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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.
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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)
2
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
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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
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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
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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
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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).
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[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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