Introduction * Jeffrey A. Barrett

,While the standard theory ofquantum mechanics has proven

I jtself remarkably useful, it is apuzzling physical theory. I Wliatmakes it puzzling is that measure-ment occurs as a primitive term.The theory tells us that a physicalsystem evolves one way unless it isbeing measured; in which case, itevolves another way,. a way that ismathematically incompatible with~e first. Consequently, one needs to~ow what constitutes a measure-~ent in order to know how a system;/will evolve. But since measurementis a primitive term, this is just whatthe standard theory fails to tell us.When we supplement the theoryWith our intuitions concerning what

I~vents ought to count as measure-ments, we get very good empirical

for the sort of experi-! so far. But

a point where our intuitionsto be useful, and

a project by sketching the basic prin-ciples of what would eventuallybecome his matrix mechanics.

In January 1926 SchrOdinger(1926a) picked up on de Broglie'sidea that all matter has wave-likeproperties, and he published thefirst in a series of papers thatpresented the mathematical founda-tions of wave mechanics. Using hisnew theory he was able to deriveboth the discrete and the continuousspectra of hydrogen. In June ofthe same year Schrodinger's paper(1926b) containing his famous waveequation was received for publica-tion. He believed that the waveequation provided the basis fora deterministic quantum theorywhere the waves represented some-thing like classical charge-matterdistributions, which he thoughtwould properly replace particlesas the stuff the physical world ismade of: As Born later remem-bered:

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how we ended up with a physicaltheory where measurement occursas a primitive term, the sense inwhich the measurement problem isa serious problem for quantummechanics, and some of the currentoptions for solving it.

The standard theory of quantummechanics was largely developed in. the years 1923 to 1932.2 In 1923 de

Broglie suggested that Einstein'sassociation of photons (light-quanta)with electromagnetic wave phe-nomena ought to be generalizedto include all material particles.He suggested that E - hv (energyequals Planck's constant timeswave-length) holds for a "fictitious"wave associated with electrons aswell as photons (de Broglie, 1923).He proposed that one might con-sequently expect electrons, likephotons, to exhibit diffraction phe-nomena, which eventually turnedout to be right.

In 1925 Heisenberg published apaper where he argued that "it isbetter. . . to admit that the partialagreement of the quantum rules [ofthe old quantum theory] with exper-iment is more or less accidental, and

is known as the to try to develop a quantum theo-Each of the retical mechanics, analogous to the

in this collection is con- classical mechanics in which onlywith some aspect of the relations between observable quan-

As. an intro- tities appear" (Heisenberg, 1925,to these papers, I will very translated and quoted in Pais, 1988,say something here about. p. 253). He then set out on just such

theory for help since itabout what constitutes [ScbrOdinger] believed. . . that be bad

accomplished a return to classicalthinking; be regarded the electron not asa particle but as a density distributiongiven by the square of his wave-function/'Iff. He argued that the idea of particlesand of quantum jumps be given upaltogether; be never faltered in thisconviction. . . . I, bowever, was wit-nessing the fertility of the particleconcept every day. . . and was convincedthat particles could not simply beabolished. A way bad to be found for

problem of finding a co-

1-6,1995.Publishers. Printed in the Netherland.s.

JEFFREY A. BARRETT2,~I

reconciling particles and waves. I saw the

connecting link in the idea of probability

(Born, 1968, p. 35).,

~~1

It was Born who formulated thestandard interpretation of SchrOdin-ger's waves.

Born suggested that quantummechanics keep SchrOdinger's math-ematical formalism but drop hisclassical interpretation. Rather thanreplacing particles as the funda-mental objects that constitute theworld, Born took Schrodinger'swaves to be descriptive of particles,which were then taken to befundamental. After briefly takingSchrodinger's wave-function 'I' tobe a measure of probability,. Bornfinally concluded that it was actuallythe norm squared of the wave-function l'lff that determined theprobability for the presence of aparticle. Born immediately realizedthat on his interpretation quantummechanics does not tell one in whatstate a system will be found; rather,it tells one what the probabilities arefor finding the system in variousmutually incompatible states. Stilllater, Born stated that "the motion ofparticles follows probability lawsbut the probability itself propagatesaccording to the law of causality"(Born, 1926, translated and quotedin Pais, 1988, p. 258). Here was thefirst explicit statement that quantumtheory needed two dynamics; oneprobabilistic, and the other deter-ministic. When Schrodinger sawthe interpretation that Born had inmind for his wave mechanics, hesaid that he regretted publishinganything on the subject (Pais, 1988,p. 261). Schrodinger thus becamean opponent of what was soon thealmost universally accepted inter-pretation of his wave mechanics.s

Born's position eventually ledto what might be called the stand-

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enough of a consensus on both theformalism and the interpretationof quantum mechanics that vonNeumann (1955) was able to writedown a more or less axiomatizedversion of the theory. We will referto von Neumann's careful formula-tion of quantum mechanics as thestandard theory.

Following Born, von Neumanndescribed "two fundamentally dif-ferent types of interventions whichcan occur in system S" (von Neu-mann, 1955, p. 351). The first ofthese are "the arbitrary changes bymeasurements" when, by what vonNeumann referred to as process 1,the state of a system instantaneouslyevolves to one of the eigenstates ofthe observable being measured. Thisis Born's probabilistic, nonlineardynamics. The second was "theautomatic changes that occur withthe passage of time," which arecharacterized by a set of unitaryoperators on the state space 'If thatrepresent the smooth time evolutionof quantum-mechanical states inthe absence of measurement; vonNeumann referred to this as process2. This is SchrOdinger's determin-istic, linear dynamics.

According to von Neumann, then,a measurement has the consequenceof causing the physical systemmeasured to depart from its usuallinear evolution and to jump into aneigenstate of the observable beingmeasured a state where the systemhas a determinate value for theobservable. Specifically, the proba-bility p ~ of a system in the staterepresented by '11 jumping into thestate represented by c!I~, where c!I~

is an eigenvector of the operatorcorresponding to the observablebeing measured, is given by

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ard interpretation of states. On thestandard interpretation of states asystem determinately has a propertyif and only if it is in an eigenstateof having that property. A particlefails to have a determinate position,for example, if it is not in an eigen-state of having a particular position.It is not that we do not know whatits position is. Rather, the particlesimply fails to have a determinateposition - it is in a superposition of

mutually incompatible positions,and according to Born's rule forassigning probabilities, and ourexperience, this state has empiricalproperties that distinguish it fromany state where the particle has adeterminate position (that is, anyparticular eigenstate of position).

It is often said that the theory ofquantum mechanics is fme, that it isthe interpretation that has problems.But here we see that the two are notso easy to separate. Born's interpre-tation of quantum-mechanical statesrequired him to introduce a newdynamical law, which is presumablyto be counted as a part of the formaltheory of quantum mechanics (whatjustification could one have forincluding the linear dynamics as apart of the theory but excluding thenonlinear dynamics?). If a systemonly has a determinate physicalproperty when it is in an eigenstateof having that property, then theprocess of measurement cannot belinear - if it was, then, given thestandard interpretation of states, wewould presumably have no way toaccount for how observers end upwith determinate results to theirmeasurements.

After SchrOdinger's 1926 papers,it soon became evident that hiswave mechanics and Heisenberg'smatrix mechanics were essentiallyequivalent. By 1932 there was P" - 1<'1'. cj),,)f(1)

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3INTRODUCTION

norln squared of the ampli-tude of the projection ofthe system's premeasurementstate onto the eigenstate.

which is simply an expression ofBorn's statistical interpretation ofthe wave-function.

Filling in some of the details, vonNeumann's formulation of quantummechanics is based on the following

principles:

The nonlinear collapse oftiJe quantum-mechanical state onmeasurement is curious. As vonNeumann put it:

one should expect that [the lineardynamics] would suffice to describe theintervention caused by a measurement:Indeed, a physical intervention can benothing else than the temporary insertionof a certain energy coupling into theobserved system. . . [in which case, thelinear dynamics would describe the time-evolution of the composite system) (vonNeumann, 1955, p. 352).

So what exactly causes the nonlinear. collapse when there is a measure-

ment? Here we get the fIrst glimmerof the measurement problem.Yon Neumann said that "we haveanswered the question as to whathappens in the measurement of aquantity. To be sure, the 'how'remains as unexplained as before"(von Neumann, 1955, p. 217).

While von Neumann lackedan explanation for the lineardynamics's failure to correctlydescribe measurement interac-tions, he nonetheless believed thatthe standard theory of quantummechanics was perfectly coherent Itwas probably Bohr, however, whodid the most to convince physiciststhat there were no foundationalproblems in quantum mechanics. Or,as Gel1-Mann once put it, "The factthat an adequate philosophical pre-sentation [of quantum mechanics]has been so long delayed is no doubtcaused by the fact that Niels Bohrbrainwashed a whole generation oftheorists into thinking that the jobwas done fifty years ago" (quoted in

. Sudbery, 1986, p. 178).

Lets consider the process ofmeasurement in more detail. A par-ticularly simple quantum mechan-ical observable is called x-spin.There are only two possible resultsof an x-spin measurement: up anddown. A system 5 in the up eigen-state will be represented by li)s anda system in the down eigenstate willbe represented by 1J.)S.6 Suppose thatM is an ideal x-spin measuringdevice - that is, M's pointerbecomes perfectly correlated withthe x-spin of its object system 5without disturbing 5's state. If Mbegins in a state where it is ready tomake a measurement Ir)" and 5begins in an up eigenstate of x-spinli)s, then the composite system M +5 will end up in a state were Mreports that its result was x-spin upand 5 is left in an up state li)M li)s.And if M begins in a state where itis ready to make a measurement Ir)Mand 5 is in a down eigenstate 1J.)s,then M + 5 will end up in the stateIJ.)M 1J.)s' This is what it means tosay that M is an ideal measuringdevice here.

What happens when M measuresthe x-spin of a system 5 that is notinitially in an eigenstate of x-spin?Suppose, for example, that 5 is ini-tially in the state

t. Every physical state is repre-sented by an element of unitlength in a Hilbert space.

2. Every complete physical ob-servable is represented by aHermitian operator on the Hilbertspace, and every Hermitianoperator on the Hilbert space

; corresponds to some physical

observable.3. A system S has a determinate

value for some observable if andonly if it is in an eigenstate ofthat observable: if the state of Sis represented by an eigenvectorof a Hennitian operator corre-sponding to eigenvalue A, onewill get the result A when onemakes the observation corre-sponding to the particular Her-mitian operator.

4. The dynamics has two parts:"I. The linear dynamics: if no

measurement is made ofa physical system, it willevolve in a deterministic,linear way that depends onlyon its energy properties.

II. The nonlinear collapse dy-namics: if a measurementis made of the system, it willinstantaneously, and non-linearly, jump (or collapse)

~.2.. to an eigenstate of theobservable being measured(a state where the systemhas a detenninate value forthe physical quantity beingmeasured). The probabilityof collapsing to a particulareigenstate is equal to the

I(2) ~ (ms + IJ.)S>

The standard theory tells us thatS will instantaneously jump toeither ms or 1J.)s when it ismeasured. The probability of thefinal state being m., ms is 1/2 andthe probability of the final statebeing IJ.)., 1J.)s is also 1/2, which isjust what we find empirically. Ineither case, since S ends up in aneigenstate of x-spin, the standardtheory predictsth3tM wjll get t))esame result if it remeasures the

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JEFFREY A. BARRETT4

bility. The standard theory tells usthat there is some physical observ-able, call it A, that has equation (3)as an eigenstate. Since neitherInAi Ins nor IJ,)AI 1J,)s could also beeigenstates of A (the eigenstates ofan observable are always mutuallyorthogonal), there is, at least in prin-ciple, an experiment that woulddetermine whether M had in factcaused a collapse of S's state: makea series of A-measurements. If youget the measurement result corre-sponding to the state represented byequation (3) every time, then therewas no collapse; if you get thisresult only half of the time, thenthere was a collapse. This meansthat in the standard theory there areempirical consequences to whichsystems are measuring devices andwhich are not, and yet the theory isperfectly silent as to which is which.

Since the standard theory'sdynamical laws constitute mutuallyincompatible descriptions of thetime-evolution of physical systems,they threaten to render the theoryinconsistent unless one can specifystrictly disjoint conditions for wheneach applies. Since the standardtheory does not provide theseconditions, and our intuitions aretoo vague to provide precise condi-tions (Does a cat cause collapses?What about a mosquito? A largeprotein molecule?), we do not knowwhen to apply the linear dynamicsand when to apply the collapsedynamics. This means that thestandard theory is at best incompletein an empirically significant waysince A-type measurements woulddistinguish between the two dy-namics. This is made especiallypuzzling by the fact that wheneverwe have had the technologicalability to perform A-type measure-ments on a system, we have found

x-spin of S without disturbing itbetween measurements.

While the linear dynamicsaccounts for the wave-like proper-ties pf matter, the collapse dynamicsplays an important role in thestandard theory. If there were qocollapse of the wave-function, itfollows from the linear dynamicsthat the final state would be

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1(3) ~ - Oi" li)s + 1.1.)" 1.1.)s)

Note that on the standard interpre-tation of states this is not a statewhere M makes a detenninate reportone way or the other: it is not a statewhere M reports up, it is not a statewhere M reports down, and it is nota state where M reports both orsays it doesn't know either indeed,strictly speaking M does not evenhave a state of its own here since itis entangled with S. So not only doesthe collapse dynamics provide aprescription for calculating proba-bilities but it also accounts for thefact that measuring devices makedeterminate x-spin reports and thefact that an observer will get thesame x-spin result on a subsequentmeasurement if the object system isundisturbed between measurements.

The standard theory's twodynamical laws, however, aremathematically incompatible. Thisincompatibility can be seen by thefact that in the above measurementstories the two dynamics yield verydifferent final states. The collapsedynamics tells us that the final statewill be the statistical mixture: prob-ability 1/2 that the state is Ii)" li)sand probability 1/2 that the state is1.1.)" 1.1.)s. The linear dynamics,however, tells us that the final stateis the superposition represented byequation (3). This is more than justsome abstract or formal incompati-

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that the system always follows thelinear dynamics..'

Von Neumann's question ofhow a collapse occurs also causesexplanatory problems for thestandard theory. Measuring devicesare presumably built exclusivelyfrom fundamental particles, and ourempirical evidence provides us withvery good reasons to believe thatfundamental particles obey thelinear dynamics. so how could ameasuring device behave in agrossly nonlinear way? Is the objectsystem S somehow supposed toknow when it is being measured?What does it take to be a measuringdevice anyway? The standard theoryprovides no answers.s

At this point one might argue thatthe measurement problem cannot bevery serious - after all. most physi-cists have successfully used thestandard theory for over seventyyears barely noticing the problem.The reason is that if one takes theevents that one would naturally callmeasurements to be the eventsreferred to by the theoretical termmeasurement. then the standardtheory makes the right empiricalpredictions for the sort of experi-ments we have performed so far. Butin order to be satisfied with this, onewould have to have very weak stan-dards for physical theories indeed.In addition to giving up on any realunderstanding of the physical world.one would have to be content withthe fact that there is a class of exper-iments (which would be extraordi-narily difficult to perform but are atleast in principle possible) for whichthe standard theory (even whensupplemented with our intuitionsconcerning what constitutes a mea-surement) can make no empiricalpredictions whatsoever.

There have been several attempts

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INTRODUCTION 5

where environmental decoherence issomehow supposed to account foran observer ending up with deter-minate measurement results. Od1ers,like Bohm (1952) and Vmk (1993),have taken the quantum-mechanicalstate to provide an incompletephysical description and have sup-plemented it with so-called hiddenvariables that then describe mea-suring devices as reporting deter-minate results. Still others, likeAlbert and Loewer (1988) have sug-gested supplementing the quantum-mechanical state with a nonphysicalstate, specifically the mental state ofeach observer, that always has adetenninate value and explains howobservers get determinate measure-ment results.

So far, no proposed resolutionto the measurement problem hasproven entirely satisfactory. Whatdoes seem increasingly clear is thatwhatever formulation of quantummechanics we end up with willviolate some cherished intuitionsconcerning what a satisfactoryphysical theory should look like.9

to resolve the measurement prob-lem. Broadly speaking, these havefollowed one or the other of twogeneral sttategies. One sttategy isto provide a criterion for whencollapses occur. Along these lines,some, like Wigner (1961), have triedto find a satisfactory criterion forwhat constitutes a measurement -

Wigner's proposal was that a systemcollapses to an eigenstate of theobservable being measured when aconscious entity apprehends thestate of the system. Ghirardi,Rimini, and Weber (GRW) (1986)have made a very different proposal.They have suggested replacing thelinear dynamics with a dynamicswhere each elementary particle hasa nonzero probability per unit timeof collapsing to an eigenstate ofposition, which has the effect ofkeeping macroscopic pointers closeto eigenstates of position.

The other general sttategy is todeny that collapses ever occur, thentry to find a satisfactory theorywhere the linear dynamics is takento be an accurate description of thetime-evolution of every physicalsystem under all circumstances.Some, like Everett (1957), havetaken the linear dynamics to be acomplete as well as an accuratedescription of the time-evolution ofevery system. The problem then isto explain how measuring devicescould make (or seem to make?)determinate reports when the lineardynamics tells us that they wouldgenerally be in complicated super-positions of making mutually in-compatible results, of being broken,

having been constructed,Providing such an explanation

to be the sttategy of Gell-and Hartle (1990), among

Notes

. This inttoduction and most of the papers

in this volume assume some understandingof how the standard fonuulation of quantummechanics works and of the basic mathe-matics involved. There are, of course, several

very good, acceSsible inb'Oductions toquantum mechanics. Albert (1992) ~little mathematical backgroooo am is a clearintroduction to quantum mechanics and someof the problems involved in fonnulating asatisfactory theory. Also wOl'th mention areShimony (1989), Hulhes (1989), and vanFraassen (1991). As long as we are on thetopic of literablre, there are two collectionsof papers that are particularly gond: Wheelerand Zurek (1983) and Bell (1987). The IJrstis a general collection on the measurementpooblem am the interpretation of quantum

mechanics and the secorxi contains some ofBell's papers. For detailed descriptions of thestandard formulation of quantum mechanics,its historical development. bow it is used,and the mathematics involved, one mightconsult von Neumann (1955), Dirac (1958),Messiah (1965), Mackey (1963), and Baym(1969). These are all standard texts and eachprovides a slightly different perspective.Finally, some of the works that have beenpanjc:u1arly influential in the development ofquantum mechanics and its interpretation arementioned in the following bibliography.I What I am calling the standard theory of

quantum mechanics throughout this intro-duction is the formulation described by vonNeumann (19S5).2 Much of this bistoricaI skek:h follows Pais

(1988, pp. 244-261).J See, for example, SchrOdinger (1926<:).. Pais (1988, p. 259) tries to reconstruct

what might have led Born to make thiscurious mista)(e., ScbrOdinger published his famous criti-

cism of the standard theory of quantummechanics in 1935. The paper, "Die gegen-wlnige Situation in der Quantenmechanik,"wu published in thRe parts in NatllF-w;.s.senschaften D, 801-812, 823-828, and844-849. Ii has since been translatedand reprinted as "The present situationin quantum mechanics: a translation ofScbrOdinger's 'cat paradox' paper", inWheeler and Zurek (1983, pp' IS2-167).6 These are unit-1enath vectors in 8 Hilben

apace. There is almost 8 one-to-ooe corre-spondence between unit-lengtb vectors andpossible physical slateR. The exception is thatIv) and 1V) are usually taken to specify thesame physical state.7 One might wonder why we don't perfOml

the appropriate experiments and determinewhat systems cause collapses. It turns out tobe very difficult. however, to control theenvironment of most macroscopic systemswell enough to perfonn an A-type measure-ment. For a discussion of how A-type mea-surements are performed and the varioustechnical diff"lCulties involved see Leggett(1986) and Clark et 01. (1991).. While the standard theory provides

no explanation, there has been significantprogress toward formulating 8 physicaltheory that would explain why measurementsyield determinate results in just this way. Onthe GRW theory, the cumulative effect ofvery slight nonlinearities in the evolutions offundamental particles ensures that macro-of quantum mechanics

;!P

~t"'0JEFFREY A. BARRETT6

scopic pointers will almost always be close

to eigenstates of position.9 I would like to thank Martha Tueller for

her help in editing.

~Bibliography

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Penrose, R. and Isham, C. (eds.): 1986,Quantum Concepts in Space andTime, Oxford University Press,Oxford.

Rosenfeld, L.: 1971, 'Men and ideas in thehistory of atomic theory', Arch. Hist.Exact Sci. 7, 69-90.

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~

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~ R.: 1992, 'Consistent interpretationsof quantum mechanics', Reviews ofModern Physics 64(2), 339-82.

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