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On the Relativistic Invariance of a Quantum

Theory Based on Beables ∗

D. Bohm and B.J. Hiley

Birkbeck College

Abstract

We discuss the question of the relativistic invariance of a quantum

theory based on beables and we suggest the general outlines of one

possible form of such a theory

It gives us great pleasure to write this article on the occassion of John Bell’s

sixtieth birthday, especially because he has shown a sustained and critical

interest in ﬁnding an interpretation of the quantum theory that is not based

on measuring instruments and observables, but rather on what he has called

beables. Non-relativistically it appears that there is no serious diﬃculty in

doing this [1, 2, 3]. However because of quantum nonlocality, the extension

to special relativity is not obvious. Nevertheless it raises some interesting

questions which we shall discuss in this paper and which we hope may help

resolve some of the problems pointed out by Bell [4].

We base our discussion on our proposed ontological approach [1, 2].

Brieﬂy we assume the electron is a particle guided by the quantum ﬁeld

according to the equation

pi=∇iS(1)

where piis the momentum of the ith particle and xiis its coordinate, while

Sis determined by

ψ(x1...xn, t) = R(x1. . . xn, t) exp ·i

¯hS(x1...xn, t)¸

∗Published in Foundations of Physics 21, 1-9, 1991.

1

With these assumptions it follows from Schr¨odinger’s equation that each

particle is acted on not only by the classical potential, but also by the

quantum potential

Q=−¯h2

2mX

i

∇2

iR

R(2)

It is further assumed that in a statistical ensemble of systems with the same

wavefunction, i.e. in the same quantum state, the probability is given by

P(x1...xn, t) = R2(x1. . . xn, t) (3)

It also follows from Schr¨odinger’s equation that |ψ|2satisﬁes the conservation

equation

∂|ψ|2

∂t +X

i

∇i·∇iS

m|ψ|2¸= 0 (4)

This means that if P=|ψ|2initially, this will hold for all times.

We then extend our model to incorporate an additional stochastic com-

ponent of the velocity of the particles, in which vi= (∇iS)/m will now be

the mean velocity of the ith particle, while the stochastic component will be

ξiwhich varies at random. We show [5] that an arbitrary initial probability

distribution will then approach |ψ|2. With these assumptions it can then be

shown that all the usual statistical results of quantum theory follow. We are

able to do this without giving a primary role to measuring instruments or to

observables. Rather, everything, including the process of measurement it-

self, is discussed in terms of beables. These beables are the particle variables

and the wave function.

Because the quantum potential does not depend on the intensity of the

wave function but only on its form, it is possible to have a strong and direct

connection between distant particles. It is this connection that, in our inter-

pretation, explains the nonlocality implied by the failure of the predictions

of quantum theory to satisfy the Bell inequality [6]. This nonlocality further

implies an instantaneous connection, as has indeed been conﬁrmed in the

experiments of Aspect et al. [7, 8]. The question at issue here is then that

of how such nonlocality is to be reconciled with relativity.

In order to bring this question into sharper focus, we outline here our

extension of the ontological interpretation of the quantum theory to a rel-

ativistic context [2, 5]. We begin with the one-particle Dirac equation. It

follows from this equation that there is a conserved charge-current density

whose time component is positive deﬁnite. This is

j=ψ∗αψ;j0=ψ∗ψ(5)

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with ∂j0

∂t +∇·j= 0 (6)

As in the nonrelativistic case, we assume a deﬁnite particle trajectory, but

now we deﬁne the velocity as

v=ψ∗αψ

|ψ|2(7)

(It can easilty be shown that |v| ≤ c.)

Noting that j0is equal to the particle density, ρ, we obtain for the

conservation equation ∂ρ

∂t +∇·(ρv) = 0 (8)

Equation (7) may be taken instead of equation (1) as the guidance relation.

Then if we assume probability density P=ρat any time, equation (8) will

guarantee that it will hold for all time.

The assumptions given here are suﬃcient to determine the one-particle

theory completely. It is evident that this theory is Lorentz invariant in

content because the Dirac equation is covariant and because from this it

follows that equation (8) is also covariant.

There is thus no real diﬃculty in ﬁnding a relativistically covariant ex-

tension of our interpretation of the quantum theory for the one-particle sys-

tem. However it does not follow that our further extension to a stochastic

model is also Lorentz invariant [5]. Indeed it is well known that there are, as

yet, unresolved diﬃculties in obtaining a consistent and covariant stochastic

process because the non-compactness of the Lorentz group prevents us from

deﬁning a covariant velocity distribution in the way in which this can be

done nonrelativistically. However the interesting point here is that for an

equilibrium distribution in which Phas become equal to ρ=|ψ|2, we do

not need a relativistically covariant stochastic process. For it follows from

the analysis given in Bohm and Hiley [5] that P=|ψ|2will result even from

non-covariant assumptions on the stochastic process. All the experimen-

tal results available thus far follow from the equilibrium distribution alone.

This means that we can, in principle at least, obtain a statistically covariant

theory of the single particle with experimental results equivalent to those

given by the current relativistic quantum mechanics. Moreover we can do

this on the basis of an underlying stochastic process that is not relativisti-

cally covariant. At present there is evidently no way experimentally to rule

out this sort of theory.

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To raise the question of nonlocality, we have to go to a many-particle

system [5]. Firstly we deﬁne the wave function of the many-body system to

be

ψ=ψi1...in...(x1. . . xn. . . t)

where inrepresents the four possible values of the index for the nth particle.

We take all particles at the same time t, noting that it can be shown that

this can be done in a covariant way [2, 5].

From the N-body Dirac equation it follows that

∂ψ ∗ψ

∂t +X

n

∇n·ψ∗αnψ= 0 (9)

where in the above we also sum over all the indices of all the particles. We

deﬁne the density ρ(x1. . . xn. . . t) = |ψ|2in the conﬁguration space of all

the particles. The guidance relation for the nth particle is then

vn=ψ∗αnψ

ρ(10)

If P=ρinitially, it follows that this will hold for all time. And if we assume

a suitable stochastic process, it further follows that an arbitrary initial value

of Pwill eventually approach ρ.

As long as we stay in a particular Lorentz frame, the above assump-

tions provide a consistent ontological interpretation of the many-body Dirac

equation. We should however add that, the wave function of all the fer-

monic particles (which are the only ones obeying the Dirac equation) will

have to be completely antisymmetric, so that the Pauli exclusion principle

will apply. Then as Dirac indeed proposed in his original work on this sub-

ject, the negative energy states of the vacuum are assumed to be full. ( It

can be shown that this approach is completely equivalent to the one that

treats fermions in terms of anticommuting annihilation and creation opera-

tors [9].) Pair production is then, of course, brought about by the transition

of a fermion from a negative to a positive energy state leaving behind a

‘hole’ with opposite quantum numbers, e.g. charge.

Although this theory is completely consistent when considered in any

one Lorentz frame, it is not completely covariant in content. However note

that the Dirac equation itself and the continuity equation (9) are Lorentz

covariant. The reason for this is that the guidance condition (10) has a

nonlocal signiﬁcance. That is to say, changes in position of any one particle

4

will in general imply the possibility of corresponding instantaneous changes

in the velocities of all the particles no matter how far apart they may be.

This would seem at ﬁrst sight to prevent us from extending our interpre-

tation to a relativistic context. However, one can see this is not so, if one

considers the fact that the statistical results of our theory depend only on

the invariance of the Dirac equation (and with it the equation (9)). Thus

we could assume a special favoured frame in which nonlocal contacts could

be instantaneous and in which there was a noncovariant stochastic process.

Nevertheless since the equilibrium probability distribution, P=|ψ|2, is the

one that would hold for all present practical purposes, it would follow that

the statistical results of all the quantum mechanical experiments that we

can now do would be the same as in the usual interpretation and therefore

be covariant. Evidently it is impossible at this stage to give any evidence

that the above view is not correct.

In eﬀect we are in this way led to a new model of quantum processes.

We begin by assuming what may be called a sub-relativistic level of stochas-

tic process which is also sub-quantum mechanical (in the sense that the

details of the stochastic process can never be revealed directly in any quan-

tum mechanical experiment). The current quantum theory and relativity

then follow only for the statistical results of the experiments. These do

not depend sensitively on the details of the underlying model, rather as the

laws of thermodynamics do not depend sensitively on the details of the un-

derlying atomic model. So at present, a wide range of sub-relativistic and

sub-quantum mechanical models is open to us.

To make our ideas a bit more deﬁnite, we could propose that the stochas-

tic process holds only down to some deﬁnite length, l0, which plays a role

similar to that of the mean free path for ordinary particles undergoing Brow-

nian motion. (As we shall bring out later, l0is probably of the order of the

Planck length.) For distances much greater than l0, the stochastic process

will be a good approximation. But for distances of the order of l0or less,

the stochastic treatment will fail and new qualities will reveal themselves.

One may expect, for example, for waves of length of the order of l0, the

current laws of quantum mechanics and relativity would break down even in

a statistical context (e.g. as the laws of macroscopic physics breaks down for

distances of the order of the mean free path.) This is at least one example

of the new kinds of experimental results that would be possible and that

could reveal the sub-relativistic and sub-quantum mechanical level.

Another possibility is that in the special frame in which instantaneous

contact is possible, the speeed of transmission of impulses is not inﬁnite but

5

still very much greater than that of light. In this case even the statistical

predictions of the quantum theory would break down in measurements that

were accurate enough to respond to such very short time diﬀerences. (This

would almost certainly require accuracies far greater than available in the

Aspect et al. experiments [7, 8].) In this case one would ﬁnd in an EPR

experiment, for example, that Bell’s inequality would be satisﬁed with the

use of instruments that responded to such short times, while they would not

be satisﬁed in measurements that did not do this. Measurements made in

such short times would in eﬀect reestablish locality in the sub-relativistic

and sub-quantum level.

To carry out experiments of this kind one would have to do something

analogous to a Michelson-Morley experiment which however would involve

an EPR-type situation. Let us suppose the velocity of the earth relative to

the preferred frame is v. By measurements of EPR correlations at diﬀerent

times of the year we could not only ﬁnd out which is the preferred frame,

but also demonstrate the existence of the sub-relativistic and sub-quantum

mechanical level.

But what would a good surmise be for the value of v? One suggestion

is that the preferred frame should be the one that is at rest in a system

of coordinates in which the time axis connects the point in question to the

origin of the universe. It seems reasonable to suggest that this would be

close to the frame in which the 3◦Kradiation is isotropic. This is the sort

of velocity we have to measure relative to the earth.

Are there any reasons available at present which would lead us to suppose

the domain of relativity is limited? Actually there are some. Firstly it is

clear that Lorentz invariance fails for distances so large that the curvature of

space becomes signiﬁcant. Generally this is dealt with by saying that Lorentz

invariance only holds in a tangent space mapping onto an inﬁnitesimal part

of the actual manifold. But in this approach, Lorentz invariance should

become more and more applicable for shorter and shorter distances. However

one can give arguments showing that Lorentz invariance will also fail to hold

for very short distances as we approach Planck length.

To see how this comes about, let us consider a vector, vµ, of ordinary

length and regard it as made up of many smaller vectors δvµwhich are on

a straight line

vµ=X

i

δvµ

i

Suppose that we make a Lorentz transformation. Strictly speaking this

cannot be deﬁned without giving the metric tensor gµν . But at very short

6

distances this tensor has large quantum ﬂuctuations and does not reduce to

its simple diagonal form. The Lorentz transformation will then give us

v0µ=X²µαgαβ (xλ)δvβ

i

It is clear that what begins as a simple vector becomes transformed into a

rather foggy distribution of constituent elements which are not well deﬁned.

So the whole idea of a simple Lorentz transformation of one line into another

breaks down and this especially so as we go to very short distances for which

the notion of tangent space has been assumed to apply. The idea of Lorentz

invariance becomes relevant only over intermediate distances for which the

ﬂuctuations of gµν will average out to something relatively small, implying

that gµν can be taken as eﬀectively constant.

All of this suggests that the Lorentz group may not be as fundamental

as has generally been assumed. If it is not valid in the domain of very large

distances, nor in the domain of very small distances, it may well cease to

be valid in yet other domains. The analysis given earlier suggests that even

at intermediate distances, it may cease to be valid for individual processes

involving beables wherever quantum nonlocality is important.

It is understandable that there may be a great deal of reluctance even

to contemplate giving up the notion that relativity and quantum mechanics

are universally valid. To do so however may give us signiﬁcant advantages.

Firstly it provides a way of developing a point of view in which physical pro-

cesses are intuitively intelligible so that we do not have to restrict ourselves to

using mathematical algorithms for computing probabilities of experimental

results without giving any idea of how these results come about. Secondly

it permits us properly to confront the question of making a cosmological

theory that is compatible with quantum mechanics. In the conventional in-

terpretation there is no way to do this because the apparatus would have

to be outside the cosmos. (For further discusssion of these two points see

Bohm and Hiley [2] and Hiley [10]).

Finally it may help to resolve some of the diﬃculties in the quantum

treatment of special and general relativity. Recall that special relativity

leads to inﬁnities. To be sure, these can be avoided by the algorithm of

renormalisation but this algorithm contradicts the initial starting point. So

the theory is ultimately not based on a set of logically coherent fundamental

principles, but rather on technical procedures brought in from outside such

principles merely for the sake of avoiding certain diﬃculties. On the other

hand if there is ultimately a preferred frame then there is no inconsistency in

7

assuming that there is a ‘cut oﬀ’ somewhere near the Planck length beyond

which the extrapolation of current laws is not appropriate. With regard

to general relativity, we also avoid unresolved diﬃculties arising from the

inﬁnite ﬂuctuations in gµν at short distances.

It therefore seems to us that it is worthwhile to keep open the options

that relativity and quantum theory may have limited domains of applica-

bility. At present there is, of course, no way to settle this issue, but still

keeping an open mind on this question seems to be the best approach.

We would further add that it is really a continuation of one of the basic

traditions of physics to consider the possibility of a deeper level out of which

emerges a very diﬀerent level conforming to relativity and quantum theory.

A case in point is the assumption of a deeper atomic level as something

very diﬀerent from the continuous macroscopic level of the older physics.

Einstein’s attempt to explain particles as singularities or pulses in a ﬁeld is

another such case.

The new level always looks very diﬀerent from the older one yet con-

tains the latter as a limiting case or as an approximation. Our proposed

new level is also very diﬀerent from previous levels of physics. We explain

these diﬀerences elsewhere [1, 2] but here we shall mention that it contains

concepts that are very strange and new such as nonlocality and the activity

of information. It need hardly be surprising that behaviour as strange as

that of the quantum theory (and to some extent that of relativity too) would

have to come out of a deeper level that was very diﬀerent in crucial ways

from those that have already been known.

References

[1] D. Bohm, B.J. Hiley and P.N. Kaloyerou. Phys. Reports 144, 321

(1987).

[2] D. Bohm and B.J. Hiley. The Undivided Universe (An Ontological In-

terpretation of Quantum Mechanics), Routledge, London (1991).

[3] J.S. Bell. Speakable and Unspeakable in Quantum Mechanics, Cam-

bridge University Press, Cambridge (1987).

[4] J.S. Bell. Op. Cit. Chapter 14.

[5] D. Bohm and B.J. Hiley. Phys. Reports 172, 93 (1989).

8

[6] J.S. Bell. Op. Cit. Chapter 2.

[7] A. Aspect, P. Grangier and G. Roger. Phys. Rev. Lett. 49, 91 (1982).

[8] A. Aspect, J. Dalibard and G. Roger. Phys. Rev. Lett. 49, 1804 (1982).

[9] S.S. Schweber, H.A. Bethe and F. de Hoﬀmann. Mesons and Fields,

Chapter 13, Row Peterson and Co., Illinois (1955).

[10] B.J. Hiley. “Cosmology, EPR Correlations and Separability”, in Bell’s

Theorem, Quantum Theory and Conceptions of the Universe, K.

Kafatos, Ed., Kluwer Academic Publishers, Dordrecht (1989).

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