtheoryatface of faster than light communication [1, 47]. value, without the vague collapse postulate, leads to the Remain to find what the operators L could be. Here many-worlds: everythingthat canhappenhappens. The j comes the beautiful finding of Ghirardi, Rimini and We- problem, besides hyper-determinism, is that “I” am ex- ber [52]15. Assume the L are proportional to the posi- cluded from the many-worlds. In order to re-introduce j tions of all elementary particles, with a proportionality the“I”,onehastointroducesomeeffectivecollapsesthat coefficient small enough that it barely affects the evolu- happen when “I” interact with the world. Note that tion of systems made out of one or only a few particles. this step is usually not taken explicitly by the many- Hence, microscopic systems would essentially not be af- worlds followers,except when they compute predictions, fected by the modified Schrödinger equation (2). How- i.e. whentheydophysics. Thisisabitsimilartothewell- ever,if a pointer is in superposition ofpointing here and known Wigner friend story [56], though Wigner never pointing there, then, since the pointer is made out of an presented it in a many-worlds context. Hence, it seems enormous number of particles, let’s say about 1020, the thatinordertomakephysicalsenseofmany-worlds,one needs some form of dualism: “I” trigger effective collapses. Before me, everything co-existed. Now that I am here, in order to make predictions, I have to condi- 15 In 1988 Professor Alberto Rimini visited Geneva to present a tion these “co-existing things”, on those that correlated colloquium. He presented the famous GRW paper [52] in the to me, using some effective collapses. Since it is a fact version Bell gave of it [53]. In the GRW theory, the non-linear that “I” exist, wouldn’t it be much simpler and cleaner stochastic terms added to the Schro¨diger equation lead to solu- to assume that the effective collapses are truly real and tionswithdiscontinuous jumpsof thewave-packet, i.e. tosome to include them in our physics theory? sortofspontaneouscollapsestriggeredbynothingbutmereran- dom chance, as time passes. Near the end of his colloquium, Bohmian mechanics is a nice and constructive exis- Rimini mentioned that an open question was to massage the tence proof of non-local hidden variables. But it suf- stochastic modifications insuch a way that the solutions would fers from similar drawbacks than the many-worlds. It is becontinuoustrajectories(inHilbertspace). Healsoemphasized hyper-deterministicandinordertomakepredictionsone the need for an equation that would preserve (anti-)symmetric has to introduce an “I” that does some conditioning by states. He may have added that, with Philip Pearle [54], they have a solution, but for sure he had no time to explain it. Im- de facto effective collapses. Hence, again, it is cleaner to mediately after the colloquium I went to Alberto and told him assume real collapses in our physics theory. Moreover, thatIknewhowtoanswerhisquestions. Heencouragedmeand doing so we may at least be wrong, i.e. at least we may I immediately added a small section to a paper already quasi- predict new phenomena. finished [1]. There is no doubt that Philip Pearle found CSL independently. Lajos Diosi, by the way, did also find it [48]. Remains the question of what triggers the collapses. Actually, everyone who, at that time, knew both GRW and Itoˆ Should we formulate the measurement problem as a stochasticdifferentialcalculuswouldhavefoundit,becauseitis searchforthoseconfigurationsofatomsandphotonsthat quitetrivial,onceyouknowthetoolsandtheproblem. Anyway, trigger a collapse, as formulated in section I? This is an Ghirardi and Pearle got very angry that I published my result interesting line of experimental research. first and I decided to leave that field. I didn’t like fights and wantedacarrier. Dualism is a very natural position in our culture. Ac- 8 tually,Idon’tseehowtoavoiditforourSciencetomake without collapses, doesn’t predict any events, hence has sense. But I believe much premature to jump to the zero explanatory power. conclusion that it is the interaction between “I” and the One additional value of collapse theories is that they outside world that triggers the collapses of the quantum naturally incorporate the passage of time. I am well states. Other forms of dualism, actually trialism: “I” aware that it is fashion in physics to claim that time is plus two sorts of stuff out there, are logical possibilities, anillusion[57]. Admittedly, timeisacomplexnotion,or but there are no good candidates and introducing some series of notions with many facets, time may be relative, new stuff seems too high a price to pay, especially when difficult to grasp, etc. But time exists. Moreover, time it is not (yet?) needed. passes [14, 16, 17]. Remains spontaneous collapses, described for instance With spontaneous collapse theories, time exists and bysomemodifiedScho¨dingerequationtowhichoneadds passes, the world out there exists and undergoes a somenon-linearstochasticterms. Theseadditionalterms stochastic evolution. And “I” exist, outside the theory, lead continuously and spontaneously, i.e. by mere ran- able to contemplate it, to develop it and act as an ob- dom chance, to collapses that barely affect microscopic server. systems, but quickly localize macroscopic objects. It seems to me that the scientific method that has been that efficient so far tells us that this spontaneous col- Acknowledgment lapseapproachisbyfarthemostpromisingone. Forme, it is also the only one that is consistent with what I ex- This work profited from stimulating discussions with Flo- pectfromphysics. Indeed,attheendoftheday,atheory rian Fröwis and Renato Renner. [1] N.Gisin, “Stochastic quantumdynamicsand relativity” [17] L. 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