No title

NON-LOCALITY AND LOCALITY IN THE STOCHASTIC INTERPRETATION OF QUANTUM MECHANICS

D. BOHM and B.J. HILEY Department of Physics, Birkbeck College (London University), Malet Street, London WC1 E 7HX, UK

I NORTH-HOLLAND AMSTERDAM -

PHYSICS REPORTS (Review Section of Physics Letters) 172, No. 3(1989) 93—122. North-Holland, Amsterdam

NON-LOCALITY AND LOCALITY IN THE STOCHASTIC INTERPRETATION OF QUANTUM MECHANICS D. BOHM and B.J. HILEY Department of Physics, Birkbeck College (London University), Malet Street, London WCI E 7HX. UK Received August 1988

Contents: I. 2. 3. 4. 5.

The necessity of non-locality in the quantum theory Brief résumé of the causal interpretation The stochastic interpretation: the single particle The many particle system Nelson’s theory of stochastic quantum mechanics

95 97 99 104 105

6. Extension to the Pauli equation 7. Extension to relativity 8. On the meaning of the non-Lorentz invariant background field References

111 114 118 121

Abstract: We review the stochastic interpretation of the quantum theory and show that, like the causal interpretation it necessarily involves non-locality. We compare and contrast our approach with that of Nelson. We then extend the stochastic interpretation to the Pauli equation. This lays the ground for a further extension to the Dirac equation and therefore enables us to discuss this interpretation in a relativistic context. We find that a consistent treatment of non-locality can be given and that it is indeed possible further to regard this non-locality as a limiting case of a purely local theory in which the transmission of what we have called active information is not restricted to the speed of light. In this case both quantum theory and relativity come out as very good statistical approximations. However, because this basically local theory implies that these latter are not exactly valid, it is possible to propose tests that could in principle distinguish such a theory from the current theories.

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1. The necessity of non-locality in the quantum theory The consideration of Bell’s inequality [1] and the Aspect experiments [2] has recently brought the question of the non-local implications of the quantum theory to the fore. Physicists have generally had a great deal of difficulty with non-locality and indeed often feel a certain revulsion towards it which is so strong that they would prefer not to consider the idea even as a possibility. Nevertheless, it is not sufficiently realised that all the commonly accepted interpretations of the quantum theory that have been proposed thus far imply some kind of non-locality [3]. The interpretation that is most broadly used in the actual work of physicists is probably that of von Neumann [4]. The essential feature of this interpretation is that if an observable 0 is measured, the wave function “collapses” to a particular eigenfunction, qi,,, of the corresponding operator with a probability 2 where C,, is the coefficient in an expansion of the wave function iji = E C,, ui,,. In the case of the Einstein—Podolsky—Rosen experiment we begin with a wave function of two particles in correlated states. For example, in a state of overall spin zero the total wave function is ~1’(x1,x2)

=

[~+(x1)~~(x2)

-

~(x1)~+(x2)].

When we measure the spin of the first particle, this wave function will “collapse” with equal probability into either &~t,/ç(x1)t~r_(x2)or e’~’2,/,_(x1)t/,+(x2), where e1”l and e’~2 are arbitrary phase factors. Although only particle 1 has been affected during the measurement, the quantum state, ~li(x 2),of particle 2 has been detennined in a correlated way in spite of the fact that, by hypothesis, there has been no connection of any kind between them. It is as if, when the spin of the first particle was measured the wave function of the second has also collapsed into the corresponding state (and it may be verified in an ensemble of measurements that in all cases in which the first particle is left in a particular state, the second particle will have all the statistical properties that it would have had if it had been measured directly.) A similar conclusion clearly follows in the Everett many universes theory [5],a measurement carried out on one particle will bring about a set of universes in which the second distant particle has “collapsed” into a corresponding quantum state. The assumption of many universes still does not change the fact that in the initial universe the process of bifurcation was initiated by~anaction on only one of the particles. This means that in the many universes theory what happens to any one particle may profoundly affect other particles, no matter how distant they may be. But this is just what is meant by non-locality. Likewise in the interpretation of Wigner [6] in which the collapse is attributed to an interaction of mind and matter (whether this be the individual human mind or some universal mind). Thus if mind interacts only with particle 1, the wave function of particle 2 will still collapse into a definite quantum state. Moreover, even the statistical interpretation has been shown to contain a similar non-locality [7]. All of these interpretations imply that the quantum state corresponds to some kind of at least relatively independent reality (which may, however, be fundamentally altered in a measurement). But in the Bohr interpretation a very different approach is adopted. Here the treatment is quite subtle and we cannot go into it in detail in this paper. However, we feel it important to call attention to Bohr’s notion that because of the indivisibility of the quantum of action, any attempt to attribute an independent reality to a particle, for example, will have an ambiguous significance, the minimum extent of the ambiguity being determined by the Heisenberg uncertainty relationships. To deal with this

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situation consistently Bohr proposes, in effect, that an experiment deals only with a whole phenomenon which cannot properly be divided into parts and which cannot be followed in detail in its inner movements. It is not immediately clear from Bohr’s writings whether or not in any way at all he admits the notion of an independent “quantum reality” (see Folse [8]). However, even if it should be true that he tacitly accepts such a notion, it is implied that nothing whatsoever can be said about this reality and that indeed, as d’Espagnat has suggested, it must be regarded as eternally “veiled” [9]. If nothing more than this can be said about an indej~endentquantum reality then it is clear that the question of non-locality of connections between distant particles cannot even arise, because there is no way to discuss the properties of the second particle except in the context of an experiment in which these are actually measured. It is here that Bohr differs fundamentally from von Neumann and all the other interpretations that we have discussed so far. However, it should be noted further that if Bohr makes the discussion of the question of non-locality meaningless, he does exactly the same with the question of locality. For those who find the concept of non-locality too mysterious, or to use Einstein’s word, “spooky”, for their tastes, Bohr’s interpretation can bring little comfort. For it implies that independent reality at the quantum level is so mysterious that it cannot be described at all, either as non-local, or as local, or indeed in any other way. Nevertheless, it is not clear that the Copenhagen interpretation has completely banished the question of non-locality. Thus Pauli, who was originally one of the main architects of this interpretation, has said that this approach does indeed have a kind of non-locality in it. As Laurikainen [10] remarks, “The famous Einstein—Podolsky—Rosen paradox was no problem at all for Pauli. That locally isolated parts of a closed system seem to have instantaneous actions at a distance is not surprising if we think that the concepts of space and time are not applicable to microphysical systems in the same way as we are used to applying them in the macroworid.” The causal interpretation of the quantum theory [11, 12] brought out the question of non-locality much more sharply than is possible in the other interpretations, as this was explained by the assumption of a new kind of quantum potential allowing for a strong interaction of distant particles. In this interpretation the EPR experiment presents no serious problem, as in any measurement process that disturbs the first particle, it can be shown that the quantum potential will produce a corresponding state of the second particle. What remains to be considered is the stochastic interpretation. By its very name this interpretation seems to imply a theory in which quantum mechanics could be explained in terms of particles undergoing independent random processes and thus, one could easily be led to expect that at least in this theory there would be no need for non-locality. For example Nelson [13] who has worked extensively on this interpretation has written, “But the whole point (of the stochastic theory) was to construct a physically realistic picture of microprocesses, and a theory that violates locality is untenable.” From this it is clear that Nelson initially expected the stochastic theory to be local even though he finally came to the conclusion that it has to be non-local. The main purpose of our paper is first of all to give a systematic and coherent account, which makes it clear that there is no way to avoid non-locality in the stochastic interpretation. Second our purpose is to extend the stochastic interpretation to include the Pauli non-relativistic theory of spin and then to go on to a relativistic theory covering both fermions and bosons, so that in principle all known particles could be included. In doing this we have been led to suggest a way of overcoming one of the main objections to non-locality, i.e. that it violates relativity. In our model, however, it is possible that both quantum

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theory and relativity are merely very good approximations to a deeper theory in which certain connections that are faster than light are possible. If these connections are very fast, yet not instantaneous, then our model becomes local, in the sense that EPR-type measurements made at very precisely defined times may satisfy the Bell inequality while measurements that are less precise will not satisfy it. In this way (as well as in other ways) it becomes possible to suggest experimental tests that could show the difference between our theory and currently accepted theories.

2. Brief résumé of the causal interpretation In the causal interpretation the primary concept was introduced that a particle has a definite path which is determined by a suitable equation of motion and that this path is fundamentally affected by a new kind of field tfr which satisfies Schrödinger’s equation ih

=

—

~—

V2t~+ Vçli.

To develop this model we start by considering the classical limit of this equation which is given by the WKB approximation. This is obtained by writing the wave function in polar form =

R e’~

Schrödinger’s equation then becomes (1)

9t

m

(2)

In the classical limit the last term in eq. (1) can be neglected because R changes very little in comparison with S. This leaves us ~

(3)

The above is just the classical Hamilton—Jacobi equation. This equation is commonly interpreted as determining the path of a particle to be normal to the wave front S = const. with momentum p = VS. It actually determines an ensemble of possible trajectories, only one of which represents the actual path of the particle. Equation (2) which contains the quantity P = R2 could then be regarded as describing the conservation of probability P in this ensemble of trajectories. So our picture is that we have a wave with a probability distribution of particle trajectories that are normal to it. We are led to the causal interpretation of the quantum theory if we notice that the same picture can hold if we do not make the WKB approximation in eq. (1), but simply suppose that in addition to the classical potential V, the particle is acted on by a further quantum potential

98

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics ‘,

(4) In a number of papers [11, 14, 15] it has been shown that this picture gives a consistent account of the quantum theory in every respect that is physically meaningful. Although at first sight it seemed that it might be a return to the old classical ideas this is not actually so, because the quantum potential has a number of striking new qualitative features. These include the following: (1) The quantum potential does not depend on the strength of the Schrddinger wave ~i, but depends only on its form. This means that its full effect may be operative even where the wave is very weak. We have proposed to explain this with the aid of the concept of active information. The particle is assumed to move under its own energy but (as was indeed suggested first by de Broglie) [16]it is “guided” by the wave so that its velocity is maintained as V=VSIm. Evidently this guidance condition also does not depend on the amplitude of the wave but only on its form. To bring out the novel implication of this idea we made a comparison to a ship on automatic pilot which is guided by information in radio waves. In its self-produced motion it will undergo accelerations which could be compared to the effect of a potential that did not depend on the amplitude of the radio wave. Similarly the electron in its self-motion, which is directed by the “guiding” information, will likewise undergo an acceleration which we now attribute to the quantum potential and which also does not depend on the amplitude of the wave. In this way we understand how the particle may be strongly linked to distant features of its environment, even by wave fields’ that are very weak. We have discussed this general question extensively in our previous paper [15]. In particular, because of the implications of the model described above we have been led to question the generally accepted and traditional assumption that as matter is analysed to smaller and smaller parts, the behaviour of these parts must necessarily become simpler and simpler. We have proposed that this need not always be so and that, on the contrary, it is quite possible that the finer parts might still have a complex and subtle behaviour perhaps even comparable to that of a ship guided by a computer (though not necessarily containing similar mechanisms). The statistical behaviour of systems constituted of a large number of such parts may indeed be much simpler than is the behaviour of the individual parts themselves (as the statistical behaviour of large populations of human beings may be simpler than that of individuals). The laws of physics as applied to macroscopic systems may thus be based on much more subtle and complex laws that apply to individual systems on a finer level. (2) When this model is applied to the many body system we find that, as in the one body system, the effects of the quantum potential as implied by the guidance condition do not necessarily become very small when the wave function is small. (This, too, can be explained by the notion of active information, implying a subtle and complex structure to each of the particles). The result of this is that the particles have the possibility of a non-local interaction. Nevertheless, in the classical limit where the guidance conditions imply a negligible quantum potential, the interactions are all local so that the usual notion of the separability of the universe into different parts is recovered as an approximation. (3) The guidance conditions and the quantum potential depend on the state of the whole system in a way that cannot be expressed as a preassign,ed interaction between its parts. As a result there can arise a new feature of objective wholeness. This is not only a consequence of the non-local interactions but

D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

99

even more it follows from the fact that the entire system of particles is organised by a common “pool of active information” which does not belong to the set of particles, but which, from the very outset, belongs to the whole. Therefore at the quantum level a whole is not merely a convenient way of looking at a set of interacting parts, but has a new quality of intrinsic wholeness (which Bohr recognized from a different point of view). Trajectories have been worked out for a number of interesting cases of the causal interpretation including the two-slit system and the particle penetrating a barrier [17—19]. These give a simple and vivid illustration of how such quantum processes work. However for stationary states with real wave functions we obtain S=const.

and p=VS=0.

Thus, for example, an electron in the s-state of the hydrogen atom is standing still somewhere in the

region where the wave function is appreciable. The reason for this is that in such a state the quantum potential exactly balances the classical potential so that there is no net force on the particle. This feature has been regarded as unsatisfactory by many physicists including Einstein even though it has been demonstrated that it is perfectly consistent. For example one can show that in a measurement of momentum, the disturbance produced by the apparatus would give rise to the usual probability distribution of momenta. Nevertheless, it must be admitted that this feature of the interpretation is counter to our physical intuition, i.e. that the equilibrium state in an atom should be the result of a dynamic process and should not represent a completely static situation. In the stochastic interpretation

this unsatisfactory feature will be seen not to arise.

3. The stochastic interpretation: the single particle The stochastic interpretation of the quantum theory was first introduced by Bohm and Vigier [20]. Later Nelson and others [21—24] developed a somewhat similar model which was, however, also different in certain significant ways that we shall discuss later. The basic ideas were first applied to a single electron. In particular Bohm and Vigier assumed that the electron is a particle suspended in a Madelung fluid whose general motion is determined by the Schrödinger equation with the density p(x) = ~(x)I2 and the local velocity v = VS/rn. The particle suspended in this fluid would be carried along with the local velocity. It was then assumed that the fluid has a further random component to its local velocity which could arise from a level below that of the quantum mechanics. This random motion will also be communicated to the particle so that it will undergo a stochastic process with a trajectory having the average local velocity and a random component. No detailed assumptions were made about this random component but it was shown that under certain fairly general conditions an arbitrary probability distribution, P(x, t), would approach the quantum mechanical distribution p = I l/1(x)~2as a limit. As pointed out by Bohm [25],this general result does not require the assumption of a Madelung fluid. All that is needed is to suppose that there is a mean velocity given by v = VS/rn, together with an additional stochastic contribution. For example, this latter may come, as proposed by Nelson [22],from a randomly fluctuating background field. Or else it may be regarded as an implication of some kind of

“vacuum fluctuations” similar in some ways, to the effects of a space-filling medium or “ether” which is

100

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

undergoing internal random motions. This notion has been emphasised by DeWitt [26] and has also been used by Vigier [27] in the context of the stochastic interpretation. The original model of Bohm and Vigier [201was not developed in further detail as this did not seem to be called for at that time. However, it now seems appropriate to examine this again. To this end we assume that whatever the origin of the random motion may be, it can be represented by a simple diffusion process. To illustrate this, let us consider the Brownian motion of particles in a gravitational field using the simple theory of Einstein. If P is the probability density of particles, then there is a diffusion current, (5)

—DVP,

j(d)

where D is the diffusion coefficient. If this were all there was, the conservation equation would be

ÔP/ôt=—DV2P

(6)

and clearly this would lead to a uniform equilibrium distribution. However, as Einstein showed, in a gravitational field there is what he called an osmotic velocity, mg uo=D~jz.

(7)

The conservation equation then becomes

~=-DV[~zP+VP].

(8)

For the equilibrium distribution, c?P/ôt = 0, and VP -~-

mg =

—

z + const.

or P = A e

-

~/kT

(9)

mg~

which is the well-known Boltzmann factor. The picture implied by the Einstein model of diffusion is that the particle is drifting downward in the gravitational field and that the net upward diffusive movement balances this to produce equilibrium. It is worth while to provide a still more detailed picture of this process. To do this let us consider a simple one dimensional model in which there is a unique free path A and a unique speed v. We assume, further that after collisions the velocities are randomly distributed in positive and negative directions while the speed is still ü. Let us now consider two layers which are separated by A. The net diffusion current between these layers will be j(d)

~

~

A.

‘

Between collisions the average velocity gained from the gravitational field is

(10)

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

101

(11)

~

The above is evidently the osmotic velocity. The net current is ~

gA1,

2 ~i kT/m and D

—

Writing

VtPA 2 9Z

~.2=

=

~ —

2

2

L~

12

~

~JAI2we have

J=_D[~P+~]

(13)

and this is just the equation that Einstein assumed. It is important to emphasise that at least in this case the osmotic velocity is produced by a field of force. Without such a field there would be no reason for an osmotic velocity. For the stochastic interpretation of the quantum theory we would like to have a random diffusion

I tiiI~

process whose equilibrium state corresponds to a probability density P = = p and to a mean current j = pu = pVSIm. Such a state is a consistent possibility if ~ti= \/p e~ satisfies Schrödinger’s equation because this implies the conservation equation 9p/9t+V~j0.

(14)

In order to have p = I 1/112 as an equilibrium density under such a random process, we will have to assume a suitable osmotic velocity. We do not have to suppose, however, that this osmotic velocity is necessarily produced by a force field, similar to that of the gravitational example, but rather it may have quite different causes. (Thus as suggested by Nelson [13], some kind of background field that would produce a systematic drift as well as a random component of the motion.) At this stage it will be sufficient simply to postulate a field of osmotic velocities, u 0 (x, t), without committing ourselves as to what is its origin. We therefore assume an osmotic velocity u0=DVp/p

(15)

and a diffusion current (16)

j(d)_Dvp

The total current will be j=~VS+DP~i_DVP. rn

p

(17)

The conservation equation is then rn

p

(18)

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D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

In the above equations there is a systematic velocity =

VS m

—

+

D

Vp p —.

This is made up of two parts, the mean velocity v = VS/p and the osmotic velocity u0 = DVp/p. The mean velocity v may be thought to arise from de Brogue’s guidance condition. As explained earlier the osmotic velocity will arise from some other source, but the main point is that it is derivable from a potential D ln p where p is a solution of the conservation equation

(~!.~)

=0

It follows from eq. (18) that there is an equilibrium state with P = p in which the osmotic velocity is balanced by the diffusion current so that the mean velocity is ü =VSIm. But now we must raise the crucial question as to whether this equilibrium is stable. In other words will an arbitrary distribution P always approach p? To simplify this discussion let us first consider the case of a stationary state in which VS = 0. Writing P = Fp we obtain the following equation 2F. (19) p aFh9t = D div(pVF) = DVp ~VF+ DpV We shall show that F must eventually approach 1 everywhere by proving that the maxima of F must always decrease and the minima must always increase. Maxima and minima are characterised by VF = 0. At such points we have pôFIôt—DpV2F.

(20)

At a maximum V2F is negative and clearly F must be decreasing at this point. At a minimum V2F is positive and therefore F must be increasing. This can only cease when F = 1 everywhere. This proof can be extended to the case where VS 0. To do this, let us first note that, as has been shown earlier, P = p is still an equilibrium distribution. It has been shown by Bohm and Vigier [20] however that if P = p is such an equilibrium distribution and if p satisfies a conservation equation, then for the general case, P approaches p. For times much longer than the relaxation time, r, of this process we will therefore quite generally encounter the usual quantum mechanical probability distribution. It is implied, of course, that at shorter times this need not be so and therefore the possibility of a test to distinguish this theory from the quantum theory is in principle opened up. As to the conditions under which such a test may be possible we shall discuss this question in the last section of this article. For the equilibrium case the mean velocity is, as has been demonstrated, ü=VS/m. Clearly this relation by itself does not determine the acceleration. To find this we need a differential equation for S. Thus, if 1/s = Vp e’5~satisfies Schrodinger’s equation then, as we have seen in eq. (1), S will satisfy the extended Hamilton—Jacobi equation and from this it follows that the mean acceleration will be mdU/dt= —V(V+

Q).

(21)

D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

103

It is clear that in the present approach the quantum potential is actually playing a secondary role. The

fundamental dynamics is determined by the guidance condition and the osmotic velocity along with the effects of the random diffusion. All these work together to keep a particle in a region in which p = 1/’ 2 is large and where its average velocity fluctuates around VS/rn. As explained earlier the quantum

potential merely represents the mean self-acceleration of the particle under the influence of the de Broglie guidance conditions, and this will be valid only if 1/s satisfies Schrödinger’s equation. If i/i had satisfied another wave equation, the mean acceleration would have been different. In fact we shall later extend this theory so that the wave function satisfies the Dirac equation, for which a very different mean acceleration is implied and in which the quantum potential does not apply. To illustrate the stochastic model, let u~consider the two slit interference experiment. A particle

undergoing random motion will go through one slit or the other, but it is affected by the Schrödinger field coming from both slits. In the causal interpretation this effect was expressed primarily through the quantum potential. In the stochastic interpretation it is primarily expressed through the osmotic velocity which reflects the contributions to the wave function 1/’ coming from both slits. Near the zeros of the wave function the osmotic velocity approaches infinity and is directed away from the zeros. Thus a particle diffusing randomly and approaching a zero is certain to be turned around before it can reach this zero. This explains why no particle ever reaches the points where the wave function is zero. And, as we have indeed already pointed out, the osmotic velocity is constantly pushing the particle to the regions of highest I 1/42 and this explains why most particles are found near the maxima of the wave function. Without assuming an osmotic velocity field of this kind, there would be no way of explaining such phenomena. As a result of random motions for example a particle just undergoing a random process on its own would have no way of “knowing” that it should avoid the zeros of the wave function. To obtain a consistent picture we must consider the random background field. This is assumed, as we have already pointed out, to be the source of the random motions of the particle, but in addition it must determine a condition in space which gives rise to the osmotic velocity. Indeed, a single particle in random motion cannot contain any information capable of determining, for example, that every time it approaches the zero of a wave function it must turn around. This information could be contained only in the background field itself. Therefore, as has already been pointed out in section 1, the stochastic model does not fulfil the expectations that would at first sight be raised by its name, that is to provide an explanation solely in terms of the random movements of a particle without reference to a quantum mechanical field (which may be taken as p and S or as i/i = V~5e’~). In the causal interpretation this field has the property that its effect does not depend on its amplitude. As we have suggested earlier, this behaviour can be understood in terms of the concept of active information i.e. that the movement comes from the particle itself, which is however “informed” or “guided” by the field. In the stochastic interpretation there is a further effect of the quantum field through the osmotic velocity, which is also independent of its amplitude. We can therefore say that along with the mean velocity field, VS/rn, the osmotic velocity field constitutes active information which determines the average movement of the particle. This latter is however modified by a completely random component due to the fluctuations of the background field. Clearly then, there are basic similarities between the causal interpretation and the stochastic interpretation (some of which will, however, be discussed only later). Nevertheless, there are also evidently important differences. One of the key differences can be seen by considering a stationary state with S = const. e.g. an s-state. In the stochastic model the particle is executing a random motion which

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D. Bohm and B.J. Hiley. Non-locality and locality in the stochastic interpretation of quantum mechanics

would bring about diffusion into space, but the osmotic velocity is constantly drawing it back so that we obtain the usual spherical distribution as an average. But now the basic process is one of dynamic equilibrium, the average velocity, which is zero, is the same as the actual velocity in the causal interpretation. Such a view of the s-state as one of dynamic equilibrium seems to fit in with our physical intuition better than one in which the particle is at rest.

4. The many particle system The extension of this model to the many particle system is straightforward. The wave function, 1/i(x~,,),which is defined in a 3N-dimensional configuration space, satisfies the many-body Schrödinger equation. We assume that the mean velocity of the nth particle is =

~j

~-~—

S(x,,).

(22)

In addition we assume an arbitrary probability density P(x,,) and a random diffusion current of the nth particle =

—D dPiax~,,

(23)

.

We then make the further key assumption that the osmotic velocity component of the nth particle is ~

=

(Dip) 8p19x1,,

,

(24)

where p= q1~2. From here on the theory will go through as in the one-particle case and it will follow that the limiting distribution will be P = 1/’ 2 Equation (23) describes a stochastic process in which the different particles undergo statistically independent random fluctuations. However, in eq. (24) we have introduced an important connection between the osmotic velocities of different particles. For the general wave function that does not split into independent factors, the osmotic velocities of different particles will be related and this relationship may be quite strong even though the particles are distant from each other. This means that we cannot eliminate quantum non-locality by going to the stochastic model. For in this model there is tacitly assumed an effectively instantaneous non-local connection which brings about the related osmotic velocities of distant particles (and in addition, of course, the mean acceleration equation (21) will still be determined by a non-local quantum potential). To bring out the full meaning of this non-locality it is useful to discuss the Einstein—Podolsky—Rosen experiment in terms of the stochastic interpretation. This requires, however, that we extend the theory of the measurement process which was developed in connection with the causal interpretation to the stochastic model. We therefore first give a brief résumé of how the causal interpretation treats the measurement process [14]. The point that is essential here is that in a measurement process the “particles” constituting the measurement apparatus can be shown to enter a definite “channel” corresponding to the actual result of the measurement. After this they cannot leave the channel in question because the wave function is

D. Bohm and B. I. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

105

zero between the channels. From this point on the wave function of the whole system effectively reduces to a product of the wave function of the apparatus and of the observed system. It was demonstrated that thereafter, the remaining “empty” wave packets will not be effective. It follows that the net result is the same as if there had been a collapse of the wave function to a state corresponding to the result of the actual measurement. It is clear that a similar result will follow for the stochastic interpretation because here too there is no probability that a particle can enter the region between the “channels” in which the wave function is zero. Let us now consider the EPR experiment in the stochastic interpretation. We recall that in this experiment we have to deal with an initial state of a pair of particles in which the wave function is not factorizable. A measurement is then made determining the state of one of these particles, and it is inferred from the quantum mechanics that the other will go into a corresponding state even though the Hamiltonian contains no interaction terms that could account for this. In the causal interpretation the behaviour of the second particle was explained by the non-local ‘features of the quantum potential which could provide for a direct interaction between the two different particles that does not necessarily fall off with their separation and that can be effective even when there are no interaction terms in the

Hamiltonian. In the stochastic interpretation all the above non-local effects of the quantum potential are still implied, but in addition there is a further non-local connection through the osmotic velocities. And when the properties of the first particle are measured the osmotic velocity will be instantaneously affected in such a way as to help bring about the appropriate correlations of the results. It is clear that the stochastic interpretation and the causal interpretation treat the non-local EPR correlations in a basically similar way. The essential point is that in an independent disturbance of one of the particles, the fields acting on the other particle (osmotic velocities and quantum potential) respond instantaneously even when the particles are far apart. It is as if the two particles were in instantaneous two-way communication exchanging active information that enables each particle to “know” what has happened to the other and to respond accordingly. Of course in a non-relativistic theory it is consistent to assume such instantaneous connections. We shall, however, show later how these considerations can be extended to take into account the fact that the theory of relativity has been found to be valid in a very broad context.

5. Nelson’s theory of stochastic quantum mechanics We give here a brief résumé of Nelson’s theory of stochastic quantum mechanics in order to bring

out its similarities and differences to the theory presented in this paper [21, 22]. Nelson begins in essence by describing the path of a particle in terms of a Markov process (defined in more detail as a Wiener process). To describe this Markov process, let us write K = K(i~x,r, x, t) as the probability that a particle at (x, t) will arrive at x + L~xat the time t + T. By definition

J

K(i~x,r,x,t)d&r1.

We also define the net distribution of particles P(x, t) with

(25)

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D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

J

P(x,t)dx=1.

(26)

We now consider how the distribution P(x, t) changes with time. It follows from eq. (25) that P(x, t + T)

=

f

K(Ax,

T,

x

—

Ax, t)P(x

—

Ax,

t)

d Ax.

(27a)

In a Markov process, Ax contains a term proportional to T which represents the average velocity and a term proportional to T1~which represents the random diffusion. Under the assumption that P and K are continuous functions of x, we make the following Taylor expansion K(Ax, r, x

—

Ax, t)P(x — Ax, t)

=

K(Ax, r, x, t)P(x, t) +

—

~ Ax

[K(Ax, T, x, t)P(x, t)]

~ Ax~Ax 1

[K(Ax, T, x, t)P(x,

t)] ~

(28)

We then define a mean velocity b(x, t) b~(x,t)r

=

K(Ax, r, x, t) Ax~d Ax

(29)

and a diffusion coefficient D~1= ~

f

Ax~Ax1 K(Ax, T, x, t) d Ax.

(30)

Equation (27a) then becomes ~=-~--(b~P)+~

Under the simplifying assumption that D.1 2P=0. ~

(31)

~(D~1P).

=

D811 the above reduces to the Fokker—Planck equation, (32)

+V.(bP)—DV

This is indeed the same as eq. (17) with b=

m

+

D

p

so that the mean velocity represented by b is the quantum mechanical mean velocity VS/rn plus the osmotic velocity D Vp/p. Let us recall that in our treatment p is not the probability density but rather that in general it is a field quantity satisfying

D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

107

+V.j=0 and in particular for non-relativistic quantum mechanics it is assumed that p = 1/12. Equation (32) then guarantees, as we have shown, that P = p in the equilibrium distribution. It is important to emphasise once again that P and p are conceptually different and that they are equal only in the special case of equilibrium. In our approach one assumes boundary conditions at a certain moment of time corresponding to a specified probability distribution P(x, t), and one follows the change of P with time through eq. (17) or eq. (32) which describes an irreversible development towards equilibrium. Nelson, however, adopts a rather different approach. He feels that “Nature operates on a different scheme in which the past and the future are on an equal footing” [28]. Therefore he postulated that a complete symmetry between past and future has to be maintained. If we go back to eq. (27a) we can see that the probability distribution K(Ax, r, x, t) corresponds to

an ensemble of those particles that pass through the point x at the time t and reach the point x + Ax at the time t + r. To maintain symmetry between past and future we must put into a corresponding role that ensemble of particles which passes through x at the time t and which comes from x Ax at time —

t

—

T.

This is described by the function K~(Ax,r, x, t) and the propagation equation for the probability P(x, t — r) =

J

K*(Ax;

T,

x

—

Ax, t)P(x — Ax, t) d Ax.

(27b)

The above equation implies that one starts from x and projects the motion backward in time. Because the stochastic process is not continuous, the forward propagator K and the backward propagator K~are not necessarily the same. K represents the ensemble of particles that diverges from x and K~the one that converges on x. Therefore if we use the probability distribution of paths that converge on x we will come to a second Fokker—Planck equation ôPh9t + V• (b~P)+ D~V2P= 0.

(33)

The appearance of the plus sign in front of D * shows that the diffusion may be described as if it took place in the reverse direction in time. But, of course, in the forward direction we actually have antidiffusion. Under reasonable conditions we can readily obtain D * = D. However, the mean velocity b * is not in general the same as the mean velocity b. We can find the relationship between them by noting that the function P appearing in eq. (33) is the same as that appearing in eq. (32). By subtracting eq. (33) from (32) we obtain

V• [P(b be)] —

2DV2P.

From this it follows that b—b~=2DVP/P. Evidently the osmotic velocity is given by

(34)

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D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

(35)

U=!~~*=DY~.

Here we see the first important difference between Nelson’s treatment and our own. Whereas we assume an osmotic velocity D Vp/p, Nelson deduces this as D VP/P. He is able to do this because he has assumed an equilibrium distribution that both the forward and reverse Fokker—Planck equations can be applied to the same system. Nelson’s next step is to define the mean acceleration of a particle. The most natural way to define the acceleration would be to start with what Nelson regards as the appropriate mean velocity -

b+b~ 2

VS m

This evidently treats both backward and forward distributions symmetrically. To find the mean rate of change of this velocity we have first to consider how any quantity f(x, t) is transported in the diffusion process. Let us begin with the forward diffusion process. The change of f(x, t) resulting from a change of position Ax during the time interval ‘r will bef(x + Ax, t + r) f(x, t). This must be averaged over all those particles that have come through x at the time t and are diverging into a distribution at the time t + T. We thus obtain for Af, the net rate of transport of f —

Af

J

K(Ax,

T,

x, t)[f(x

+

Ax, t + r) — f(x, t)] d Ax

(36)

and this is approximately equal to (37) According to the principle of time symmetry we can define the “reverse” transport equation (38) Let us now apply these considerations to the definition of the acceleration. At this point, however, we encounter an ambiguity in the definition of the acceleration, i.e. should we use A, or A~or some combination of these two? With the use of Nelson’s principle of symmetry, the ambiguity can be decreased but it cannot be eliminated altogether. Thus to obtain the acceleration we could put v = f in eqs. (37) and (38). The most obvious definition of acceleration satisfying Nelson’s symmetry requirement would be a This leads to the equation oi~ fb+b~\ a=-~---+~ 2 )(v~~)v -

-

-

(39)

D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

109

This is of course the usual expression for the acceleration. What it signifies is that the mean velocity ii is transported by ü itself. With this formula, Newton’s law becomes rna = F. In our treatment we have taken F=V[V+Q]

and thus we are in effect simply assuming the quantum potential along with the classical potential (as is indeed also done in the causal interpretation). However, Nelson proposes that the acceleration should not be defined by eq. (39) but rather in another way. Using eq. (36) he defines the forward velocity as Ax(t)

=

J

K(Ax, r, x, t) [x(t + r)— x(t)] d Ax

(40a)

and he gives a corresponding definition for the backward velocity,

A*xO)

=

J

K*(Ax, r, x, t) [~t)

—

x(t

—

r)] d Ax.

(40b)

He then proposes to define the acceleration as a’

=

~[AA~+ A~A]x.

(41)

What this means is that the forward velocity distribution is being transported by the backward distribution and vice versa.

We have been unable to obtain a clear picture of what such a distribution would look like. Aside from this lack of clarity in the kinematic significance of this definition of acceleration there is considerable mathematical arbitrariness as to what the acceleration should be. Thus even if we accept Nelson’s argument of symmetry in time this would still leave open the possibility that acceleration could be given by a” = Aa + ~a’ where A + ,a = 1.

Certainly the simplest definition would be to take the acceleration to be as defined in eq. (39). One can see that Nelson is motivated to take it as a’ given by eq. (41) because if one does this then, as we shall see, we can obtain the quantum potential from Newton’s law as modified by the new definition of acceleration. Thus, by means of a simple calculation Nelson finds that ~ (V)2] p

If we write p

=

2p

R2 and ü = VS, we obtain

loS (VS)2 2 V2R\ rna’=Vt—+ —2rnD —J. \Ot 2rn Ri

Newton’s law rna’ = —VV then reduces to

(42)

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D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

(43)

In addition P = R2 satisfies the conservation equation (44) If we choose D = /l/2m then eqs. (43) and (44) reduce to (1) and (2), which are equivalent to Schrödinger’s equation. To sum up then, Nelson was able to derive the quantum potential (and Schödinger’s equation) by changing the definition of acceleration through the use of eq. (41). But as we have seen, this definition has considerable arbitrariness from the mathematical point of view while it is unclear what it means kinematically or physically. If it could be made clear that this definition is physically or kinematically plausible then Nelson’s approach would evidently have an important advantage. But without such clarification we might as well simply assume Schrödinger’s equation and use the ordinary definition of acceleration whose meaning is clear (and is the most natural one even in Nelson’s theory). A further key difference between our approach and that of Nelson has to do with the question of reversibility. Indeed, as we have already pointed out, Nelson’s treatment is restricted by its very formulation to the assumption of universal reversibility in quantum processes. This means that if there should be a deviation from equilibrium at any time t, it is most likely that this has arisen as a fluctuation from the equilibrium distribution. A similar situation arises in classical statistical mechanics in the proof of the Boltzmann H-theorem. Boltzmann defines a statistical function H and demonstrates that as a result of random processes it is most likely to decrease to a minimum, corresponding to the maximum of the entropy. But because the basic laws of mechanics are reversible, it follows that H is also most likely to decrease to a minimum if it is projected backward in time, as pointed out in the literature [291. This apparently paradoxical behaviour is understood, as we have indeed suggested for the quantum mechanical case also, by noting that in an equilibrium distribution, any deviation from equilibrium is most likely to have arisen from a situation that was closer to equilibrium in the past, and to return to a situation that is closer to equilibrium in the future. Nelson’s approach effectively restricts us to the assumption that it is impossible to have any distribution in nature other than one that has arisen by a random fluctuation from the equilibrium distribution. It is clear, however, that in ordinary statistical mechanics it is most common that a system far from equilibrium did not get there by such a random fluctuation process. Rather one will almost always find that it was produced by some systematic disturbance from outside the system or by a systematic inner change in the system itself (e.g. a mixture of hydrogen and oxygen ignited by a spark). In such cases a reversible treatment such as that of Nelson would not be applicable. Rather one would have to start with some non-equilibrium distribution P(x, t) and show by a treatment similar to ours that P(x, t) approaches an equilibrium distribution p(x, t) in an irreversible way. In the case of classical statistical mechanics as described above, the fundamental and universal laws are, of course, reversible. Nevertheless, in particular experimental contexts an initial large and systematic deviation from equilibrium will in general not arise from a random fluctuation, but will be a result of contingent boundary conditions that implies a state that is far from equilibrium. As a result of the stochastic process within the limited system itself, these states will relax irreversibly to the state of equilibrium.

D. Bohm and B. I. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

111

The question is then whether the reversible statistical laws that Nelson is assuming to underly the current quantum theory are universal and necessary or whether there is not some broader context of law operating at a deeper level which will allow for sudden changes in the now limited context of the quantum mechanical probability distributions (as happened with the sudden ignition of the mixture of hydrogen and oxygen). In this case the reversible approach of Nelson would be inapplicable and a treatment such as ours, which includes irreversible processes within this limited context would be needed. We shall discuss such a case in the last section of this paper. There is a further conceptual difficulty in Nelson’s approach, in that his treatment of osmotic velocity is not very clear. Thus in our approach the concept of osmotic velocity is sharply distinguished from the probability concept. Indeed, as we have seen earlier, the equilibrium probability density P = p is explained as a result of the balance between the tendency of the osmotic velocity to bring particles to regions of greater p and the random diffusion which opposes this. On the other hand,~in Nelson’s treatment it would at first sight appear that the probability density itself was the cause of the osmotic velocity. We have shown in section 3 that this cannot consistently be maintained because, for example, the mean behaviour of an ensemble of different systems cannot determine why in an individual system the particle keeps away from places where p = 0. Nor can one understand this behaviour from the average properties of an individual system over time. Rather, as we have pointed out, the osmotic velocity has to be regarded as a field representing a condition in space. As a matter of fact Nelson himself also feels there has to be a field behind the stochastic process in order to make physical sense [13]. However, he does not seem to give any clear notion of what this field could be. It is our proposal that this is just the field of osmotic velocities.

6. Extension to the Pauli equation As an intermediate step to the development of a relativistic theory we shall give a brief summary ‘of the stochastic interpretation of the Pauli equation. Indeed there already exists a causal interpretation for the Pauli equation [30]in which the further property of intrinsic angular momentum is attributed to the particle. This interpretation has been extended to the two body system [31]. However, in the many-body system this model loses its intuitive simplicity and involves the introduction of a great many further intrinsic properties of the particles other than the spin, whose meaning is not clear. An even greater difficulty, however, is that with any reasonable size of the particle which may be anywhere between 10_17 cm and i0~cm, the matter at the outer edge of the particle would have to be moving with a speed enormously greater than that of the speed of light if it is to have an angular momentum /1/2. This difficulty was not relevant in the previous papers dealing with this model because they were concerned only with a non-relativistic context in which this question does not arise. But it is relevant here as it is our specific intention in the next section to extend the theory to relativistic quantum mechanics. To avoid these difficulties we shall use an idea originally proposed by Bell in which he generalised the de Brogue pilot wave theory to apply to the Pauli equation [32].To develop this theory, we shall write the Pauli wave function as ,/~(x,t) where the index runs from 1 to 2. We then put this in polar form i~(x,t)

=

R. e1~.

(45)

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D. Bohm and B.J. Hiley, Non-locality and locality’ in the stochastic interpretation of quantum mechanics

The Pauli equation admits a conserved density

and a probability current

~ (1/~’V1/’~ ç1s1V1/i~’) —

~—

= ~

1/’~l~ ~.

(46)

Let us then define a net velocity v=j/p,

(47a)

which evidently satisfies the conservation equation ~+V.(pv)=0.

(47b)

The basic assumption is that the particle is specified completely by its position and momentum and that it is “guided”, not by the de Broglie relation v = VS/rn, but rather by equation (47a) which gives rise to 1 ~VS. v=—~~ç1i~j

(48)

—~.

In effect the guiding velocity is a kind of average over the two components of the wave function. But from this point on the theory goes through in a way that is very similar to that of the de Broglie theory (as explained in more detail by Bell) [32].

Let us now extend this model to the stochastic interpretation. As with the simple Schrödinger equation without spin, we assume that the particle is subject to a Markov process with an osmotic velocity u0=DVp/p

(49)

and a diffusion current (50) where P is the probability distribution of the particle in its random motion. The equation for the conservation of probability becomes

2PO.

OP/Ot+VJP(v—DVp/p)]+ DV

(51)

This problem is the same as the one we have already treated so that it follows that P = p will be an equilibrium distribution. Since this is the usual quantum mechanical distribution in the case of spin, all

the usual results of quantum mechanics will follow. To demonstrate the above in full detail would take us away from the main purpose of the present

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

113

article. However, we can illustrate what is meant here in terms of the simple example of a particle subject to an inhomogeneous magnetic field in the z-direction. Such a field adds a term to the Hamiltonian operator eh IOB\ H1=—u~—j z. 2mc z\Oz/o This term produces opposite effects on the components of the wave functions 1/’~ and 1/’2. 1/’~ gains a positive velocity and begins to move in the + z direction, while 1/’2 moves in the z direction. Thus the two parts of the wave function separate and cease to overlap. While they are separating, the particle in its random diffusion may enter one or other of the packets. But after they have separated it must remain in one packet or the other with a probability proportional to the strength of that packet. (Of course, it is the osmotic velocity that keeps it in its packet in the way that has been explained earlier.) From this point on the particle will respond to a magnetic field in a way that corresponds to the “spin” that we customarily attribute to its packet. For example, if we restrict ourselves to those particles that have undergone a positive deflection in the z-direction and if we once more apply the —

same inhomogeneous magnetic field, these particles will again be deflected positively, just as is predicted by the usual approach to the quantum theory. If however the inhomogeneous magnetic field is applied in the x-direction, this ensemble of particles will be split into two parts of equal probability with opposite deflections, once again just as is predicted in the usual application of quantum theory. If, on the other hand, a constant magnetic field, in the x-direction, is applied to those particles that have

been deflected upwards, the wave function will begin to mix together components of opposite spin quantum number, and these particles will then behave in every way as if their spin had been determined in some direction between that of x and that of z. This too, of course, is what is obtained in the usual

approach to the quantum theory. Along lines of this kind it can be made clear that we are able to explain all the manifestations usually attributed to spin without bringing in anywhere that the particle has an intrinsic angular momentum. What then can be the meaning of this new property of spin? What we have done in effect is to introduce a concept, new in this context, of a property that depends on the relationship between the particle and the field of active information implied by the wave function. The particle itself has no properties other than position and velocity. However, even the velocity depends on the information field and this, in turn, depends on both components of the wave function which combine, as shown in eq. (46), to

determine the information field which “guides” the particle. The existence of properties that depend on the relationship between the system and its environment is not completely new in physics. For example in thermodynamics the property of free energy is not entirely intrinsic, since it corresponds to the work that can be done in a reversible isothermal process, to which the heat bath makes an essential contribution. In a certain sense the Bell model treats “spin” in a similar way because this property depends as much on, for example, the osmotic velocity field, (49), and the guidance velocity field, (48), as it does on the particle itself. Because these fields can produce strong effects where the intensity is weak, such relational properties are much more striking than they are in classical examples such as those given by thermodynamics. This is also true in the causal interpretation in which many properties of the particle, classically regarded as

intrinsic, actually depend crucially on the quantum potential which strongly reflects the environment of the particle. In the Pauli equation such a relational character of particle properties is carried yet further because even the entire spin angular momentum is understood as such a property. Indeed even orbital

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D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

angular momentum can also be shown to be a relational property, e.g. dependent on the quantum potential and on the osmotic velocity, so that conservation of total angular momentum (spin plus orbital) has to be seen as a relational quality rather than an intrinsic property. When we apply the Pauli equation to the many-body system we can see a still greater extension of this principle of relational properties. Thus we write for the wave function for a system of N particles xN) =

If

~

exp

5’l”N

h

(52)

satisfies the many-body Pauli equation, there is a conserved probability density (53)

~

~ ‘l’N

If we define the probability current of the nth particle by in

~

=

R,~... 1 V~S~,..1im ,

(54)

the conservation equation will be (55) The velocity of the nth particle will then be v,, =i~ip.

(56)

To develop the stochastic interpretation we will merely suppose a random process in the configuration space of the N-particles with an osmotic velocity of the nth particle v~=DV~p/p

(57)

and a diffusion current =

—DV~P,

(58)

where P is the probability distribution for the N-particles in their random motion. It is clear that with these assumptions P will approach p and the usual quantum statistical results will 2N components. The information field has thus become much more complex than is required for a particle without spin, but clearly the basic principle of relational properties is still the same.

follow. But of course the osmotic velocity depends on a wave function having

7. Extension to relativity

The natural relativistic extension of the Pauli equation is the Dirac equation. The step from the Pauli equation to the Dirac equation is quite straightforward.

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

115

Let us begin with a one particle system which obeys the Dirac equation. It is well known that this implies the conservation equation Op/Ot+V~j’0

(59)

p=~j1/~~2

(60)

1. t=1,~ j=1

(61)

with

and ~

The similarity to the case of the Pauli equation is very clear. Of course the two cases are different in that the wave function for the Dirac equation has four components rather than two. In addition the current depends directly on the wave function and not on its gradient. These differences are indeed what make possible the relativistic invariance of the Dirac equation. But they are not important for the stochastic interpretation. Thus we can say that the particle is guided not by the de Broglie relation v = VS/rn but rather by the Dirac relation defining a mean velocity u=j/p

(62)

which will satisfy the conservation equation OpIdt+V~(pv)0.

(63)

This is essentially the same form as we had in the non-relativistic case, but the difference is that, because p and v were derived from the Dirac wave function, the content of this equation is now Lorentz covariant. To make a stochastic interpretation we could try to introduce a Lorentz invariant Markov process with a probability density P for the particle. To this end Vigier [27]has suggested, as we have indeed mentioned earlier, that this may be achieved on the basis of an extension of Dirac’s notion [33]of a covariant ether which would have a Lorentz invariant distribution in its local velocity. A basic difficulty with this notion is that there can be no upper limit to the four-velocity in a covariant distribution. Therefore this ether would have to have an infinite energy density. This concept does not seem satisfactory since it only adds to the problem of infinities that are already present in current field theories. A number of theories have been proposed, which aim to provide a covariant stochastic explanation of the Klein—Gordon equation (considered as applying to a particle). All of these imply the possibility of backward motion in the time. This is however subject to the paradox that a change at a given time can change what comes earlier, while this in turn will change what happens at the originally considered moment, etc. To be sure Feynman has used such backwards tracks and has reinterpreted them in terms

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D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

of pair creation, but this was only to obtain a mathematical formula for the calculation of matrix elements, and was in no sense the basis of a causal or stochastic picture of an actual process. Covariant theories of diffusion of this kind have several further difficulties. One of these is that they all [34—36]imply a process of diffusion in the time direction. In terms of the proper time, r, of the particle this implies that in addition to a systematic drift component in the motion of the order of Ar, there will be a random component of the order of (Ar)”2 Therefore for short intervals there is bound to be backward motion in the time with all the problems that we have seen to be implied by this. In addition there appears to be the possibility that the osmotic velocities will also be backward in the time. In view of all these difficulties and unclarities we are doubtful that a satisfactory covariant theory of diffusion is possible. But is it actually necessary for the stochastic process to be covariant? All that we really require is that the equilibrium distribution be covariant, because this is all that is relevant for the explanation of the current relativistic quantum theory. We are therefore led to adopt a different approach to the problem. We will assume a field that fluctuates in a way that is random but with a distribution of fluctuations that is not covariant. This implies that there is a certain general frame in which the diffusion current can be written as ~(d)

=

—DVP,

(64)

while the osmotic velocity is u 0=DVp/p.

(65)

The conservation equation then becomes ~+V.[P(v+DVp/p)1_DV2P=0.

(66)

From the arguments we have already given, the equilibrium probability distribution will be P = p and the mean velocity will be v as given in eq. (62). The whole meaning of the equilibrium distribution will, however, be covariant because it is determined ultimately from solutions of the Dirac equation which is Lorentz invariant. We have therefore achieved our objective of obtaining a covariant equilibrium distribution even though the random processes that establish it are not. This is a very important point as it implies a model of a very different kind from those which have been suggested thus far. To emphasise this, we shall give the main features of this model once again. Firstly, we assume that there is a very complex randomly fluctuating background field which is not Lorentz covariant. We then assume that this field also determines the osmotic velocity u0 = D(Vp) /p. This latter is evidently conditioned by a field p that is covariant in its content. As a result of this non-covanant random process which is linked to a covariant field, there arises a covariant equilibrium distribution. It is indeed common for equilibrium distributions to be essentially independent of a wide range of processes by which they may be established. So there is no essential difficulty in having a non-covariant stochastic process establish a covariant equilibrium distribution. We can readily extend this theory to the many body system. As is well known, the Hamiltonian is the sum of the free particle Hamiltonians, together with local field interactions. The total wave function of the system is defined by the components 1I’~ (x1 . . . xN) where each i in the set runs from 1 4 and the —~

D. Bohm and B.J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

set runs from i, to

~N.

117

We can consider all the particles at a common time. Let us write (67)

and define

~

,

~{1}(a~){~}{)}1/J{J}

= {l}

{J)

(68)

(where a~is the Dirac matrix corresponding to the nth particle). We then obtain (69) As in the Pauli theory we define a probability density, P, in the configuration space of the N

particles. We then go on to define an osmotic velocity of the nth particle u0~DV~pIp

(70)

and a diffusion current for the nth particle J~=—DV~P.

(71)

From here on the problem is essentially the same as for the Pauli equation and the proof that P approaches p goes through in the same way.

The basically new feature brought in by the many-particle Dirac equation is pair creation. At present this is most commonly treated using the fermionic annihilation and creation operators. However, Dirac originally treated the same problem in terms of the particle theory. If the wave function is completely antisymmetric, the exclusion principle will be valid. It is therefore possible to assume that the negative

energy states are filled in the vacuum state. Pair creation is then described as the jump of one of the vacuum state electrons to a positive energy state leaving behind a “hole” which acts like a positron. All the probabilities of transition and the differences of all energies from that of the vacuum state are exactly the same with this treatment as they are when fermionic operators are used. (The complete

equivalence of the fermionic field treatment and the particle treatment using antisymmetric wave functions is well known.) As with the one-particle system we obtain a Lorentz invariant distribution of equilibrium probabilities even though the random process that gives rise to these possibilities is not Lorentz invariant. In addition there will be non-locality of the kind that we have discussed earlier, in the sense that the osmotic velocity u0,, and average velocity v,, of the the nth particle may be instantaneously connected to that of distant particles in a significant way. A corresponding stochastic model may be made for bosons. However, it will not be our approach to

regard bosons as particles in the way that is done in the treatments of the Klein—Gordon equation that we have discussed earlier. Rather we shall treat bosons as quantized states of a quantum mechanical field theory. We have already given a causal interpretation of such theories elsewhere [37].To obtain a stochastic interpretation, all that is needed is to add to the field motions a suitable Markov process

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D. Bohm and B. J. Hiley, Non-locality and locality in the stochastic interpretation of quantum mechanics

(which will, as with fermions, not be Lorentz covariant). It is clear on the basis of what we have done already, that the equilibrium distribution in this process will give rise to the usual quantum mechanical probabilities, which are covariant. It follows then that we can obtain a stochastic interpretation for both bosons and fermions, and since these cover all systems that are known, the stochastic interpretation may be taken as universally valid. Since the stochastic interpretation, as developed here, implies that the deeper laws of the background field are not Lorentz invariant, it follows that Lorentz invariance will be restricted to the statistical laws of quantum theory and, of course, to their classical deterministic limit. All experiments to date have been carried out in this domain. In the next section we shall discuss the possibility of new kinds of experiments that will go beyond this domain.

8. On the meaning of the non-Lorentz invariant background field Many physicists would feel that our explanation of Lorentz invariance as a statistical simplification of a deeper non-Lorentz invariant theory is a step backwards. For example they may be afraid that we would necessarily come to something like the Lorentz ether theory which did in fact explain the observed Lorentz invariance of physical processes on the basis of a set of fundamental laws of the ether that were not Lorentz invariant. There are serious objections to such an approach because it seems strange to have a Lorentz invariance of all observable phenomena which makes it impossible for the supposedly fundamental non-Lorentz invariant laws to manifest themselves in experiment. Einstein’s proposal that the Lorentz invariant laws could stand on their own without being grounded in a non-Lorentz invariant ether was surely an important step forward. Yet with the further progress of physics some evidence has emerged that even Lorentz invariance may be a principle of limited validity. For example, everyone admits that Lorentz invariance breaks down on a sufficiently large scale (where the curvature of space is significant). Indeed, modern formulations of relativity emphasise instead the requirement of local Lorentz invariance. But even this becomes doubtful at the order of a Planck length or less. Thus, because of the quantum fluctuations of the metric tensor g~,the meaning of a Lorentz transformation becomes ambiguous at such short distances. Consider for example a vector of length L made up of N parallel parts, each denoted by Ax~t~ and ea~hhaving a magnitude of the order of the Planck length or less. Under an infinitesimal Lorentz transformation parametrized by s~the vector Ax~~ undergoes a transformation a A A; =r g~~(xfl )&~ .

(72)

As a result of quantum fluctuations of g~an original vector made up of N parallel parts will be turned into something like a Brownian curve with a certain probability distribution. Therefore the idea of a simple geometrical interpretation of the Lorentz transformation as going from one straight line to another is no longer applicable. Many physicists might find it not entirely unacceptable to consider giving up Lorentz invariance at very short distances. We are proposing, however, that it may fail at any distance because as we have seen, the explanation of quantum mechanics requires non-local connections which are not limited by the speed of light. Even more, the Aspect experiment interpreted through Bell’s theorem gives evidence that if there is any explanation of quantum mechanics it will require this sort of non-locality. This question can be discussed at several levels. Firstly, we have shown elsewhere [141that because

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the quantum potential has a highly non-linear behaviour, it is too “fragile” and unstable to allow the transmission of a signal faster than light. In the stochastic interpretation this argument becomes even stronger because the particle has a random motion which will guarantee, for example, that it cannot carry a modulated signal without scrambling up its order in a way that could never be recovered. But this still leaves open the question ofwhat is the meaning ofthe connections brought about by the deeper background fields at speeds greater than that of light. It could be said that these will involve subtler properties of matter which were missed in the statistical treatment that is given by the quantum laws. Indeed as we have already indicated, statistical laws are generally abstractions and simplifications from more complex behaviour of a large number ofindividuals. Therefore it need not be surprising that these simplified statistical laws will be invariant to Lorentz transformation,. while the deeper and more subtle laws are not. In order to fit the quantum mechanics exactly, it is necessary for the non-local connection to be instantaneous in the time order of development at the deeper level. However, one could consider the consequence ofthe assumption that these connections were not instantaneous, but took place at speeds very much greater than that of light. If the speeds are great enough then it is clear that the current laws

of the relativistic quantum theory would be very good approximations. However, there would be two important new features that would come in. The first of these would involve a question of principle and the second would have to do with a test for the theory. With regard to the basic principles, it is clear that if the speed of propagation ‘of the connections were very large but finite, then the theory would no longer be non-local. Actually the connections would be local, because the osmotic velocity, for example, would be propagated continuously through space and not instantaneously. Nevertheless, this kind of locality would have qualitatively new features because what is propagated is active information whose effectiveness does not depend on the intensity of the wave. Therefore distant systems could still affect each other profoundly. To illustrate this point we may consider as an analogy a set of computers separated by a few

kilometers which are connected by radio communication. These computers can, evidently affect each other so strongly that they function virtually as one whole system, at least for all processes that are slow enough so that the time delay in communication can be neglected. Similarly the set of quantum particles may function effectively as a single whole provided that the time of transmission of the active information is negligible. If the velocity of transmission is infinite, so that the laws of quantum mechanics are exactly valid, then Bell’s inequality would have to be violated. Similarly in the computer analogy in so far as the time of transmission of the radio waves can be neglected, Bell’s inequality would also be violated by the behaviour of this system of computers. It is clear then that the violation of Bell’s inequality is not necessarily restricted to microscopic systems, or to those that operate at a quantum mechanical level. Rather it may take place wherever there is an effectively instantaneous connection through active information in which the objects respond with their own energy to this information.

It must be emphasised at this point, however, that speeds greater than that of light are considered to be possible only in the transmission of active information, as represented by the osmotic velocity and the mean velocity. It could not be possible, however, for matter as we know it, e.g. in the form of

particles, to have a systematic motion faster than that of light. It was indeed for this reason that we did not pursue the model of an instrinsic angular momentum of the particle, because as we have already pointed out, this would mean that the matter at the edge of the particle would have to be moving systematically at velocities immensely greater than that of light. It follows from what has been said above, that the quantum mechanics may still be consistent with

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locality provided that we give up the requirement that the laws of the deeper background field have to be Lorentz invariant. But it follows also that the laws of quantum mechanics and relativity may only be an approximation, albeit a very good one. At first sight, the above might suggest that the laws of this deeper background field as well as its particular conditions would define a unique absolute order of time, and thus play a role similar to that of the Lorentz ether. However, this need not be so. For example it may turn out that, as has indeed already been suggested, by the behaviour at short distances, the very notions of time and space as we

know them would cease to be valid at a sufficiently deep level [38].Rather, as Wheeler [39]has also suggested in another context, the notions of space and time will emerge as limiting cases of what may be

called “pre-space” in which an entirely different fundamental order prevails. It would not be appropriate to go into detail here but it may be said that this order is the implicate order [40] in which the basic movement is that of enfoldment and unfoldment. Two points that are separate in the ordinary unfolded space and time may be in contact in the enfolded order. This notion is indeed close to the idea of Pauli on non-locality, as discussed in section 1. Thus ultimately we will explain all non-local interactions as a result of such contact. However, when these contacts are viewed in the unfolded order, the net result will be as if there had been the kind of non-local connections that we have been attributing to the background field. There is no intrinsic reason why these connections when expressed in the explicate order may not take a particularly simple form in a certain frame of coordinates. This is indeed what is implied by our stochastic model.

If there is a background field of the kind we have been describing then it will not play a role analogous to the Lorentz ether because, as we shall see, it is in principle possible to have an experimental test revealing the presence of this field. Thus it does not simply vanish from all observable

phenomena, as does the Lorentz ether. In essence this test would consist of doing the EPR experiment under conditions in which the relative times of detection of the two particles was extremely accurately determined. In principle the Bell inequality would then no longer be violated. For, there would be no time for the effects of a disturbance on one particle to propagate to the other, so that the latter would no longer go into a corresponding state of close correlation with the first. Of course this will require a measurement of extremely high accuracy because the speed of transmission of active information is assumed to be very much higher than the speed of light. It is clear that accuracies far beyond that of the Aspect experiment would be needed ~to make such a test possible. In certain ways this test is reminiscent of the Michelson—Morley experiment, although it differs in the crucial respect that what is at issue here is not the measurement of the speed of light but rather a measurement of the immensely greater speed of active information as represented by the fields of mean velocity and osmotic velocity. Nevertheless, as in the Michelson—Morley experiment, we would have to take into account that the speed of the earth relative to the background field is unknown. Thus one would have to make measurements in different directions and see whether there was any change in the results. If such a change were found this would indicate a failure of both quantum mechanics and relativity. The ideas proposed here could be subject to additional tests especially in connection with processes that would involve the Planck length, where we might in any case expect the current laws of quantum mechanics and relativity to break down. In our model it seems reasonable to suppose, for example, that the effective free path for diffusion is of the order of the Planck length. For wave packets that begin to approach this dimension the statistical laws would start to break down (as would happen in a gas for sound waves of wavelength approaching the mean free path). With sufficiently high energy processes,

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the diffusive movement would thus no longer be in statistical equilibrium. In this case probability distributions different from those of quantum mechanics could emerge. Of course, as we have shown, these would relax in some time to the normal distribution. But with measurements that could deal with very short times, one could “catch” the non-quantum mechanical distributions before they could relax into those of the quantum mechanics. It is clear then that the stochastic model allows for additional tests which could show its validity in sufficiently fast processes. In this respect the situation is similar to that which prevails in supersymmetry theory and string theory. These too could be properly tested only with processes which would involve something of the order of the Planck length and corresponding times. For the time being their main advantage is that they provide a unified and relatively coherent treatment of a wide range of processes which have otherwise to be treated separately. Similarly the stochastic interpretation of the quantum theory (indeed also the causal interpretation) can provide a relatively coherent account of how different aspects of quantum processes are related in an intelligible way, which is not supplied by other interpretations.

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