bohm hiley kaloyerou 1986

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PHYSICS REPORTS (Review Section ofPhysics Letters) 144, No. 6 (1987) 32L-375. North-Holland, Amsterdam

AN ONTOLOGICAL BASIS F O R T H E QUANTUM THEORY

D. BOHM, B.J. HILEY and P.N. KALOYEROU

Depa rtme ntof Physics, Birkheck College. (London University), Ma/ct Street, London WC 1 E 7HX, GreatBritain

Received July 1986

Contents:

I. Non-relativistic particlesystems, D. Bohm, B.J. Hi/er II. A causal interpretation ofquantum fields, D. Bohm, B.J.

I. Introduction 32 3 Hiley, P.N. Ka/oyerou

2. New ontological implications of the causal interpreta- I. Introduction 34 9

tion. The single-particle system 32 4 2 . The causalinterpretation of thescalar field 34 93. Extension to themany-body system 330 3. Analysis into normal modesan dthe groundstate of the

4. A brief resum of the theory ofmeasurements in the field 35 2

causal interpretation 332 4 . The excited state of thefield 35 6

5. Quantum transitions discussed independently of mea- 5 . The conceptof aphoton 36 3

surement 33 7 5.1. The absorption of a quantum ofenergy 36 3

6. The quantum processes offusion and fission 339 5.2. The absence ofphoton trajectories 36 6

7. Quantum wholeness and the approach to the classical 6. Interference experiments 36 8

limit 340 6.1. The treatment ofinterference 36 8

8. Summary a nd conclusions 34 5 6.2. The two-slit interference pattern 3 68

References 34 7 6.3. The PfleegorMandel experiment 36 97. The EPRexperiment 37 0

8. Generalization of the interpretation and extension to

fermions 37 3

References 37 5

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AN ONTOLOG ICAL BASIS FORTH E QUANTUM THEORY

D. BOHM, B.J. HILEY and P.N. KALOYEROU

Dep artm en tofPhysics, Birkbeck College, (London University), MaletStreet,

London WC1 E7HX, GreatBritain

INORTH-HOLLAND - AMSTERDAM


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PHYSICS REPORTS (Review Section ofPhysics Letters) 144 , No. 6 (1987) 3 23 34 8 . North-Holland, Amsterdam

I. Non-relativistic Particle Systems

D. Bohm andB.J. Hiley

Abs trac t:Inthis paper we systematically developan ontology that is consistent with thequantum theory. W e startwith the causal interpretation of the

quantum theory, which assumes that the electron is a particle always accompanied by a wave satisfying Schrodingers equation. This wave

determines a quantum potential, which hasseveral qualitatively new features, that accountfor the differencebetween classical theory and quantum

theory.

Firstly, it dependsonly on theformof the wave functionand not on its amplitude, so that its effect does not necessarily fall offwith thedistance.From this, it follows that asystem maynot be separable fromdistant featuresofi ts environment, an d may be non-locally connected to other systems

that arequite far away from it,Secondly, in a many-body system, the quantum potential depends on the overall quantum state in a wa y that cannot be expressed as a

preassigned interaction amongthe particles, Thesetw o features of thequantum potential together implya certainn ew quality ofquantum wholeness

which i s brought out in some detail in this article.

Thirdly, the quantum potential can develop unstablebifurcation points, which separate classesofparticle trajectories according to thechannelsinto which they eventually enter and within which they stay, This explains how m easurement is possible without collapse ofthewave function,

and how all Sorts ofquantum processes, such astransitions between states, fusion oftwo systems into onean d fission of onesystem into two, areable to take place without the need for ahumanobserver. Finally, we show how the classical limit is approached in a simple way, whenever the

quantum potential i s small compared with thecontributions to theenergy thatwould be present classically. This completes ourdemonstration that

an objective quantum ontology is possible, in which the existence of theuniverse can be discussed,without the needfor observers or for collapse of

the wave function.

1. Introduction

Quantum theory has been presented almost universally as a theory giving nothing but statisticalpredictionsof the results ofmeasurements. The fact that the predictions are statistical is, ofcourse, notinitself very novel, but the basically new feature on which most interpretationsagree i sthat the theorycan be formulated only in terms of the results of a measurement process, and that thereis no way evento conceive of the individual actual system, except insofar as it manifests itself through thephenomenona that are to be observed in such a processof measurement.*

In contrast, previous theories (e.g., Newtonian mechanics or classical statistical mechanics) startedwithassumptions about what each individual systemis anddiscussed both measurement andstatistics as

having a secondary kind of significance that is ultimately based on assumptions as to what is, withregard to individual systems. That is to say, previous theories dealt basically with the ontology ofindividual actual systems while the quantum theory appears to have a fundamental and irreducibleepistomologicalandstatistical component. Indeed, without discussing the statistical results of measure-ments there would be nothing to talk about in the theory at all except for pure mathematics with nophysical interpretation.

In 1952 one of us (DB) [3] proposed a causal interpretation of the quantum theory based on theassumption that an individual electron, for example, is constituted of a particle satisfying certain

*Ballantine [1]has proposed what he regards as a statistical interpretation which is Consistent with the assumption that the concept of an

individualelectron in its actual movement is meaningful. However, he discussesonly ensembles, and never makes a ny statement as to what the

individualelectron actually is, The same i s true ofLandes 1 2 ] theory.


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324 tIn onto/ogical basisfort he quantumtheory

equations ofmotion anda wave satisfying Schrdingers equation. Both particle and wave aretakentohe objectively real whether they are observed or not. This has raised the possibility that quantummechanics can be understood essentially in terms of ontological assumptions concerning the nature of

individual systems so that the epistemological and statistical content would then take on a secondaryrole as in Newtonian mechanics.

In support of this view, we have analysed the measurement processin greatdetail in afurtherpaperusing the causal interpretation [4],and we have shown that the objective reality of the individualmeasurementprocesscan be maintained consistently, without the need either for collapse of the wavefunction or forconsciousness to play a fundamental role at this level. However, Shimony [5] haspointed out that in the very act ofdiscussing in this way in terms ofa measurementprocess, we are still,in effect,giving epistomologya basic role, atleast tacitly. Perhaps onecould say that Shimony is askingus explicitly to formulate a theory solely in terms of what Bell [6. 7] has called beables, withoutbringing in observables except as a special case ofwhat is happening among the beables.

To meet thischallenge we shall i n this paper furtherdiscuss the ontological significance of thecausal

interpretation of the quantum theory, in terms of a few illustrative examples, that serve to bring outwhat is implied in thisapproach. These will include theprocesses of quantum transition, offusion oftwo particles into a single combined system, and of fission of such a combined system into two

particles. (The latter cases have already beendiscussed tosome extent in an earlier paper [81.)We feel that these examples are sufficiently broad in theirimplications toindicate how the ontology

worksout moregenerally, and to enable us to understand individual quantum processes without havingto bring in measurement. Indeed, we shall see that what is now called a measurement i s itself only a

special case of the above-mentioned processes of fusion and fission,i n which the thing measuredand the measuring apparatus may now be seen to constitute an indivisible whole. And as Bell [71hassuggested, the term measurement i s therefore somewhat inappropriate, since what is measuredis, toa considerable extent, formedwithin the activity of the process itself.

Although we do thus interpret the quantum theory in terms of a particular kind of ontology, thisrequiresus to introduce several fundamentally new notions as to the nature of being (orof beables)that may be admitted into physical theories. However, as these notions are of aquite differentkind to

those that havegenerally beenaccepted in physics thus far, it will not be useful to summarize them atthis point. Rather, it will be best to let them come out in full detail at appropriate stages of thediscussion. All that we wish to emphasize herei s that the quantum theory can be understoodintuitivelyon an ontological basis only ifwe are ready toconsider the assumption of fundamentally new kindsofqualities and properties for matter rather than merely to continue the development of past lines ofthought.

Finally, we show that the causal interpretation leads to a simple approach to the classical limit inwhich, on the basis of a single notion of reality that is valid at all levels, one can see that classicalbehaviourresults when a certain typically quantum mechanical contribution tothis reality (the quantum

potential) can be neglected.

2. New ontological implications of th e causal interpretation. The single-particle system

W e shall now develop in some detail the main new implications of the causal interpretation of

quantum theory. Firstly, as has indeed been suggested in the Introduction, we suppose that theelectron, for example. actually is a certain kind of particle following a continuous and causally


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D. Bohm and B.J. Hiley, Non-relativistic particle systems 325

determined trajectory. As we havealso suggested, however, this particle is never separated from anewtype of quantum wave field that belongs to it and that fundamentally affects it . This quantum field,i/i(x, t), satisfies Schrdingers equation, just as the electromagnetic field satisfies Maxwells equations.It too is therefore causally determined.

In classical physics, a particle moves according to Newtons laws of motion, and, as is well known,

the forces that enterinto these laws can be derived fromthe classical potential, V.The basicproposalinthe causal interpretation is that the quantum theory can be understood in a relatively simple way byassuming that the particle is also acted on by an additional quantum potential, Q , given by

Q=-~~h~~whereR=I~2

and h is Plancks constant, while m is the mass of the particle. Evidently, the quantum potential isdetermined by the quantum wave field, I/I.

To justify thisproposal, we begin by considering Schrdingers equation for a single particle

(2)

We write

=Re5~

and obtain

(3)

where Q is given in eq. (1) and

with P=R2. (4)m

Clearlyeq. (3) resemblestheHamiltonJacobiequation exceptfor an additional term,Q . Thissuggeststhat we may regard the electron as a particlewith momentum p V S subject not only to the classicalpotential Vbut also to the quantum potential Q . Indeed the action ofthe quantum potential will thenbe the major source of thedifference between classical and quantum theories. This quantum potentialdepends on the Schrdinger field t/iand is determined by the actual solutionof theSchrodingerequationin anyparticular case.

Given that the electron is always accompanied by its Schrodinger field, wemay then say that thewhole system is causally determined; hence the name causal interpretation.

Equation (4) can evidently be regarded as a continuity equation with P =R2 being a probabilitydensity,as Born suggested. The function Phas, however, two interpretations, one through the quantumpotential and the other through theprobability density. It is our proposal that the fundamental meaningofR (and therefore indirectlyof P) is that it determines the quantum potential. Asecondary meaning isthat it gives theprobability density for the particle tobe at a certain position. Here we differ from Bornwho supposed that it was the probability of finding the particle there in a suitable measurement.


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326 An ontological basis for the quantum theory

Indeed, as has been pointed out in the Introduction, in the causal interpretation, the measurementprocess itself has to be interpreted as a particular application of the theory, which is formulated

basically in terms of beables rather than of observables [7] (while the observables are treated asstatistical functions of the heables).

Equation (4) implies that it is consistent to interpret P as a probability density in a statistical

ensemble ofwell-defined trajectories, each following thecausal lawsdescribed above. For ifthe densityholds initially,then this equation guarantees that it will hold fora ll time. W e shalldiscuss the question

ofwhy there is such astatistical ensemble further on,as well as why its probability density will approach

no matter what the initial form of thisdensity may have been.As the theory develops, we shall find that the electron is by no means a structureless particle.

Rather, what is suggested by its behaviouri s that it is a highly complex entity that is deeply affected byits quantum field in an extremely subtle anddynamic way. Moreover, this entity is not to be regarded(as is done in theusual interpretations) as somehow directly possessing both particle-like and wave-likeproperties. Rather, the observed wave-like properties will follow, as we shall see, from the generaleffect of the quantum wave field on the complex structure of the particle.

At first sight, it may seem that to consider the electrons as somekind ofparticle that is affected by

the quantum field, tb . is a return to older classical ideas. Such a notion is generally felt to have longsince been proved to be inadequate for the understanding of quantum processes. However, closerinspection shows that this i s not actually a return to ideas ofthis sort. For the quantum potential has anumberof strikingly novel features,which do not cohere with what is generally accepted as the essentialstructure of classical physics. As we shall see, these are just such as to imply the qualitatively newproperties ofmatter that are revealed by the quantum theory.

The firsto f these new propertiescan be seen by noting that the quantum potential is not changedwhen we multiply the field intensity ~]iby an arbitrary constant. (This is because b appears both i n the

numerator and the denominator of Q .) This means that the effect of the quantum potential isindependent of the strength (i.e.. theintensity) of the quantum field but depends only on itsform. Bycontrast, classical waves, which actm echanically (i.e., to transfer energy and momentum, forexample,to push a floating object) always produce effects that are more or less proportional to the strength ofthe wave.

To give an analogy, we mayconsider aship on automatic pilot being guided by radio waves. Here

too, the effect of the radiowaves is independent of theirintensity anddepends only on theirform. Theessential point is that the ship is moving with its own energy, and that the information in the radiowaves is taken up to direct the much greater energy of the ship. We may therefore propose that anelectron too moves under its own energy, and that the information in theform of the quantum wavedirects the energyof the electron.

This introducesseveral new features into the movement. First of all, it means that particles movingin empty space under the action of no classical forces still neednot travel uniformly in straight lines.This is a radicaldeparture from classical Newtonian theory.M oreover, since the effect of the wave doesnot necessarily fall offwith theintensity, even distant featuresof the environmentcan profoundly affect

themovement. As an example, let usconsider the interferenceexperiment [9].This involves asystemof two slits. A particlei s incident on thissystem, along with its quantum wave. While the particlecanonly go through one slit or the other, the wave goes through both. On the outgoing side of the slitsystem, the waves interfere to produce a complex quantum potential whichdoes notin generalfall offwith the distance from the slits. This potential is shown in fig. 1 . Note the deep valleys and broadplateaux. In the regions where the quantum potential changes rapidly therei s a strong force on the


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D. Bohm and B.J. Hiley, Non-relativistic particle systems 327

Fig. 1. The quantum potential for the two-slit experiment.

particle.The particlei s thus deflected, even though no ordinary typeof force is acting. The movementof the particle is therefore modified as shown in fig. 2 (which contains an ensemble of possibletrajectories).

In this explanation of the quantum properties of the electron, the notion ofinformationplays a keyrole. Indeed it is helpful to extend this notation of information and introducewhat could be called

active information. The basic idea of active informationi s thata formhaving very little energyentersinto anddirects amuch greater energy. The activity of the latter is in thisway given aform similar tothat ofthe smaller energy. It is thereforeclear that the original energy-formwill inform (i.e. putform

into) theactivity of the larger energy.As an example, consider a radio wave whose form carries asignal. The energyof the sound thatw e

hearfrom our radio comes, however, not from this wave but from its power plug or batteries. This isessentially an unformedenergy, that takesup theform or information carriedby the radiowave. Theinformation in the radio wave is, in fact, potentially active everywhere, but it is actually active, onlywhere andwhen its form enters into theelectrical energy within theradio. A moredeveloped exampleofactive informationi s obtainedby considering thecomputer. The information content in a silicon chipcan similarly determinea whole range ofpotential activitieswhich may be actualized through having theform of this information enter the electrical energy coming from a power source. Which of thesepossibilities will in any given case be actualized depends on awider context thatincludes thesoftwareprogrammes and the responses of the computer operator at any given moment.

Although the computerdoes thus indicate akind ofobjective significance for information, neverthe-less like the radio setit depends on a structureset up through the thought ofhuman beingsand so it stillretains a trace ofsub jectivity. An example thatdoes not involve structuresset up by human beings isthefunction of the DNA molecule. The DNA is said by biologists to constitute a code, that is to say, alanguage. The DNAmolecule is considered as information content for this code, while the meaningis expressed in terms ofvarious processes; e.g., those involving DNA molecules, which read the


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328 An ontological basisfort he quantum i/u r %

I 1 1 1 1 (1

I it. 2 . Ti ,i~eciotic~ho the tsw-dii experiment.

DNA code, and carry out the activities that areimplied by particular sections of the DNAmolecule.The comparison to ournotion ofobjective andactive informationi svery close. Thus, in the processof

cellgrowth it is only theformofthe DNAmolecule that counts, while the energy is supplied by the restof the cell (and indeed ultimately by the environment as a whole). Moreover, at any moment, only apart of the DNAmolecule is being read andgiving rise toactivity. The rest i s potentially active andmay become actually active according to the total situation in which the cell finds itself.

The notion ofactive informationi s not only relevant in objectivecontexts as those described above,but it evidently also applies to subjective human experience. For example, in reading a map weapprehend the informationcontext through our own mental energy. Andby awhole set of virtual orpotential activities in the imagination, we can see the possible significance of this map. Thus theinformation is immediately active in arousing the imagination but this activity is still evidently inwardwithin the brain and nervous system. Ifwe are actually travelling in the territory itself then, at anymoment, some particular aspect m ay be furtheractualized through ourphysical energies, acting in thatterritory (according to a broader context including what the human being knows and what he isperceiving at that moment).

Itis clear that the notion ofactive information already has a widespread application in many areasof

human experience andactivity, as well as beyond these ( in the case of DNA). Our proposal is thentoextend the notion of the possibility of theobjective applicability of the notion of active information tothe quantum level. That is to say the information in the quantum potential is potentially activeeverywhere, but actually active only where this information enters into the activity of the particle.

This implies, however, that as we have already suggested, an electron, or any other elementaryparticle, has acomplex and subtle inner structure (e.g., at least comparable to that of a radio). This


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330 An ontological basis br the quantion theory

analysable in thought (e.g., through themovement of the particle acted on by the quantum potential).However, in Bohrs approach, the entire experimental situationi s an unanalysable whole, about whichnothing more can be said at all.

3. Extension to the many-body system

W e shall now go on toconsider themany-body system in which we shall see,in several striking ways,

afurther development of this difference between classical and quantum ontologies.We shall begin by considering the two-body system. The wave function t~i(r1,r2 , t) s a t i s f i e s t h e

Schrdinger equation

ih~=[_ _(V~+V~)+V]~ (5)

whereV 1 and V . refer toparticles 1 and2 respectively. Writing c l i ReiS I anddefining P R2 = IJ~IIJ

we obtain

~S (VS)2 (VS)2+ +V+Q=0 (6)

~t 2m 2rn

where

h ( V 2 - l - V 2 ) R

- 2m R

a n d

aP/at+ div1(PV1S!m) +div2(PV1S/m) = 0. (8)

As i n t h e c a s e o f t h e o n e - b o d y s y s t e m , e q . ( 6 ) c a n be interpretedas the HamiltonJacobi equation

with the momenta of the two particles being respectively

p1=V~S a n d p2=V2S.

But this time it i s t h e H a m i l t o n J a c o b i equation for as y s t e m o f t w o p a r t i c l e s responding not only to the

classical potential, V, but also the quantum potential, Q . This latter now depends on the position ofbothparticles in away that does not necessarily fall offwith thedistance. We thus obtain thepossibilityof anon-local interaction between the two particles.

Going on to consider the N-body system we will have

Q = Q(r~,r, r5, t)

so that the behaviour of each particle may depend non-locally on all the others,no matterho w faraway

t h e y may b e .Ifwe take P =R

2 as t h e p r o b a b i l i t y d e n s i t y in the configuration space of the two particles, then


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D. Bohm an d B.J. Hiley, Non-relativistic particle systems 33 1

clearly eq. (8) is the equation for the conservation of this probability. With this definition ofprobability, it follows, as has been shown, by meansofa detailed andextensive treatment [3, 4 1 , that allthestatistical results ofmeasurements are (as in theone-body case) the same in thecausal interpretationas they are in the usual formulationof the theory.

For several centuries, there has been a strong feeling that non-local theories are not acceptable in

physics. It is wellknown, for example, that Newton feltuneasy about action-at-a-distance t16] and thatEinstein regarded this action as spooky [171.One can understand this feeling, but if one reflectsdeeply andseriously on thissubject onecan see nothing basically irrational about such an idea. Ratherit seems to be most reasonable to keep an open mind on thesubject and therefore to allow oneself toexplore this possibility. If the price of avoiding non-locality is to make an intuitive explanationimpossible, one has to ask whether the cost is not too great.

The only serious objection we can see to non-locality is that at first sight it does not seem to becompatible with relativity because non-local connections in general would allow a transmission ofsignals faster than the speed of light. However, we have extended the causal interpretation to arelativistic quantum field theory. Although this interpretation now implies that events outside each

others light cones can be connected but the additional new features of the quantum potential do notpermit a signal to be transmitted faster than light [4]. Itfollows then that the causal interpretation is

compatible with theessential implications of the theory of relativity.While non-locality as described above as an important new feature of the quantum theory,there is

yet another new feature that implies an even more radical departure from the classical ontology, towhich little attention has been paid thus far. This is that the quantum potential,Q , depends on thequantum state of the whole system in a way that cannot be defined simply as a pre-assignedinteraction between all the particles.

To illustrate what this means in more detail, we may consider the example of the hydrogen atom,whose wave function is a product of function f(x), where x is thecentre-of-mass coordinate, andg(r)

where r is the relative coordinate,

~P=f(x) g(r).

The quantum potential will contain a term representing the interaction of electron and proton

2 2I i Vg(r)

~ 2~i g(r)

In the s s t a t e , Q 1 is afunction only of r itself, while in the p state,i t depends on the relativeangles, Uand ~, a s w e l l . E v i d e n t l y , it i s i m p o s s i b l e t o f i n d a s i n g l e pre-assigned function of r, w h i c h w o u l dsimultaneously representthe interaction ofelectron and proton in both s and p states. And, ofcourse,the problem would be still more sharply expressed ifw e brought in all the other states (d, f, etc.).It is clear from the above that the wholeness of the entire system has a meaning going beyond

anythingthat can be specifiedsolely in terms of the actual spatial relationshipsbetween particles.This isindeed the feature which makes quantum theorygo beyond mechanism ofany kind. For theessence ofamechanicalbehaviouri s that the parts interact in some pre-assignedway to make up thewhole. Evenifthe interaction is non-local, thesystemis still mechanical aslong as theinteraction potential is afixed

and pre-assigned function of the particle variables. But in the causal interpretation of the quantumtheory, this interaction depends upon the wave function of the entire system, which is not only


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332 An ontological basis fort he quantum theor~

contingent on the state of the whole but also evolves with time according to Schrddingers equation.Something with this sort of independentdynamicalsignificance that refers to the whole system and thatis not reducible to a property of the parts and their inter-relationships is thus playing a key role in thetheory. As we have stated above this is the most fundamentally new ontologicalfeature implied by thequantum theory.

The above-described feature should, in principle, applyto the entire universe. However, as has beenshownelsewhere [18],when the wave function of asystem factorises into two parts, the corresponding

subsystems will behave independently. In this case, each subsystem can be treated on its own. It i s thisfeature which provides the ground for the possibility of understanding how, in spite of quantumwholeness, the world still behaves in the classical limit as a set of relatively independent parts thatinteract mechanically.

Such considerations are crucially important to understand theapplication of quantum mechanics tothe many-body system. For example, a chemical bond in amolecule in a certain quantum state can beseen to be a consequence of the quantum potential for the whole molecule, which is such that in thisstate it tends to hold the electronsin places where they contribute to the bonding. On the other hand.in a different quantum state,the quantum potential will be different, so that the molecule maynot bestable.

Similarly, in the superconductingstate of a many-electron system, thereis a stable overallorganizedbehaviour, in which the movements are coordinated by the quantum potential so that the individual

electrons are not scattered by obstacles. One can sayindeed that in such a state, the quantum potentialbrings about a coordinated movement which can be thought of as resembling a ballet dance.

As the temperature goes up, the property ofsuperconductivity disappears. In termsof ourapproach,this is because the wave function breaksup into a set of independent factors which can be thought of asrepresenting independent pools of information, so that the electrons will cease to be guided by acommon pool of information and will instead respond to independent poolsof thiskind. Therefore, theelectrons begin to behave like an unorganized crowd of people who are all acting more or lessindependently and thus begin to jostleeach other, so that the propertyof superconductivity disappears.

This suggests that the quantum potential arising under certain conditions has the novel quality of

being able to organize theactivity of an entire set ofparticles in away that depends directlyon thestateof the whole. Evidently, such an organization can be carried to higher andhigher levels andeventually

maybecome relevant to living beings [191.The possibility that information can be in onecommon pool or divide into independentpoolsin the

way described above arises basically from the multi-dimensional nature of the wave function, which

constitutes one pool when it is not factorizable and many when it is.This is what enables us to answer one of the principle objections that has been made to causal

interpretation, i.e., that the wave function, being in aconfiguration space, cannot be understood in afield in a 3 +1-dimensional space time. For we are now regarding the wave function as belonging to aninformation structure that can quite naturally be considered to be multi-dimensional(organized into asmany sets of dimensions as may be needed), rather than as a simple source of a mechanical force.

4. A briefresum of the theory of measurements in th e causal interpretation

We shall now give a brief resum of our treatment of the quantum measurement process [41,as thistreatmentintroduces a number ofkey new concepts that will be carried over into our discussion of the

quantum potential approach.


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D. Bohma nd B.J. Hiley, Non-relativistic particle systems 333

Our first stepis, for the sake of convenience, todivide theoverall process of measurement intotw o

stages. In thefirst stage, the observed system interactswith the measuring apparatus in such a waythat the wave function of the combined system breaks into a sum of non-overlapping packets, eachcorresponding to a possible distinct result of the measurement. Let the initial wave function of the

observedsystem be

where the cli~(x)are the possible eigenfunctions of theoperatorsthat are beingmeasured. The initialwave function of the combined system is then

= q 50 ( y )E C~~~(x)

where ~( y) is the initial wave function of the measuring apparatus. After the two systems haveinteracted ~ goes over into

= ~ nt/i~(x)cb~(y)

where the 4~(y) are the different wave packets of the relevant parameters of the apparatus thatcorrespond to the possible results of the measurement. For a proper measurement to be made, thepackets 4..~(y), must, of course, be distinct and non -overlapping.

As we show in Bohm and Hiley [4], during the period of interaction, the wave functions, andtherefore the quantum potential, become very complex and rapidly fluctuating functions of the time.When the packets have separated, the apparatusparticles must haveenteredone of them (say m)andwill have zero probability of leaving (because thereis no probability of entering the spaces in betweenpackets). From then on, the quantum potential acting on theparticles will be determined only by thepacket cl~m(x) b~( y), becauseall the otherpackets(which do not overlapthisone) will not contribute toit. So, at least as far as theparticles are concerned, we mayignore all the other packets, andregard

them as, in the sense discussed in section 2, constitutinginactive or physically ineffective information.Here, it must be emphasized that thiswill still happen evenwhen thereissome spatial overlap between

~frtn(X) and the remaining packets, cli~(x).This is because of the multi-dimensional nature of themany-body wave function, which implies that the packet, li~(x)4~(y),and any other packet, say~/i~(x)/.~(y),will fail to overlap as long as oneofits factors fails to overlap, even though the other factorwould still have some overlap.

One maydescribe what happens in another way by saying that each packet, b~(y)forms a kind ofchannel. During the period of interaction, the quantum potential develops a structure ofbifurcationpoints, such that trajectories of the apparatus particles on one side of one of these points enter a

particular channel (say 4m(y)), while the others do not. Eventually, each particle enters one of the

channels to the exclusion ofall the othersand thereafter stays in this channel.When this hashappened,the observed particlewill behave from thenon, as ifits wave function were just lfJ~(x),even ifc l im (x)and the rest of the clsr~(x)should still overlap. The fact that the apparatus particles must enterone oftheir possible channels andstay there is thus what is behind the possibility of a set ofclearly distinctresults of a quantum measurement.

At this point, however, one may ask what is therole of the inactive packets, not containing theparticles. Can we be sure that they must necessarily remain permanentlyinactive?Th e answeri s that in


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33 4 .4n ontological basis fir the quantum theory

p r i n c i p l e , i t i s i n f a c t s t i l l p o s s i b l e t o b r i n g a b o u t a c t i v i t y o f s u c h p a c k e t s . F o r e x a m p l e , o n e may a p p l y

an interaction Hamiltonian to one of these inactive packets, say cl~r(X), such that it comes to coincideonce again with cli,~(x),while leaving ~~~(y) unchanged. The two packets together will then give us

+ li~(x)).If cli~1(x)and clhr(X) overlap, there will be interferencebetween them, and this will

give rise to a new quantum state, in which the previously inactive packet. tIr(X), will now affect the

quantum potential, so that it will once again be active.But here, it is necessaryto note that thus far, the measuring apparatus and the observed system

have been treated on an essentially symmetrical footing. W e have not, as yet, brought into thetheoreticaldescription anything that would assign aspecial role to the state of the measuring apparatusas something that w as actually capable of being known by a human being. It was here that weintroduced our second stage of the measurement process, which contained adetection or registrationdevice capable of amplifying the distinctions in the states of the apparatus particles to a large scalelevel that is easily observable by ordinary means. Such a registration device will contain a very large(macroscopic)number, N, of particles. When thisdevice interacts with the apparatus particles, y, itswave function A(Z Z,~)will have to be brought into the discussion. To each distinct state, n, ofthe

apparatus particles, there will he a corresponding state, A,,(Z1,... ZN) of the registration devices.

The wave function of the relevant system will then he

~C~,,(y)~(x)A,(Z1,... Z5).

Each of the A , will also not overlap with the others, so that even if the q~(y) s h o u l d l a t e r c o m e t o

overlap, thiswould still not affect the quantum potential. asthe particles of theregistration devicewillnow be in distinct channels.

Could the channels of the registration device in turn be made to overlapagain? In Bohm and Hiley[4] it was emphasized that this would have essentiallyzeroprobability, because in registration,therehasoccurred a thermodynamically irreversible process. (So that, for example, tohave overlap here wouldbe as improbable as for a kettle ofwater placed on ice to boil.)

It follows then that once registration has occurred, the packets (orchannels) not containing all the

relevant particles (including those constituting the registration device) will indeed be permanentlyinactive or physically ineffective. From this,w e can see that therei s no need to introduce aco llapseof the wave function in ameasurement. For, because of thevery behaviourof all the relevant particlesand of the wave functions, the observed system will from now on act entirely as if it were in thequantum state, cl~(x).This state will correspond to the channels determined by the wave function of allthe constituents of the registration device, that are actually occupied by the particles constituting thisdevice. So it follows that everything will from this point on take place as ifthe wave function hadcollapsed to the actual result, without the need forany such collapse ever to occur. And thiscomesabout without the need to bring in the consciousness of an observer, in arguments such as have beenproposed by Wigner [20]. As in thecase of superconductivity discussed in section 3 , we can considerthis processin terms of the dance of the particles, as guidedby pools of information, represented

by the wave function. The two systems begin by executing independent dances, because the wavefunction then factorises, and corresponds to separate pools of information. During the period ofinteraction, both systems move according to a single and more complex pool of information, so thatthey are carrying out a common dance. It then follows that what happens in the over-all processdescribed above can better he regarded as a mutual transformation of observed system and observing

apparatus, rather than as a measurement in which the two systems are governed by separate and


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D. Bohm an d 8.1. Hiley, Non-relativistic particle systems 335

independent poolsof information.Thusduring the period ofinteraction this single morecomplexpoolofinformation determines (among other things), aset of bifurcationpoints. Finally, as theparticles enterone of thechannels, as determined by the bifurcation points, the two systems are moving once againaccording to independent pools of information, and therefore executing separate dances. But theirmovementsare correlated,i n thesense that each stateofmovement of the observedsystemgoes togetherwith acorresponding state of the apparatus. In this process ofmutual transformation, onecan indeedhardly say that anything has been measured forwhat are called the results of a measurement are notpresentto begin with, but come about in theprocess ofinteraction itself.

While it is clear that what is automatically called the measurement process has thus been given anoverall causal and objective description, one may nevertheless ask what is the meaning of all theempty wave packets (i.e., those not containing particles). Thosestill satisfy Schrdingers equation,but are nevertheless permanently inactive, in the sense that they never manifest themselves in themovements of the particles at all. Such packets seem to be floating, almost like wraiths in a strange

multi-dimensional world. One can see, forexample, in the many worlds interpretation of Everett[211, the problem is dealt with in a certain way, as each packet would correspond to a differentuniverse, with its different measuring instrument (along with its different human observer as well).Whatdo all these empty packetssignify in the causal interpretation?

To help makeclear what the permanent loss of potential for activity of such packets means,we mayconsider as an analogy adevicethat w as used to illustrate what has beencalled theimplicate order [22].In thisdevice, a droplet ofinsoluble ink was inserted into aviscous fluid such as glycerine, whichwasthensubject to ashearing rotation in a controlledway. The ink droplet gradually became invisible as itwas drawn out into afine thread. When the rotation was reversed, thistime the thread wasgraduallydrawn together,until the ink dropletsuddenly emergedagain into visibility. While it w as drawn out andinvisible, the distribution of ink particles appeared to be at random with no order. Yet it had ahidden order,which was revealed when the droplet came together again. This order was described asimplicate or enfolded, while the ink droplet itself was explicate or unfolded.

Let us now imagine printing amessage in ink droplets suspended inglycerine. As these are enfolded,the information in the message becomes inactive, within the field of anythingthat is sensitive only to a

concentrationof ink beyond a certain minimum threshold. Buti t is stillpotentially active there, ascanbe seen by the fact that it can be unfolded again into its original form. If, however, the fluid had notbeenviscous, therewould have been an irreversible diffusion. Aftersuch aprocess, the informationin

themessage would be permanentlyinactive in the field in question, as theoriginal form could never bereconstituted again. Nevertheless, in a certain sense, some highly enfolded transformation of theoriginal structure of the printed message is still in principle present, regardless of how much diffusionhastaken place.

The analogyto the quantum theory is clear. As long as the measuring apparatus interactsreversiblywith the classical system, channels that are inactive with regard to the particles are still potentiallyactive. But as soon as the irreversible interaction with the registration devicetakes place, the channelsnot containing particles are permanently inactive. In the usual language, we would say that the

information has been lost, but as with the diffusion of ink particles, it has merely ceased to becapable of acting in the manifest domain.

From the above, we see that the information content in the wave function is quite generally in anon-manifest (implicate) order (generally multi-dimensional), while the particles are in the ordinary(explicate) order ofspace time. Indeed, it can be shown [22]thatthe Greens function, which describesthe movementof thiswave function, corresponds to just aprocess ofenfoldment andunfoldment. The


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3 36 An ontological hasi.s forth e quantumtheory

movements of the particles then express the meaning of the information content in a manifest(explicate) order.

In the above example, the informationw as ultimately carried or held by ink droplets. In otherapplications of information theory, therehas always been some material systemor field that carries theinformation. This implies that such information has a certain usually small energy, but that the energyin theactivity which is its meaning ismuch larger and has an independentsource. Can we look backat the informationin the quantum field in this way?

As yet, we have no theory as to what is the origin of the quantum potential. However, we maysuppose that the information it represents is carried in some much more subtle level of matter andenergy, which has not yet manifested in physical research. In this connection, we may recall that insection 2, we proposed that the behaviour of matter does not always become simpler as we go tosmaller low dimensions, and that a particle may have a structure (somewhere between iO~cm and

i0~cm ) which is complex and subtle enough to respond to information in ways that might evenresemble, for example, the activity of aship guidedby radar waves. It is i n this structure (which couldinclude generalized fields of a very subtle nature as well as particles)that we may look for the basis ofthis informationin physical structure.

Another analogy to theprocess in which information becomes inactive can be obtained by thinkingofwhat happens when we make adecisionfrom a number of distinctpossibilities. Before the decisioni smade, each of these possibilities constitutes akind of information. This may be displayed virtually inimagination as the sort of activities that would follow if we decided on one of these possibilities.Immediately after we make such a decision, there is still the possibility of altering it. However, as weengage in more and more activities that are consequent on this decision, we will find it harder andharder to change it . For we are increasingly caught up in its irreversible consequences and sooner orlater we would have to say that the decision can no longer be altered. Until that moment, theinformation in the otherpossibilities was still potentially active, butfrom that point onsuch information

is permanently inactive. The analogy to the quantum situation is clear for the information in theunoccupied wave packet becomes more and moreinactive as more and moreirreversible processes are

set in train by the channel that is actually active.

In the case of our own experience of choice, the inactive possibilities may still have a kind ofghostly existence in the activity of the imagination, but eventually this too will die away. Similarly,according to our proposal, the inactive informationin thequantum potential existsat a verysubtle levelof the implicate order. W e may propose, however, that perhaps this too will eventually die awaybecause of as yet unknown features of the laws of physics going beyond those of quantum theory.

The above discussion shouldhelp to make i t clearho w it is possible to understand quantumprocesseswithout bringing the collapse of the wave function and without bringing in the consciousness of anobserver. DEspagnat [23] has however suggested in a critical vein that our ability to do without thecollapse of the wave function does in fact dependultimately on co nsciousness. For in his view, we haveassumed thatwe perceive only the supplementary(particle) variables, and forh im this implies that allthe unoccupied packets are dropped from our account, simply because they are not perceived and

thereforedo notenter ourcon sciousness. In ourapproach, however,our basic assumption hasactuallynothing to do with consciousness. Rather, it is that the particles are the directmanifest reality,while thewave function can be seen only through its manifestations in the motions of the particles. This is

similar to what happens in ordinary field theories (e.g., the electromagnetic), on which the fieldslikewise can manifest themselves only through theforces that they exerton theparticles.*Moreover the

*The main difference i s that theparticlescan besources offields, whereas, i n thequantum theory, particlesdo not serveas sourcesofthe wave

function. But evidently this doesno t affect the question ofhow the fields are manifest themselves.


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D. Bohm an d B.J. Hiley, Non-relativistic particle systems 337

conclusion that after an irreversible detection process has taken place, the unoccupied packets willnever manifestthemselves in the behaviourof the particles follows, as we have seen, from the theoryitself andhas nothing to do with our not being conscious of these packets.

We can furtherjustify ourassumption that the immediately manifest level is that of the particles bynoting that (as will be shown in more detail insection 7) in the classical limit, from which the content ofour sensory data comes, the quantum potential can be neglected. This means that all that we learnabout the world through the senses has to pass through a level in which the wave function plays noessential role. That is to say, as we have indeed already indicated, empirical knowledge referring to thewave function hasultimately to come from inferences drawn from observations of the behaviour ofstructures ofparticles, as manifested at the classicallevel. This conclusionis however a consequence ofour basic assumptions concerning the nature of reality as awhole anddoes not depend on anyfurtherh y p o t h e s i s c o n c e r n i n g t h e p a r t i c u l a r c o n te n t o f o u r consciousness.

5. Quantum transitions discussed independently ofm easurement

We are now ready to extend the notions developed here to show howspecific but typical processescan be discussedapartfromthe contextof measurement, andindeed apartfrom the need to bring in theactivity of any human beings at all (e.g., no one is needed to prepare a quantum state).

We begin this section with adiscussion of a transition process. If an atom, for example, is to jumpfrom one stationary state to another, it is necessary that there be an additional system available which

can take up the energy. This is usually the electromagnetic field within which, forexample, a photoncan be created. However, to avoid the complexities arising in the causal interpretation to theelectromagnetic field [3,4 1 , we shallsuppose that the energyi s taken up in an Auger-like effect by anadditional particlein the neighbourhood, that was originally in a bound state near the atom i n question.Clearly the principles involvedwill be thesame no matter what are thedetails of thesystem that carriesaway the energy.

Let usconsider an atom containing an electron with coordinates x and with an initial wave function

c l 0 ( x ) exp(iE0t) corresponding to a stationary state with energy E0. (We shall write /1 =1 in the restofthe paper.) The additional particles will have coordinates y and an initial wave function q~0(y,t). Ingeneral 40(y, t) will represent a state in which the y particle is bound near a centre. The combinedsystem will then have an initial wave function,

1 I A ~ J = c l i0 (x )exp(iE0t) ~0(y, t).

Through interaction between the atomic electron and the additional particle the wave function willbegin to include other stationary states of the electron. Tosimplify thediscussion, we will suppose thatonly one of these, li~(x)exp(iE1t), contributessignificantly. The corresponding wave function of theadditional particlewill be 4~(y,t),which represents a particle that is no longer bound. (Wesuppose, of

course, that the energy, E0 E~,given off by the electron is considerably greater than the bindingenergy of the y particle.) As a function of time the wavefunction of thecombined systemwill thenbe

= c l i0 ( x ) exp(iE11t) ~0(y, t) +J a(t, t) c li1 ( x ) exp(iEft) ~(y t t) dt (9)where a(t, t) can becalculated using ordinary time-dependent perturbation theory [26].


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338 An ontological basis fort he quantum theors

What theabove formulameans is that during asmallinterval of time dt acontribution to the wavefunction will be produced which is described by the integrand. That is to say, there will be a terml/J~(x)exp(iE~t)corresponding to the final state of the electronwhich is multiplied by ~ t t) dt.This latter function corresponds to a y particle that during the time interval (t i) moves away fromthe atom very rapidly because it h as ab s o r b e d t h e e n e r g y d i f f e r e n c e Ef E

1 1 . As a resulttherewill be a

negligible overlap with 411(y, t). As time goes on the total contribution to c l 1 ( x ) exp(iE1t) willaccumulate in away that is proportional to the integral in theabove equation. This integral represents asum o f a s e t o f c o n t r i b u t i o n s ~(y,t i)dt, each o f w h i c h h as m o v e d a differentdistance from thea t o m , b u t p r a c t i c a l l y a l l o f w h ich w i l l h av e a n e g l i g i b l e overlap with 4~1(y,t).

A s b r o u g h t o u t i n section 4 , t h i s f e a t u r e o f n e g l i g i b l e o v e r l a p i m p l i e s t h e establishment ofa separate c h a n n e l a l o n g w i t h b i f u r c a t i o n points that divide all trajectories into two classes, i.e., those that enterthechannel andthose that do not.For smallvalues of t,the numberof trajectoriesentering the channelis proportional to t, and as a simple calculation shows, the proportionality factor yields the correctprobability. This means that while interaction is taking place, the y particle and the electron will (as i n

the case ofsuperconductivity) beexecuting a common dance but that afterwards their dances willbe independent. However, theywill be correlated, in the sense that for all initial conditions i n which the

electron has undergone a transition to anew dance, they particle will be moving freely and far awayf r o m t h e a t o m , w h i l e a l l o t h e r i n i t i a l c o n d i t i o n s w i l l r e m a i n b o u n d .

In the way discussed in section 4 , the information in whichever is the unoccupied channel hasbecome inactive. Of course, at this stage, it is still potentially active. But, as happened in the case ofthemeasuring apparatus there is a second stage. In thisstage, the y particle interacts irreversibly with itsmacroscopic environment. The same arguments used in section 4 will show that the unoccupiedc h a n n e l s w i l l t h e n h av e l o s t t h e p o t e n t i a l f o r a c t i v i t y , a n d s o may f r o m t h i s p o i n t o n , b e d r o p p e d f r o m t h e

physical account.Clearly, the entire process has, in this way, taken place with an individual system, and there is

m o r e o v e r n o n e e d t o d i s c u s s a c o l l a p s e o f t h e w a v e f u n c t i o n t o t h a t o f t h e b o u n d s t a t e . Th er e i s a l s o n oneed to talk about the preparation of the initial state by a human being, for the x and y particles couldfall into stationary states on their own account by givingup energy (e.g., to the electromagneticfield).A similaranalysis would show how any suitable initial state (e.g., offree particles) couldgive rise, i n atransition to the state fromwhich we initially started. This means that we can regard thewhole system

as existing in its ownright whether there are humanbeings to prepare states and observe themor not.A very important feature of our interpretation is that it makes possible a simple and precise

definition of what is to be meant by the time at which a transition takes place. In terms of the wavefunction, all that we can talk aboutis the meanlife time. For example, the mean life time of a uraniumatom is ofthe order of 2 x l0~years. Nevertheless,in any specimen, some uraniumatoms are observedto decay in avery short time, which can indeed in principle be measured very precisely (e.g., within10 -~s). There is no clear way to discuss thisin terms ofthe wavefunction alone [24].However, inthecausal interpretation, thewave function does not exhaust thewhole of reality.There is also the particle,which maybe bound in a stationary state for a long time until by chance it comes near abifurcationpoint [251in which case it very rapidly moves out into an unbound state, thus implying a relativelywell-defined time of transition. This is adefinite advantage of the causal interpretation, not only overthe usual interpretation, but also over themany-worlds theory, which (as we shall discuss in moredetail in section 8) also has no clearly defined concept of the time of transition.

Finally, we can now clear up a question that has long been puzzling in terms of the usualinterpretation, i.e., that of the watched dog effect [24] (orZenos paradox). If onesupposes that an


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D. Bohm an d B.J. Hiley. Non-relativistic particle systems 339

electronis continually watched by apiece of apparatus, the probability of transition has beenshownto go to zero. It seems that the electron can undergo transitiononly if it is not watched. This appearstobe paradoxicalin the usualinterpretation whichcan only discussthe results of watching andhas noroom for any notion of the electron existing while it is not being watched. But in the causalinterpretation with its objective ontology, thispuzzle does not arise because the system is evolvingwhether it is watched or not. Indeed, as our theory of measurement shows, the watched system isprofoundly affected by its interaction with themeasuring apparatus [4] and,so we can understand why,ifi t is watched too closely, it will be unable to evolve at all.

6. The quantum processes of fusion an d fission

We now go on to extend the results ofthe previous section to the examplesoffusion andfission.We begin with a discussion of fusion. We will consider the case ofan atom containing an electronwith c o o r d i n a t e x . T h i s a t o m i s a b l e t o c a p t u r e a n i n c i d e n t p a r t i c l e w i t h c o o r d i n a t e Z t o f o r m a new

c o m b i n e d s y s t e m , w h i c h i s a n e g a t i v e i o n . A s h a p p e n e d i n t h e c a s e o f a t r a n s i t i o n , t h e e x c e s s e n e r g y

w i l l b e c a r r i e d a w a y b y a n a d d i t i o n a l p a r t i c l e with c o o r d i n a t ey b o u n d t o a c e n t r e i n t h e n e ig h b o ur h o o d .T h i s w i l l r e p r e s e n t t h e fusion of two independent s y s t e m s t o f o r m a new w h o l e .

T h e i n i t i a l s t a t e of t h e e l e c t r o n i n t h e a t o m w i l l b e d e n o t e d b y c l 0 ( x ) exp(iE0t) a n d t h e i n i t i a l s t a t e

of t h e i n c i d e n t p a r t i c l e b y ~0(Z,t). T h e i n i t i a l s t a t e o f t h e a d d i t i o n a l p a r t i c l e w i l l b e ~0(y,t), a s i n t h e

p r e v i o u s s e c t i o n . The c o m b i n e d i n i t i a l w a v e f u n c t i o n f o r t h e e n t i r e s y s t e m w i l l t h e n b e

~I~(x,y, Z, t) =~0(Z,t) 1 0 ( y , t)~/i0(x)e x p ( i E 0 t ) . ( 1 0 )

B e c a u s e o f t h e i n t e r a c t i o n b e t w e e n t h e t h r e e p a r t i c l e s w i t h c o o r d i n a t e s x , y a n d Z, t h i s c o m b i n e d

w a v e f u n c t i o n w i l l c h a n g e . We s h a l l as s um e t h a t t h e m a i n p o s s i b i l i t y f o r ch an g e i s t h e i n t r o d u c t i o n of a

b o u n d s t a t e f o r p a r t i c l e Z w h i c h w i l l c o r r e s p o n d t o a w a v e f u n c t i o n A(x,Z) exp(iE~t)w h i l e t h e

additional particle that takes up the energyreleased has a wavefunction ~ t), r e p r e s e n t i n g a s t a t ethat is no longer bound. Using time-dependent perturbationtheory in the waydiscussed in theprevioussection, it can be shown that the final wave function becomes

~(x, y,Z , t) = ~~(x,y, Z , t) +J a(t, t) ~ tt) A(x, Z)exp(iEft) dt. ( 1 1 )

The f i n a l w a v e f u n c t i o n c o n t a i n s a c h a n n e l c o r r e s p o n d i n g t o a f r e e y p a r t i c l e a n d a s t a t e i n w h i c h

t h e x a n d Z p a r t i c l e s h av e f u s e d i n t o a s i n g l e w h o l e . W h i l e t h e p a r t i c l e s a r e a l l i n t e r a c t i n g , t h e y a r e

e x e c u t i n g a common d a n c e , i n w h i c h t h e r e a r e c r i t i c a l ( b i f u r c a t i o n ) points, such that those on one

s i d e e n t e r t h e c h a n n e l c o r r e s p o n d i n g t o fusion while those on the other do not. Finally, the y particlee x e c u t e s a d a n c e i n d e p e n d e n t o f t h e o t h e r t w o , b u t c o r r e l a t e d t o t h e i r s t a t e . Wee m p h a s i z e o n c e

a g a i n t h a t a l l t h i s happens without the presence of any apparatus either to prepare or measure thes y s t e m , a n d w i t h o u t a n y collapse of the wave function.Wenow c o m e t o t h e fission processwhich is essentially theinverse of the fusion process. It i s o n l y

necessaryfor us to give a brief sketch ofho w this takes place as the treatment is similar in both cases.


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340 An ontological basisforth e quantum theory

The main difference is that the y particle must now supply t h e e n e r g y needed for fission, whereaswith fusion, it hadto absorb the energy released. We therefore suppose that the y particleis incidentwith a high energy, in a state represented by q~0(y,t), and that after this particle interactswith theatom, it is scatteredinto one ofa set of statesof lowerenergy, representedby 4~(y,t). The initial state

of the combined system can be representedby the wave function

~P0(x,y,Zt) =~y, t)A(x,Z) exp(iE1t)

Through interaction this becomes

~(x, y, Z, t) =~(x, y,Z , t) + 1 a(t,t) cli~(x)exp(iE~t)~ ~(Z, tt) ~ tt)dt. (12)

The sum over states appears above because the outgoing particle can move inmany directions and

t h e a d d i t i o n a l p a r t i c l e w i l l m o v e i n a c o r r e l a t e t h w a y . H o w e v e r , i n t i m e , t h e v a r i o u s d i r e c t i o n s b e c o m eseparatedin space and the functions ~(y, t t)~villcease to overlap. The y particlewill enterone ofthechannels corresponding to a particular 4~(y,t t) a n d w i l l r e m a i n i n i t . A l l t h e o t h e r c h a n n e l s w i l lbe empty andcan be left out of the discussionfrom this point on.

The probabilities for transition, whether for fusion or fission, can be obtainedby integrating all theparticle coordinates over the relevant packets and as has been shown elsewhere [3]thiswill come outthe same as for the usual interpretation.

It can be seen from this discussion that the measurement process treated in section 4 is actually acombination of fusion and fission. At first, themeasuring apparatusundergoes fusion with the observedsystem, and this is followed by fission. This meansthat what has commonly beencalled measurementis actually a transformation of the entire system. In this transformation the apparatus and theobserved system enter into a common dance for a short time and then separate to give rise toindependent movements in which the observed system is executing anew dance corresponding tothewave function cli~(x).

W e therefore emphasise, once again, Bells suggestion that the process described above should noteven be called measurement, except perhaps in the classical limit where the effect of these transformations is negligible. In every such transformation, that which is measuring and that which ismeasured are both altered in a certain irreducible way. Ofcourse, in a rough treatment that isadequate for the large scale level the alterations may generally be neglected,but anysuitable refinedtreatment must take such alterations into consideration.

7. Quantum wholeness an d the approach to the classical limit

We have seen thus far that the quantum behaviour ofmatter shows a certain kind of wholeness,brought about by the quantum potential. This latter functions as active information that may reflectdistant features of the environment and maygive rise to a non-local connection betweenparticles thatdepends on the quantum state of the whole, in a way that is not expressible in terms of the


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D. Bohm an d B.J. Hiley, Non-relativistic particle systems 341

relationships of the particles alone. How then do we account for the classical limit in which mattergenerally does not show such wholeness, but behaves as if it were constituted of independent parts,which do not depend significantly on distant features of the environment?We s h a l l b e g i n b y d i s c u s s i n g how q u a n t u m m e c h a n i c s a p p r o a c h e s t h e c l a s s i c a l l i m i t . Th er e a l r e a d y

e x i s t s a w e l l - k n o w n wayo f t r e a t i n g t h i s q u e s t i o n m a t h e m a t i c a l l y ; i . e . , t h e W . K . B . a p p r o x i m a t i o n . T h i s

i s i n d e e d o b t a i n e d j u s t b y n e g l e c t i n g t h e q u a n t u m p o t e n t i a l , w h i c h l e a v e s u s w i t h t h e H a m i l t o n J a c o b i

e q u a t i o n . S u c h n e g l e c t w i l l b e j u s t i f i e d w h e n t h e q u a n t u m p o t e n t i a l te r m i s a c t u a l l y small comparedw i t h t h e o t h e r t e r m s i n t h e H a m i l t o n J a c o b i e q u a t i o n . R o u g h l y speaking,then,classical behaviour is

a p p r o a c h e d , w h e n t h e c l a s s i c a l p o t e n t i a l d o m i n a t e s o v e r t h e q u a n t u m p o t e n t i a l ; a n d w h e n t h i s i s n o t s o ,

we o b t a i n t y p i c a l q u a n t u m b e h a v i o u r . T h e a p p r o a c h t o t h e c l a s s i c a l l i m i t h as t h e r e f o r e n o t h i n g t o d o

w i t h s e t t i n g Ii =0 , a s P l a n c k s c o n s t a n t i s a l w a y s actuallythe same, while the relativeimportance of thet e r m s i n t h e e q u a t i o n s i s w h a t may c h a n g e w i t h conditions.

The W . K . B . a p p r o x i m a t i o n i s o b t a i n e d s i m p l y b y solving for S, with Q s e t e q u a l t o z e r o . I n t h eone-dimensional case, w e o b t a i n , f o r a s t a t i o n a r y s t a t e o f e n e r g y , E,

~ dx (13

J ~/2m(E -

V(x))

From the conservation equation, we obtain P -~(E V(x))112 so that R -~(E V(x))1~4.Theapproxi-

matewave function is then

=A(E V(x))4e x p [ 2 J d x (2m(E V(X)))h/2].

The test for the validity of this approximation is simply to evaluate the quantum potential with thea p p r o p r i a t e w a v e f u n c t i o n , a n d t o s h o w t h a t i t i s s m a l l c o m p a r e d with (E V(x)). T h i s leads to the

c r i t e r i o n

2m (E V(x)~114~ (E V(x))t14 ~1. ( 1 4 )

In simple physical terms, this criterion will be satisfied when the classical potential, V(x), d o e s n o tch an g e m u c h i n r e l a t i o n t o EV(x), w i t h i n a q u a n t u m w a v e l e n g t h [26].Th is c r i t e r i o n t e n d s t o b e

satisfied in most large-scale situations where the quantum number is large, and the wave length ist h e r e f o r e s h o r t e n o u g h s o t h a t V(x)doesnot change appreciablywithin it. Nevertheless, wherevertherea r e t u r n i n g p o i n t s , a t w h i c h E V(x) = 0 , i t w i l l n e c e s s a r i l y b r e a k d o w n . B e ca u se t h e a p p r o x i m a t i o n

b r e a k s d o w n i n t h i s w a y , a s p e c i a l t r e a t m e n t i s n e e d e d t h e r e , w h i c h s h o w s t h a t i n g e n e r a l a w a v e w i l l b ereflected from such points, with a phase change, the details ofwhich depend somewhat on theprecise

sh ape o f t h e p o t e n t i a l [26].Th u s , f o r a s t a t i o n a r y s t a t e , w e mu s t t a k e a l i n e a r c o m b i n a t i o n o f w a v e srunning in opposite directions,

(E V(x)yh/4{exp[i J dx (2m(EV(X)))h/2] +exp[_iJ dx (2m(E V(x)))~112+a]}


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342 A r m ontological basi.s f orth e quantum theory

w h e r e a i s a s u i t a b l e p h a s e f a c t o r , d e t e r m i n e d i n t h e w a y i n d i c a t e d a b o v e . T h e w a v e f u n c t i o n i s now a

c o m b i n a t i o n o f s i n e s a n d c o s i n e s , s o t h a t t h e a m p l i t u d e R(x) oscillates i n e a c h w a v e l e n g t h . S i n c e , i nt h e l i m i t o f h i g h q u a n t u m numbers, the wave length is very short, the quantum potential will still bel a r g e . I n d e e d , t h i s i s j u s t a n e x a m p l e of t h e p o s s i b i l i t y o f t h e s t r o n g d e p e n d e n c e o f t h e q u a n t u m

potentialon d i s t a n t f e a t u r e s o f t h e e n v i r o n m e n t . ( I n t h i s c a s e , t h e r e g i o n w i th EV(x) = 0 w h e r e t h e

w a v e i s

r e f l e c t e d . ) T h i s

m e a n s t h a t

t h e c r i t e r i o n

(14) f o r t h e v a l i d i t y o f

t h e WKB

a p p r o x i m a t i o n d o e sn o t a l w a y s g u a r a n t e e t h a t t h e q u a n t u m potential is s m a l l , e v e n i n s t a t e s w i t h h i g h q u a n t u m numbers,and that therefore a system does not always approach the classicallimit forhigh quantum states.

The situationdescribed above, in which the w a v e f u n c t i o n i s e f f e c t i v e l y r e a l , i m p l i e s t h a t V S =0. Inthe causal interpretation, thismeans that the particles are at r e s t . S u c h a c o n c l u s i o n v i o l a t e d E i n s t e i n sphysical intuition, which was such that he felt that, at least in states with high quantum numbers, thep a r t i c l e s ou g h t t o b e m ov ing b a c k and f o r t h w i t h e q u a l p r o b a b i l i t i e s f o r each direction [27].However, itmu s t be a d m i t t e d that whether we use the usual i n t e r p r e t a t i o n o r t h e c a u s a l i n t e r p r e t a t i o n or i n d e e dany other interpretation, the quantum theory clearly predicts for this case that there will be at r i g o n o m e t r i c a l l y v a r y i n g p r o b a b i l i t y d e n s i t y f o r t h e p a r t i c l e s , with z e r o s a t t h e n o d e s o f t h e w a v e

f u n c t i o n . I f t h e p a r t i c l e s a r e m ov ing b a c k a n d f o r t h , how c a n t h e y p o s s i b l y c r o s s t h e s e m o d e s , a t w h i c h

theycan never be present? (Unless theirspeeds are infinite, which would violateclassical intuition evenmore than would the notion that their speed is zero.)

One of us [28]answered Einstein, but we would like here to present such an answer in moredetail.Firstly, we emphasize again that p =0 is, for this case, areasonable result, consistent with theexistenceof nodesi n the wave function, whereas the intuitive notion of equiprobability of opposite velocities is

not. What was behind Einsteins objection was the commonly accepted notion we have referred tobefore,namely, that high quantum numbersmust always implyclassical behaviour. But ifwe reflect onthis point further, we can see that Einsteins objection implied that the quantum mechanics must bewrong for this example, in thesense that prediction ofnodesin statesofhigh quantum numbers for theprobability density cannot be right. Therefore, Einsteins criticism against the causal interpretations ismisdirected i t should have been directed against the quantum theory itself, because it implies that thistheory must fail for high quantum numbers.

W e wish however to argue here that the quantum theory is in fact valid forhigh quantumnumbers.To show this,consider replacing the impenetrable walls (i.e. the nodesof thewave function) with highbut penetrable barriers, so that this experiment would become essentially an interferenceexperiment.Indeed, it would just be the equivalent of the FabryPerot interferometerfor neutrons or electrons.Considering the many kinds of macroscopic scale quantum interference experimentswith electrons andneutrons,one can therefore see no reason to doubt that nodes wouldbe discoveredin this FabryPerotcase, even forhigh quantum number, if the experiments were ever done.

In thisconnection thereis a common and indeed very natural tendency to form the notion of somekind of averaging process that would wipe out such nodes. For most large-scale systems thiswould, in

fact, be correct. For example, there might be averaging over random thermal disturbances due to theenvironmentor,in asimpler caseofthe isolated system, onemay take a linear combination ofsolutions

with asmall range ofenergies to form a local wave packet. In such situationsthe amplitude of the wavefunction would change slowly so that the quantum potential would benegligible and the classical limitwould therefore be approached forhigh quantum numbers. However, if the system is isolated and itsenergyi s w ell-defined so that only one energyeigenfunction will be present in the wave function, thenthe nodes predicted by the quantum mechanics will necessarily have to be present. However, such astate ofwell-defined energy requires avery special kind ofexperimental situation to bring it about and


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D. Bohma nd B.J. Hiley, N on-relativistic particle systems 343

maintain it, because it is highly unstable to small perturbations (as will indeed be brought out in mored e t a i l l a t e r o n i n t h i s s e c t i o n ) .

We therefore emphasise again that although high quantum numbers generally lead to classicalbehaviour, there are special conditions inwhich they do not do so.It is an advantage of the causal interpretation that it gives a simple and correct criterion for the

classical limit, i.e., that the quantum potential be negligible. We are thus able to treat reality on alllevels in the same way. Itwould seem that inany investigation into macroscopic quantum phenomenon,

or into the relationship betweenclassical and quantumdomains, clarity will be enhanced, if wecan thustreat the process without a break or cut between these levels, such as seems to be required in otherinterpretations.

The above discussion of the approach to aclassical limit has been in terms of aone-particle system.To extend this to the many-particle system, we have also to take into account the non-local andstate-dependent connection between particles, which is brought about by the many-body quantumpotential. In particular, this sort of connection will arise when the wave function is not factorisable(recall that when the wave function is factorisable the particles behave independently). Such anon-factorised wavefunction will generally be presentin a bound state which is stabilised by attractiveclassical potentials. Of course, for such bound states, quantum wholeness plays an essential part.

However, bound states can be broken up (i.e., they can undergo fission) if they are bombarded with

particles, including photons and phonons, havingsufficient energy. Our question is then: what happenst o q u a n t u m w h o l e n e s s w h e n t h i s b r e a k - u p o c c u r s ?

Weh av e a l r e a d y d i s c u s s e d f i s s i o n a s b r o u g h t a b o u t b y i n t e r a c t i o n w it h a n e x t e r n a l p a r t i c l e t h a t

s u p p l i e s t h e n e c e s s a r y e n e r g y . I n t h i s c a s e , a s s h o w n i n e q . ( 1 2 ) , t h e e x t e r n a l p a r t i c l e with c o o r d i n a t e y

e v e n t u a l l y g o e s i n t o o n e of i t s p o s s i b l e s e t s o f w a v e p a c k e t s . T h e r e a f t e r t h e o t h e r p a c k e t s h av e n o e f f e c t

o n t h e q u a n t u m p o t e n t i a l s o t h a t t h e e f f e c t i v e w a v e f u n c t i o n i s th en r e p r e s e n t e d b y a s e t o f f a c t o r s

c o r r e s p o n d i n g t o i n d e p e n d e n t b e h a v i o u r o f t h e v a r i o u s p a r t i c l e s (wh o se c o o r d i n a t e s a r e x , y a n d Z ) .

T h i s m e a n s t h a t d i s t a n t s y s t e m s o r i g i n a t i n g i n f i s s i o n o f a s i n g l e l a r g e r s y s t e m w i l l i n g e n e r a l b e i n s t a t e s

corresponding to independent behaviour and so quantum non-local connection will not be present.However, it is possible to have fission under special conditions in which such non-local connection

remains. This indeed is what happens in the experiment of Einstein, Podolsky and Rosen. A typicalway ofbringing this about is through thespontaneous d i s i n t e g r a t i o n o f a s y s t e m i n t o t w o p a r t s . T h u s l e ta n u n s t a b l e m o l e c u l e b e r e p r e s e n t e d b y t h e w a v e f u n c t i o n

1 1 1 ( x , Z ) = ~ C ~ c l i ~ ( x ) x ~ ( Z ) . (15)

A f t e r d i s i n t e g r a t i o n t h e w a v e f u n c t i o n w i l l be

W~(x,Z)=~C~~ ( 1 6 )

where a~an d b~r e p r e s e n t t h e r e s p e c t i v e d i s t a n c e s t h a t the two particleshavemoved. Thiscorrespondsto a function in which there are still correlations even though the particles have separated bym a c r o s c o p i c d i s t a n c e s a n d i n w h i c h t h e q u a n t u m p o t e n t i a l c o r r e s p o n d s t o a n o n - l o c a l i n t e r a c t i o n .

As we have shown in our measurement paper [4], any measurement of the properties of thesep a r t i c l e s w i l l b r i n g a b o u t f a c t o r i s a t i o n w i t h o u t t h e n e e d f o r a c o l l a p s e of t h e w a v e f u n c t i o n . H o w e v e r ,s i n c e t h e p u r p o s e of t h i s p a p e r i s t o d i s c u s s t h e o n t o l o g y i n d e p e n d e n t l y o f t h e q u e s t i o n o f m e a s u r e m e n t


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344 An ontological basis fort he quantum theory

we shall now show how factorised wave functions are brought about by almost any kind of smalli n t e r a c t i o n or p e r t u r b a t i o n t h a t may b e p r e s e n t n a t u r a l l y o r fortuitously. For example, suppose thata

particlewith coordinate y interacts with the particlex . Let the initial wave function of the y particle be

40(y). The effect of theinteractton is to change termssuch as 4~(y)qi~(x a~)into linear combinations

~ a 1 , b.(y) A 1 ( x ) where A .form acomplete set. The total wave function will become

~ (17)

The y p a r t i c l e e v e n t u a l l y g o e s i n t o o n e o f t h e s e p a r a t e p a c k e t s i n t o w h i c h 1P~finallydevelops. Let thisbe denotedby s. The other packetswill thennot contribute to the quantum potential and the effectivewave function will become

~(x,y,Z)=~,(y) As(x)~CnasnXn(Z~b,m). (18)

Clearly this reduces to a set offactorsfor a ll of theparticles andso the quantum non-local connectionsw i l l be d e s t r o y e d .

What the above shows is that quantum non-local connection is fragile andeasily broken by almosta n y d i s t u r b a n c e o r p e r t u r b a t i o n . ( Z e h [29] has come to asimilar conclusion, but on a different basis.)This kind of connection is stable only when a system is held together by classical potentials and eventhenonly when the system is not disturbed by interactions with enough energy to disintegrate it. It is

c l e a r t h e n t h a t s u c h n o n - l o c a l c o n n e c t i o n s may be e x p e c t e d t o a r i s e m a i n l y a t v e r y l o w t e m p e r a t u r e s f o r

whichthe random thermal notions do not haveenough energy to breakup thesystem. It will also ariseunder special conditions established in the laboratory in which asystembreaksup without anyexternaldisturbances(asi n the experimentof EPR). Butas a rule we mayexpect that non-local connection willnotnormally be encountered under ordinaryconditions, in which every system is bathed in electromag-netic radiation andi ssubjected to external perturbations of all kinds a s well as random thermal energies

(usually in the form of phonons).

Moreover, because typical systems are constituted of particles with classical potentials V(r) t h a t f a l lo f f with t h e d i s t a n c e , t h e y t e n d a t a p p r e c i a b l e t e m p e r a t u r e s t o f o r m r e l a t i v e l y i n d e p e n d e n t b l o c k s w i t hcorrespondinglyweak interaction between theblocks, along the lines suggested by Kadanoffet al. [10].T h e v a r i o u s d i s t u r b a n c e s m e n t i o n e d a b o v e , w i l l t h e n e n s u r e t h a t t h e b l o c k s w i l l h av e p r o d u c t w a v efunctions, so that the various parts of a system will be executing independentdances. This completesthe demonstration that the relative independence characteristic of classical mechanics will generally bemaintainedatthe large-scale level while on a sufficiently small scale, the size of which is dependent ontemperature and other conditions, there will generally be non-local connections and other forms ofquantumwholeness.

The above-described fragility of quantum behaviour at a large-scale level holds also for theo n e - p a r t i c l e s y s t e m . We h av e i n d e e d a l r e a d y i n d i c a t e d , f o r e x a m p l e , t h a t s p e c i a l e x p e r i m e n t a l

conditions are needed to establish well-defined quantum states with very high quantum numbers.Consider, for example, a particle with coordinates x, in such a stationary state that interacted with afree particlewith coordinate y , through aclassical interaction potential V(x y) o f r a t h e r s h o r t r a n g e .Atreatmentsimilarto that used for transition,fission andfusion would thenshow that a setof separatechannels would ultimately be created, in which the x particle will be in localized packets,with a spreadcorresponding roughly to the range of the potential. Since these packets are too narrow to be

bohm hiley kaloyerou 1986

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