John Bell - Against Measurement

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Against ‘measurement’John Bell

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Physics WorldAugust 1990

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Physics World August 1990 3 3

Uncertainty over terms such as 'apparatus'is still rife in serious discussions of quantum mechanics,

over 60 years after its conception

Against'measurement'

SURELY, after 62years, we should havean exact formulationof some serious part ofquantum mechanics?By 'exact' I do not ofcourse mean 'exactlytrue'. I mean only thatthe theory should befully formulated inmathematical terms, with nothing left to the discretion ofthe theoretical physicist . . . until workable approximationsare needed in applications. By 'serious' I mean that somesubstantial fragment of physics should be covered. Nonrela-tivistic 'particle' quantum mechanics, perhaps with theinclusion of the electromagnetic field and a cut-off interac-tion, is serious enough. For it covers 'a large part of physicsand the whole of chemistry' (P A M Dirac 1929 Proc. R.Soc. A 123 714). I mean too, by 'serious', that 'apparatus'should not be separated off from the rest of the world intoblack boxes, as if it were not made of atoms and not ruledby quantum mechanics.

The question, '. . . should we not have an exactformulation . . . ? ' , is often answered by one or both of twoothers. I will try to reply to them: Why bother? Why not lookit up in a good book?

Why bother?Perhaps the most distinguished of 'why bother?'ers hasbeen Dirac (1963 Sci. American 208 May 45). He dividedthe difficulties of quantum mechanics into two classes,those of the first class and those of the second. The second-class difficulties were essentially the infinities of relativisticquantum field theory. Dirac was very disturbed by these,and was not impressed by the 'renormalisation' proceduresby which they are circumvented. Dirac tried hard toeliminate these second-class difficulties, and urged others todo likewise. The first-class difficulties concerned the role ofthe 'observer', 'measurement', and so on. Dirac thoughtthat these problems were not ripe for solution, and shouldbe left for later. He expected developments in the theorywhich would make these problems look quite different. Itwould be a waste of effort to worry overmuch about themnow, especially since we get along very well in practicewithout solving them.

Dirac gives at least this much comfort to those who aretroubled by these questions: he sees that they exist and aredifficult. Many other distinguished physicists do not. Itseems to me that it is among the most sure-footed ofquantum physicists, those who have it in their bones, thatone finds the greatest impatience with the idea that the'foundations of quantum mechanics' might need some

JOHN BELL

This article is published with the permission of PlenumPublishing, New York; it will appear in the proceedings of 62Years of Uncertainty (Erice, 5—14 August 1989).

attention. Knowingwhat is right by in-stinct, they can be-come a little impatientwith nitpicking dis-tinctions betweentheorems and assump-tions. When they doadmit some ambiguityin the usual formula-

tions, they are likely to insist that ordinary quantummechanics is just fine 'for all practical purposes'. I agreewith them about that: ORDINARY QUANTUMMECHANICS (as far as I know) IS JUST FINE FOR ALLPRACTICAL PURPOSES.

Even when I begin by insisting on this myself, and incapital letters, it is likely to be insisted on repeatedly in thecourse of the discussion. So it is convenient to have anabbreviation for the last phrase: FOR ALL PRACTICALPURPOSES = FAPP.

I can imagine a practical geometer, say an architect, beingimpatient with Euclid's fifth postulate, or Playfair's axiom:of course in a plane, through a given point, you can drawonly one straight line parallel to a given straight line, at leastFAPP. The reasoning of such a natural geometer might notaim at pedantic precision, and new assertions, known in thebones to be right, even if neither among the originally statedassumptions nor derived from them as theorems, mightcome in at any stage. Perhaps these particular lines in theargument should, in a systematic presentation, be disting-uished by this label - FAPP - and the conclusionslikewise: QED FAPP.

I expect that mathematicians have classified such fuzzylogics. Certainly they have been much used by physicists.

But is there not something to be said for the approach ofEuclid? Even now that we know that Euclidean geometryis (in some sense) not quite true? Is it not good to knowwhat follows from what, even if it is not really necessaryFAPP? Suppose for example that quantum mechanics werefound to resist precise formulation. Suppose that whenformulation beyond FAPP is attempted, we find anunmovable finger obstinately pointing outside the subject,to the mind of the observer, to the Hindu scriptures, toGod, or even only Gravitation? Would not that be very, veryinteresting?

But I must say at once that it is not mathematicalprecision, but physical, with which I will be concernedhere. I am not squeamish about delta functions. From thepresent point of view, the approach of von Neumann's bookis not preferable to that of Dirac's.

Why not look it up in a good book?But which good book? In fact it is seldom that a 'noproblem' person is, on reflection, willing to endorse atreatment already in the literature. Usually the goodunproblematic formulation is still in the head of the person

3 4 Physics World August 1990

'Perhaps the most distinguished of 'why bother?'ers has been PaulDirac1

in question, who has been too busy with practical things toput it on paper. I think that this reserve, as regards theformulations already in the good books, is well founded.For the good books known to me are not much concernedwith physical precision. This is clear already from theirvocabulary.

Here are some words which, however legitimate andnecessary in application, have no place in a formulation withany pretension to physical precision: system, apparatus,environment, microscopic, macroscopic, reversible, irreversible,observable, information, measurement.

The concepts 'system', 'apparatus', 'environment', im-mediately imply an artificial division of the world, and anintention to neglect, or take only schematic account of, theinteraction across the split. The notions of 'microscopic'and 'macroscopic' defy precise definition. So also do thenotions of 'reversible' and 'irreversible'. Einstein said thatit is theory which decides what is 'observable'. I think hewas right - 'observation' is a complicated and theory-ladenbusiness. Then that notion should not appear in theformulation of fundamental theory. Information? Whoseinformation? Information about what?

On this list of bad words from good books, the worst ofall is 'measurement'. It must have a section to itself.

Against 'measurement'When I say that the word 'measurement' is even worse thanthe others, I do not have in mind the use of the word inphrases like 'measure the mass and width of the Z boson'.I do have in mind its use in the fundamental interpretiverules of quantum mechanics. For example, here they are asgiven by Dirac (Quantum Mechanics Oxford UniversityPress 1930):

'. . . any result of a measurement of a real dynamical variableis one of its eigenvalues . . . "

'. . . if the measurement of the observable . . . is madea large number of times the average of all the results obtained

will be . . .'

'. . . a measurement always causes the system to jump into aneigenstate of the dynamical variable that is being measured . . .'

It would seem that the theory is exclusively concernedabout 'results of measurement', and has nothing to sayabout anything else. What exactly qualifies some physical

systems to play the role of 'measurer'? Was the wavefunc-tion of the world waiting to jump for thousands of millionsof years until a single-celled living creature appeared? Ordid it have to wait a little longer, for some better qualifiedsystem . . . with a PhD? If the theory is to apply toanything but highly idealised laboratory operations, are wenot obliged to admit that more or less 'measurement-like'processes are going on more or less all the time, more orless everywhere? Do we not have jumping then all the time?

The first charge against 'measurement', in the fundamen-tal axioms of quantum mechanics, is that it anchors therethe shifty split of the world into 'system' and 'apparatus'.A second charge is that the word comes loaded withmeaning from everyday life, meaning which is entirelyinappropriate in the quantum context. When it is said thatsomething is 'measured' it is difficult not to think of theresult as referring to some pre-existing property of theobject in question. This is to disregard Bohr's insistencethat in quantum phenomena the apparatus as well as thesystem is essentially involved. If it were not so, how couldwe understand, for example, that 'measurement' of acomponent of 'angular momentum' — in an arbitrarilychosen direction — yields one of a discrete set of values?When one forgets the role of the apparatus, as the word'measurement' makes all too likely, one despairs of ordinarylogic - hence 'quantum logic'. When one remembers therole of the apparatus, ordinary logic is just fine.

In other contexts, physicists have been able to take wordsfrom everyday language and use them as technical termswith no great harm done. Take for example, the 'strange-ness', 'charm', and 'beauty' of elementary particle physics.No one is taken in by this 'baby talk', as Bruno Touschekcalled it. Would that it were so with 'measurement'. But infact the word has had such a damaging effect on thediscussion, that I think it should now be banned altogetherin quantum mechanics.

The role of experimentEven in a low-brow practical account, I think it would begood to replace the word 'measurement', in the formula-tion, by the word 'experiment'. For the latter word isaltogether less misleading. However, the idea that quantummechanics, our most fundamental physical theory, isexclusively even about the results of experiments wouldremain disappointing.

In the beginning natural philosophers tried to understandthe world around them. Trying to do that they hit upon thegreat idea of contriving artificially simple situations inwhich the number of factors involved is reduced to aminimum. Divide and conquer. Experimental science wasborn. But experiment is a tool. The aim remains: tounderstand the world. To restrict quantum mechanics to beexclusively about piddling laboratory operations is to betraythe great enterprise. A serious formulation will not excludethe big world outside the laboratory.

The quantum mechanics of Landau andLifshitzLet us have a look at the good book Quantum Mechanics byL D Landau and E M Lifshitz. I can offer three reasons forthis choice:

(i) It is indeed a good book.(ii) It has a very good pedigree. Landau sat at the feet of

Bohr. Bohr himself never wrote a systematic accountof the theory. Perhaps that of Landau and Lifshitz is thenearest to Bohr that we have.

(iii) It is the only book on the subject in which I have


read every word.This last came about because my friend John Sykes

enlisted me as technical assistant when he did the Englishtranslation. My recommendation of this book has nothingto do with the fact that one per cent of what you pay for itcomes to me.

LL emphasise, following Bohr, that quantum mechanicsrequires for its formulation 'classical concepts' - a classicalworld which intervenes on the quantum system, and inwhich experimental results occur (brackets after quotesrefer to page numbers):

'. . . It is in principle impossible . . . to formulate the basicconcepts of quantum mechanics without using classical

mechanics.' (LL2)

'. . . The possibility of a quantitative description of the motionof an electron requires the presence also of physical objectswhich obey classical mechanics to a sufficient degree of

accuracy.' (LL2)

'. . . the 'classical object' is usually called apparatus and itsinteraction with the electron is spoken of as measurement.However, it must be emphasised that we are here not discussinga process . . . in which the physicist-observer takes part. Bymeasurement, in quantum mechanics, we understand anyprocess of interaction between classical and quantum objects,occurring apart from and independently of any observer. Theimportance of the concept of measurement in quantum

mechanics was elucidated by N Bohr.' (LL2)

And with Bohr they insist again on the inhumanity of it all:

'. . . Once again we emphasise that, in speaking of 'performinga measurement', we refer to the interaction of an electron witha classical 'apparatus', which in no way presupposes the

presence of an external observer.' (LL3)

'. . . Thus quantum mechanics occupies a very unusual placeamong physical theories: it contains classical mechanics as alimiting case, yet at the same time it requires this limiting case

for its own formulation . . . " (LL3)

'. . . consider a system consisting of two parts: a classicalapparatus and an electron . . . The states of the apparatus aredescribed by quasiclassical wavefunctions <£„(£)> where thesuffix n corresponds to the 'reading' gn of the apparatus, and£ denotes the set of its coordinates. The classical nature of theapparatus appears in the fact that, at any given instant, we cansay with certainty that it is in one of the known states $„ withsome definite value of the quantity g; for a quantum system

such an assertion would of course be unjustified.' (LL21)

'. . . Let %(£) be the wavefunction of the initial state of theapparatus . . . and W{q) of the electron . . . the initialwavefunction of the whole system is the product tf)After the measuring process we obtain a sum of the form

where the Att{q) are some functions of q.' (LL22)

'The classical nature of the apparatus, and the double role ofclassical mechanics as both the limiting case and the foundationof quantum mechanics, now make their appearance. As hasbeen said above, the classical nature of the apparatus means

that, at any instant, the quantity g (the 'reading of theapparatus') has some definite value. This enables us to say thatthe state of the system apparatus + electron after themeasurement will in actual fact be described, not by the entiresum, but by only the one term which corresponds to the'reading' gn of the apparatus An(q)<&n(%). It follows from thisthat An(q) is proportional to the wavefunction of the electron

after the measurement . . .' (LL22)

This last is (a generalisation of) the Dirac jump, not anassumption here but a theorem. Note, however, that it hasbecome a theorem only by virtue of another jump beingassumed — that of a 'classical' apparatus into an eigenstateof its 'reading'. It will be convenient later to refer to thislast, the spontaneous jump of a macroscopic system into adefinite macroscopic configuration, as the LL jump. Andthe forced jump of a quantum system as a result of'measurement' — an external intervention — as the Diracjump. I am not implying that these men were the inventorsof these concepts. They used them in references that I cangive.

According to LL (LL24), measurement (I think theymean the LL jump)'. . . brings about a new state . . . Thusthe very nature of the process of measurement involves afar-reaching principle of irreversibility . . . causes the twodirections of time to be physically non-equivalent, i.e.creates a difference between the future and the past.'

The LL formulation, with vaguely defined wavefunctioncollapse, when used with good taste and discretion, isadequate FAPP. It remains that the theory is ambiguous inprinciple, about exactly when and exactly how the collapseoccurs, about what is microscopic and what is macroscopic,what quantum and what classical. We are allowed to ask: issuch ambiguity dictated by experimental facts? Or couldtheoretical physicists do better if they tried harder?

The quantum mechanics of K GottfriedThe second good book that we will look at here is that ofKurt Gottfried (Quantum Mechanics Benjamin 1966). AgainI can give three reasons for this choice:

(i) It is indeed a good book. The CERN library had fourcopies. Two have been stolen - already a good sign. Thetwo that remain are falling apart from much use.

(ii) It has a very good pedigree. Kurt Gottfried wasinspired by the treatments of Dirac and Pauli. His personalteachers were J D Jackson, J Schwinger, V F Weisskopfand J Goldstone. As consultants he had P Martin, CSchwartz, W Furry and D Yennie.

(iii) I have read some of it more than once.

This last came about as follows. I have often had thepleasure of discussing these things with Viki Weisskopf.Always he would end up with 'you should read KurtGottfried'. Always I would say 'I have read Kurt Gottfried'.But Viki would always say again next time 'you should readKurt Gottfried'. So finally I read again some parts of KG,and again, and again, and again.• At the beginning of the book there is a declaration ofpriorities (KG1): '. . . The creation of quantum mechanicsin the period 1924-28 restored logical consistency to itsrightful place in theoretical physics. Of even greaterimportance, it provided us with a theory that appears to bein complete accord with our empirical knowledge of allnonrelativistic phenomena . . .'

The first of these two propositions, admittedly the lessimportant, is actually given rather little attention in thebook. One can regret this a bit, in the rather narrow context


of the particular present enquiry - into the possibility ofprecision. More generally, KG's priorities are those of allright-thinking people.

The book itself is above all pedagogical. The student istaken gently by the hand, and soon finds herself or himselfdoing quantum mechanics, without pain - and almostwithout thought. The essential division of KG's world intosystem and apparatus, quantum and classical, a notion thatmight disturb the student, is gently implicit rather thanbrutally explicit. No explicit guidance is then given as tohow in practice this shifty division is to be made. Thestudent is simply left to pick up good habits by beingexposed to good examples.

KG declares that the task of the theory is (KG 16)'. . . topredict the results of measurements on the system . . .' Thebasic structure of KG's world is then W = S + R where 5is the quantum system, and R is the rest of the world —from which measurements on 5 are made. When our onlyinterpretive axioms are about measurement results (orfindings (KG11)) we absolutely need such a base R fromwhich measurements can be made. There can be noquestion then of identifying the quantum system S with thewhole world W. There can be no question - withoutchanging the axioms — of getting rid of the shifty split.Sometimes some authors of 'quantum measurement'theories seem to be trying to do just that. It is like a snaketrying to swallow itself by the tail. It can be done - up toa point. But it becomes embarrassing for the spectators evenbefore it becomes uncomfortable for the snake.

But there is something which can and must be done — toanalyse theoretically not removing the split, which cannot bedone with the usual axioms, but shifting it. This is taken upin KG's chapter 4: 'The Measurement Process . . . " Surely'apparatus' can be seen as made of atoms? And it oftenhappens that we do not know, or not well enough, either apriori or by experience, the functioning of some system thatwe would regard as 'apparatus'. The theory can help us withthis only if we take this 'apparatus' A out of the rest of theworld R and treat it together with 5 as part of an enlargedquantum system S': R=A + R'; S + A = S'; W=S' + R'.The original axioms about 'measurement' (whatever theywere exactly) are then applied not at theS/A interface, but at the AIR' interface — where for somereason it is regarded as more safe to do so. In real life itwould not be possible to find any such point of divisionwhich would be exactly safe. For example, strictly speakingit would not be exactly safe to take it between the counters,say, and the computer - slicing neatly through some of theatoms of the wires. But with some idealisation, which might'. . . be highly stylised and not do justice to the enormouscomplexity of an actual laboratory experiment . . . "(KG165), it might be possible to find more than one not tooimplausible way of dividing the world up. Clearly it isnecessary to check that different choices give consistentresults (FAPP). A disclaimer towards the end of KG'schapter 4 suggests that that, and only that, is the modestaim of that chapter (KG189): '. . . we emphasise that ourdiscussion has merely consisted of several demonstrationsof internal consistency . . . " But reading reveals otherambitions.

Neglecting the interaction of A with R', the joint systemS' = S + A is found to end, in virtue of the Schrodingerequation, after the 'measurement' on 5 by A, in a state

where the states tyn are supposed each to have a definite

apparatus pointer reading gn. The corresponding densitymatrix is

4 Un m

At this point KG insists very much on the fact that A, andso S', is a macroscopic system. For macroscopic systems,he says, (KG186) '. . . trAp = trAp for all observables Aknown to occur in nature . . . " where

P =

i.e. p is obtained from p by dropping interference termsinvolving pairs of macroscopically different states. Then(KG188) '. . . we are free to replace p by p after themeasurement, safe in the knowledge that the error willnever be found . . . "

Now, while quite uncomfortable with the concept 'allknown observables', I am fully convinced of the practicalelusiveness, even the absence FAPP, of interference be-tween macroscopically different states (J S Bell and MNauenberg 1966 'The moral aspects of quantum mechanics'in Preludes in Theoretical Physics North-Holland). So let usgo along with KG on this and see where it leads: '. . . If wetake advantage of the indistinguishability of p and p tosay that p is the state of the system subsequent tomeasurement, the intuitive interpretation of cm as a proba-bility amplitude emerges without further ado. This isbecause cm enters p only via | c j 2 , and the latter quantityappears in p in precisely the same manner as probabilitiesdo in classical statistical physics . . .'

I am quite puzzled by this. If one were not actually onthe lookout for probabilities, I think the obvious interpreta-tion of even p would be that the system is in a state inwhich the various ^ s somehow coexist: ^ P ^ * and ^ 2 ^ 2and . . .

This is not at all a probability interpretation, in which thedifferent terms are seen not as coexisting, but as alterna-tives: ^P,^* or ^2^2* or . . .

The idea that elimination of coherence, in one way oranother, implies the replacement of 'and' by 'or', is a verycommon one among solvers of the 'measurement problem'.It has always puzzled me.

It would be difficult to exaggerate the importanceattached by KG to the replacement of p by p: '. . . Tothe extent that nonclassical interference terms (such as cmc%)are present in the mathematical expression for p . . . thenumbers cm are intuitively uninterpretable, and the theoryis an empty mathematical formalism . . .' (KG 187)

But this suggests that the original theory, 'an emptymathematical formalism', is not just being approximated —but discarded and replaced. And yet elsewhere KG seemsclear that it is in the business of approximation that he isengaged, approximation of the sort that introduces irrever-sibility in the passage from classical mechanics to thermo-dynamics: '. . . In this connection one should note that inapproximating p by p one introduces irreversibility,because the time-reversed Schrodinger equation cannotretrieve p from p.' (KG188)

New light is thrown on KG's ideas by a recent recapitula-tion, referred to in the following as KGR (K Gottfried 'Doesquantum mechanics describe the collapse of the wavefunc-tion?' Presented at 62 Years of Uncertainty, Erice, 5-14August 1989). This is dedicated to the proposition that(KGR1) '. . . the laws of quantum mechanics yield the


results of measurements . . . "These laws are taken to be(KGR1): '(1) a pure state isdescribed by some vector inHilbert space from which ex-pectation values of observ-ables are computed in thestandard way; and (2) thetime evolution is a unitarytransformation on that vector'(KGR1). Not included in thelaws is (KGR1) vonNeumann's '. . . infamouspostulate: the measurementact 'collapses' the state intoone in which there are nointerference terms betweendifferent states of the mea-surement apparatus . . . " In-deed, (KGR1) 'the reductionpostulate is an ugly scar onwhat would be a beautifultheory if it could be removed

Perhaps it is useful to recallhere just how the infamouspostulate is formulated byvon Neumann (J vonNeumann 1955 MathematicalFoundations of QuantumMechanics Princeton University Press). If we look back wefind that what vN actually postulates (vN347, 418) is that'measurement' — an external intervention by R on 5 —causes the state

$ = X cn<l>n

to jump, with various probabilities into fy1 or <\>2 or . . .From the 'or' here, replacing the 'and', as a result of

external intervention, vN infers that the resulting densitymatrix, averaged over the several possibilities, has nointerference terms between states of the system whichcorrespond to different measurement results (vN347). Iwould emphasise several points here.

(i) von Neumann presents the disappearance of coherencein the density matrix, not as a postulate, but as aconsequence of a postulate. The postulate is made at thewavefunction level, and is just that already made by Diracfor example.

(ii) I cannot imagine von Neumann arguing in theopposite direction, that lack of interference in the densitymatrix implies, without further ado, 'or' replacing 'and' atthe wavefunction level. A special postulate to that effectwould be required.

(iii) von Neumann is concerned here with what happensto the state of the system that has suffered the measurement— an external intervention. In application to the extendedsystem S'(= S + A) von Neumann's collapse would notoccur before external intervention from R'. It would besurprising if this consequence of external intervention on S'could be inferred from the purely internal Schrodingerequation for S'. Now KG's collapse, although justified byreference to 'all known observables' at the S'/R' interface,occurs after 'measurement' by A on S, but before interac-tion across S'/R'. Thus the collapse which KG discusses isnot that which von Neumann infamously postulates. It is

John von Neumann - ' . . . infamous postulate: the measurement act'collapses' the state into one in which there are no interference termsbetween different states of the measurement apparatus

the LL collapse rather thanthat of von Neumann andDirac.

The explicit assumptionthat expectation values are tobe calculated in the usual waythrows light on the subse-quent falling out of the usualprobability interpretation'without further ado'. For therules for calculating expecta-tion values, applied to projec-tion operators for example,yield the Born probabilitiesfor eigenvalues. The mysteryis then: what has the authoractually derived rather thanassumed? And why does heinsist that probabilitiesappear only after the butcher-ing of p into p, the theoryremaining an 'empty mathe-matical formalism' so long asp is retained? Dirac, vonNeumann, and the others,nonchalantly assumed theusual rules for expectationvalues, and so probabilities,in the context of the unbutch-ered theory. Reference to the

usual rules for expectation values also makes clear whatKG's probabilities are probabilities of. They are probabilitiesof 'measurement' results, of external results of externalinterventions, from R' on S' in the application. We mustnot drift into thinking of them as probabilities of intrinsicproperties of S' independent of, or before, 'measurement'.Concepts like that have no place in the orthodox theory.

Having tried hard to understand what KG has written, Iwill finally permit myself some guesses about what he mayhave in mind. I think that from the beginning KG tacitlyassumes the Dirac rules at S'/R' - including the Dirac-vonNeumann jump, required to get the correlations betweenresults of successive (moral) measurements. Then, for 'allknown observables', he sees that the 'measurement' resultsat S'/R' are AS IF (FAPP) the LL jump had occurred inS'. This is important, for it shows how, FAPP, we can getaway with attributing definite classical properties to 'appar-atus' while believing it to be governed by quantummechanics. But a jump assumption remains. LL derived theDirac jump from the assumed LL jump. KG derives,FAPP, the LL jump from assumptions at the shifted splitR'/S' which include the Dirac jump there.

It seems to me that there is then some conceptual drift inthe argument. The qualification 'as if (FAPP)' is dropped,and it is supposed that the LL jump really takes place. Thedrift is away from the 'measurement' ( . . . externalintervention . . .) orientation of orthodox quantum mecha-nics towards the idea that systems, such as S' above, haveintrinsic properties - independently of and before observa-tion. In particular the readings of experimental apparatusare supposed to be really there before they are read. Thiswould explain KG's reluctance to interpret the unbutchereddensity matrix p, for the interference terms there couldseem to imply the simultaneous existence of differentreadings. It would explain his need to collapse p into p,in contrast with von Neumann and the others, withoutexternal intervention across the last split S'/R'. It wouldexplain why he is anxious to obtain this reduction from the


internal Schrodinger equation of S'. (It would not explainthe reference to 'all known observables' - at the S'/R'split.) The resulting theory would be one in which some'macroscopic' 'physical attributes' have values at all times,with a dynamics that is related somehow to the butcheringof p into p — which is seen as somehow not incompatiblewith the internal Schrodinger equation of the system. Sucha theory, assuming intrinsic properties, would not needexternal intervention, would not need the shifty split. Butthe retention of the vague word 'macroscopic' would reveallimited ambition as regards precision. To avoid the vague'microscopic' 'macroscopic' distinction — again a shifty split— I think one would be led to introduce variables whichhave values even on the smallest scale. If the exactness ofthe Schrodinger equation is maintained, I see this leadingtowards the picture of de Broglie and Bohm.

The quantum mechanics of N G van KampenLet us look at one more good book, namely Physica A 153(1988), and more specifically at the contribution: 'Tentheorems about quantum mechanical measurements', by NG van Kampen. This paper is distinguished especially byits robust common sense. The author has no patience with'. . . such mind-boggling fantasies as the many worldinterpretation . . . " (vK98). He dismisses out of hand thenotion of von Neumann, Pauli, Wigner — that 'measure-ment' might be complete only in the mind of the observer:'. . . I find it hard to understand that someone who arrivesat such a conclusion does not seek the error in his argument'(vKlOl). For vK '. . . the mind of the observer is irrelevant. . . the quantum mechanical measurement is terminatedwhen the outcome has been macroscopically recorded . . . "(vKlOl). Moreover, for vK, no special dynamics comes intoplay at 'measurement': '. . . The measuring act is fullydescribed by the Schrodinger equation for object systemand apparatus together. The collapse of the wavefunction isa consequence rather than an additional postulate . . . "(vK97).

After the measurement the measuring instrument,according to the Schrodinger equation, will admittedly bein a superposition of different readings. For example,Schrodinger's cat will be in a superposition \cat> = a\life>+ b\death>. And it might seem that we do have to deal with'and' rather than 'or' here, because of interference: '. . . forinstance the temperature of the cat . . . the expectationvalue of such a quantity G . . . is not a statistical averageof the values G« and Gdd with probabilities \a\2 and \b\2, butcontains cross terms between life and death . . . " (vK103).

But vK is not impressed: 'The answer to this paradox isagain that the cat is macroscopic. Life and death aremacrostates containing an enormous number of eigenstates|/> and \d> . . .

\cat> = X a,\l>

. . . the cross terms in the expression for <G> . . . as thereis such a wealth of terms, all with different phases andmagnitudes, they mutually cancel and their sum practicallyvanishes. This is the way in which the typical quantummechanical interference becomes inoperative betweenmacrostates . . .' (vK103).

This argument for no interference is not, it seems to me,by itself immediately convincing. Surely it would bepossible to find a sum of very many terms, with different

amplitudes and phases, which is not zero? However, I amconvinced anyway that interference between macro-scopically different states is very, very elusive. Grantingthis, let me try to say what I think the argument to be, forthe collapse as a 'consequence' rather than an additionalpostulate.

The world is again divided into 'system', 'apparatus', andthe rest: W = S + A + R' = S' + R'. At first, the usualrules for quantum 'measurements' are assumed at the S'/R'interface - including the collapse postulate, which dictatescorrelations between results of 'measurements' made atdifferent times. But the 'measurements' at S'/R' which canactually be done, FAPP, do not show interference betweenmacroscopically different states of S'. It is ax t / the 'and' inthe superposition had already, before any such measure-ments, been replaced by 'or'. So the 'and' has already beenreplaced by 'or'. It is as if it were so . . . so it is so.

This may be good FAPP logic. If we are more pedantic,it seems to me that we do not have here the proof of atheorem, but a change of the theory — at a strategically wellchosen point. The change is from a theory which speaksonly of the results of external interventions on the quantumsystem, S' in this discussion, to one in which that systemis attributed intrinsic properties — deadness or aliveness inthe case of cats. The point is strategically well chosen in thatthe predictions for results of 'measurements' across S'/R'will still be the same . . . FAPP.

Whether by theorem or by assumption, we end up witha theory like that of LL, in which superpositions ofmacroscopically different states decay somehow into one ofthe members. We can ask as before just how and how oftenit happens. If we really had a theorem, the answers to thesequestions would be calculable. But the only possibility ofcalculation in schemes like those of KG and vK involvesshifting further the shifty split - and the questions with it.

For most of the paper, vK's world seems to be the pettyworld of the laboratory, even one that is not treated veryrealistically: '. . . in this connection the measurement isalways taken to be instantaneous . . .' (vKlOO)

But almost at the last moment a startling new vista opensup - an altogether more vast one:

'Theorem IX: The total system is described throughout by thewave vector W and has therefore zero entropy at all times . . .

This ought to put an end to speculations about measure-ments being responsible for increasing the entropy of theuniverse. (It won't of course.)' (vKl l l )

So vK, unlike many other very practical physicists, seemswilling to consider the universe as a whole. His universe,or at any rate some 'total system', has a wavefunction, andthat wavefunction satisfies a linear Schrodinger equation. Itis clear, however, that this wavefunction cannot be thewhole story of vK's totality. For it is clear that he expectsthe experiments in his laboratories to give definite results,and his cats to be dead or alive. He believes then in variablesX which identify the realities, in a way which thewavefunction, without collapse, can not. His completekinematics is then of the de Broglie-Bohm 'hidden variable'dual type: (¥(t,«r), X(t)).

For the dynamics, he has exactly the Schrodingerequation for ^ , but I do not know exactly what he has inmind for the X, which for him would be restricted to some'macroscopic' level. Perhaps indeed he would prefer toremain somewhat vague about this, for

'Theorem IV: Whoever endows W with more meaning than isneeded for computing observable phenomena is responsible forthe consequences . . .' (vK99)


Towards a precise quantum mechanics

In the beginning, Schrodinger tried to interpret his wave-function as giving somehow the density of the stuff of whichthe world is made. He tried to think of an electron asrepresented by a wavepacket - a wavefunction appreciablydifferent from zero only over a small region in space. Theextension of that region he thought of as the actual size ofthe electron - his electron was a bit fuzzy. At first hethought that small wavepackets, evolving according to theSchrodinger equation, would remain small. But that waswrong. Wavepackets diffuse, and with the passage of timebecome indefinitely extended, according to the Schrodingerequation. But however far the wavefunction has extended,the reaction of a detector to an electron remains spotty. SoSchrodinger's 'realistic' interpretation of his wavefunctiondid not survive.

Then came the Born interpretation. The wavefunctiongives not the density of stuff, but gives rather (on squaringits modulus) the density of probability. Probability of what,exactly? Not of the electron being there, but of the electronbeing found there, if its position is 'measured'.

Why this aversion to 'being' and insistence on 'finding'?The founding fathers were unable to form a clear picture ofthings on the remote atomic scale. They became very awareof the intervening apparatus, and of the need for a 'classical'base from which to intervene on the quantum system. Andso the shifty split.

The kinematics of the world, in this orthodox picture, isgiven by a wavefunction (maybe more than one?) for thequantum part, and classical variables — variables whichhave values - for the classical part: ($?(t,q . . .), X(t)

). The Xs are somehowmacroscopic. This is not spelled outvery explicitly. The dynamics is notvery precisely formulated either. It in-cludes a Schrodinger equation for thequantum part, and some sort of classicalmechanics for the classical part, and'collapse' recipes for their interaction.

It seems to me that the only hope ofprecision with the dual (^ , x) kinema-tics is to omit completely the shiftysplit, and let both M*1 and x refer to theworld as a whole. Then the xs must notbe confined to some vague macroscopicscale, but must extend to all scales. Inthe picture of de Broglie and Bohm,every particle is attributed a positionx(t). Then instrument pointers —assemblies of particles have positions,and experiments have results. Thedynamics is given by the worldSchrodinger equation plus precise'guiding' equations prescribing how thex(t)s move under the influence of ^ .Particles are not attributed angularmomenta, energies, etc, but only posi-tions as functions of time. Peculiar'measurement' results for angularmomenta, energies, and so on, emergeas pointer positions in appropriate ex-perimental setups. Considerations ofthe KG and vK type, on the absence(FAPP) of macroscopic interference,take their place here, and an importantone, in showing how usually we do nothave (FAPP) to pay attention to the

whole world, but only to some subsystem and can simplifythe wavefunction . . . FAPP.

The Born-type kinematics (% X) has a duality that theoriginal 'density of stuff picture of Schrodinger did not.The position of the particle there was just a feature of thewavepacket, not something in addition. The Landau-Lifshitz approach can be seen as maintaining this simplenondual kinematics, but with the wavefunction compact ona macroscopic rather than microscopic scale. We know,they seem to say, that macroscopic pointers have definitepositions. And we think there is nothing but the wavefunc-tion. So the wavefunction must be narrow as regardsmacroscopic variables. The Schrodinger equation does notpreserve such narrowness (as Schrodinger himself drama-tised with his cat). So there must be some kind of 'collapse'going on in addition, to enforce macroscopic narrowness.In the same way, if we had modified Schrodinger'sevolution somehow we might have prevented the spreadingof his wavepacket electrons. But actually the idea that anelectron in a ground-state hydrogen atom is as big as theatom (which is then perfectly spherical) is perfectly toler-able - and maybe even attractive. The idea that amacroscopic pointer can point simultaneously in differentdirections, or that a cat can have several of its nine lives atthe same time, is harder to swallow. And if we have no extravariables X to express macroscopic definiteness, the wave-function itself must be narrow in macroscopic directions inthe configuration space. This the Landau-Lifshitz collapsebrings about. It does so in a rather vague way, at rathervaguely specified times.

In the Ghiradi-Rimini-Weber scheme (see the box andthe contributions of Ghiradi, Rimini, Weber, Pearle, Gisin

The Ghiradi-Rimini-Weber schemeThe GRW scheme represents a proposalaimed to overcome the difficulties ofquantum mechanics discussed by JohnBell in this article. The GRW model isbased on the acceptance of the fact thatthe Schrodinger dynamics, governing theevolution of the wavefunction, has to bemodified by the inclusion of stochasticand nonlinear effects. Obviously thesemodifications must leave practically un-altered all standard quantum predictionsabout microsystems.

To be more specific, the GRW theoryadmits that the wavefunction, besidesevolving through the standard Hamil-tonian dynamics, is subjected, at randomtimes, to spontaneous processes cor-responding to localisations in space ofthe microconstituents of any physicalsystem. The mean frequency of thelocalisations is extremely small, and thelocalisation width is large on an atomicscale. As a consequence no prediction ofstandard quantum formalism for micro-systems is changed in any appreciableway.

The merit of the model is in the factthat the localisation mechanism is suchthat its frequency increases as the num-ber of constituents of a composite system

increases. In the case of a macroscopicobject (containing an Avogadro numberof constituents) linear superpositions ofstates describing pointers 'pointingsimultaneously in different directions'are dynamically suppressed in extremelyshort times. As stated by John Bell, inGRW 'Schrodinger's cat is not both deadand alive for more than a split second'.

# The original and technically detailedpresentation of GRW can be found in1986 Phys. Rev D 34 470; a brilliant andsimple presentation has been given byJohn Bell in Schrodinger: CentenaryCelebration of a Polymath C W Kilmister(ed) 1987 Cambridge University Pressp41# A general discussion of the conceptualimplications of the scheme can be foundin 1988 Foundation of Physics 81 1# The GRW model has been the objectof many recent papers and a lively debateon its implications is going on. Recentlythe model has been generalised to coverthe case of systems of identical particlesand to meet the requirements of relativis-tic invariance.

G C Ghiradi, A Rimini and T Weber


and Diosi presented at 62 Years of Uncertainty, Erice, 5-14August 1989) this vagueness is replaced by mathematicalprecision. The Schrodinger wavefunction even for a singleparticle, is supposed to be unstable, with a prescribed meanlife per particle, against spontaneous collapse of a pre-scribed form. The lifetime and collapsed extension are suchthat departures of the Schrodinger equation show up veryrarely and very weakly in few-particle systems. But inmacroscopic systems, ax a consequence of the prescribedequations, pointers very rapidly point, and cats are veryquickly killed or spared.

The orthodox approaches, whether the authors thinkthey have made derivations or assumptions, are just fineFAPP — when used with the good taste and discretionpicked up from exposure to good examples. At least tworoads are open from there towards a precise theory, it seemsto me. Both eliminate the shifty split. The de Broglie-Bohm-type theories retain, exactly, the linear wave equa-tion, and so necessarily add complementary variables toexpress the non-waviness of the world on the macroscopicscale. The GRW-type theories have nothing in theirkinematics but the wavefunction. It gives the density (in amultidimensional configuration space!) of stuff. To accountfor the narrowness of that stuff in macroscopic dimensions,the linear Schrodinger equation has to be modified, in theGRW picture by a mathematically prescribed spontaneouscollapse mechanism.

The big question, in my opinion, is which, if either, ofthese two precise pictures can be redeveloped in a Lorentzinvariant way.

'. . . All historical experience confirms that men might notachieve the possible if they had not, time and time again,reached out for the impossible.' Max Weber

'. . . we do not know where we are stupid until we stick ournecks out.' R P Feynman

Further readingJ S Bell and M Nauenberg 1966 The moral aspect of quantummechanics in Prelitdes in theoretical physics (in honour of V FWeisskopf) (North-Holland) 278-286P A M Dirac 1929 Proc. R. Soc. A 123 714P A M Dirac 1948 Quantum mechanics third edn (OxfordUniversity Press)P A M Dirac 1963 Sci. American 208 May 45K Gottfried 1966 Quantum mechanics (Benjamin)K Gottfried Does quantum mechanics describe the collapse of thewavefunction? Presented at 62 Years of Uncertainty, Erice, 5-14August 1989L D Landau and E M Lifshitz 1977 Quantum mechanics third edn(Pergamon)N G van Kampen 1988 Ten theorems about quantum mechanicalmeasurements Physica A 153 97-113J von Neumann 1955 Mathematical foundations of quantum mecha-nics (Princeton University Press)

John Bell is in the theory division, CERN, CH-1211, Geneva,Switzerland i

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John Bell - Against Measurement