Collapse. What else?

0.19 MB·

by  Nicolas Gisin

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
Schr¨odingerequation.

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 super-
position 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 Schr¨odinger [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 noth-
ing, 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 uni-
verse. Indeed, sincethere can’tbe anyinfluences coming
from outside and since the Schr¨odinger 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
Schr¨odinger 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 de-
terministic 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 de-
terministic 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 Schr¨odinger 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 Schr¨odinger equation that has to be modified. First describing how isolated quantum systems evolve. Hence
attemptstomodifytheSchr¨odingerequationtriedtoex- it predicts deviations from the standard Schr¨odinger 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
Schr¨odinger 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 Schr¨odingerequation 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 Schr¨odingerequationdoes 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 Schr¨odinger 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 col-
lapses. 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 Fr¨owis and Renato Renner.
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