The pilot wave

Z think that conventional formulations of quantum theory, and of
quantum field theory in particular, are unprofessionally vague and
ambiguous. Professional theoretical physicists ought to be able to do
better. Bohm has shown us a way.?
In the very early days of quantum theory, de Broglie, who had been
the first to associate a wavefunction with a particle, suggested that,
instead of being the complete description of the system, as in conventional
quantum theory, the true role of this wavefunction might
be to guide the motion of the particles. In such a theory the
wavefunction is therefore called a pilot wave. The particles would
always have precise trajectories, which would be determined in a
unique way from the equations of the theory. It is such trajectories
that constitute the ‘hidden variables’ of the theory.
These ideas were not well received; probably they were regarded
as a step backwards from the liberating ideas of quantum theory
to the old restrictions of classical physics. Nevertheless, and in spite
of von Neumann’s theorem discussed above, interest in hiddenvariable
theories did not completely die, and in 1952 David Bohm
produced a theory based on the pilot wave idea, which was deterministic
and yet gave the same results as quantum theory. It also
provided a clear counter-example to the von Neumann theorem.
In Bohm’s theory a system at any time is described by a
wavefunction and by the positions and velocities of all the particles.
(Since it is positions that we actually observe in experiments, it is
perhaps paradoxical that these are called the ‘hidden’ variables, in
contrast to the wavefunction.) To find the subsequent state of the
system, it is necessary first to solve the Schrodinger equation and
thereby obtain the wavefunction at later times. From this wavefunction a ‘quantum force’ can be calculated. This force is
added to the other, classical, forces in the system, e.g. those due
to electric charges, etc, and the particle paths are then calculated
in the usual classical way by Newton’s laws of motion. The quantum
force is chosen so that there is complete agreement with the
usual predictions of quantum mechanics. What we mean by this is
that, if we consider an ensemble of systems, with the same initial
wavefunction but different initial positions, chosen at random but
with a distribution consistent with that given by the wavefunction,
then at any later time the distribution of positions will again agree
with that predicted by the new wavefunction appropriate to the
time considered. It is beyond the scope of this book to discuss
further the technical details, and problems, associated with these
considerations.
In comparing the Bohm-de Broglie theory with ordinary quantum
theory we note first that, since they give the same results for all
quantities that we know how to measure, they are equally satisfactory
with regard to experiments. As far as we know both are,
in this sense, correct. The former has the added feature of being
deterministic, but with our present techniques this is not significant
experimentally. The degree to which it is regarded as a conceptual
advantage is a matter of taste.
A much more important advantage of the hidden-variable theory
is that it is precise. It is a theory of everything; no non-quantum
‘observers’ are required to collapse wavefunctions since no such
collapse is postulated. All the problems of Chapter Three
disappear.
In connection with this last observation, we should note two
points. First, it may be asked how we have been able to remove the
requirement for wavefunction collapse when, in Chapter Three, we
appeared to find it necessary. The answer lies in the fact that,
whereas previously the wavefunction was the complete description
of the system, so that there was no place for the difference between
transmission or reflection (for example) to show other than in the
wavefunction, now that we have additional variables to describe
the system this is no longer the case. The wavefunction can be identical
for both transmission and reflection, since the difference now
lies in the hidden variables, in particular in the positions of the
particle.
Secondly we should note a reservation to the remark above that the two theories always agree, Readers may indeed be wondering
how this can be, when in one case we have wavefunction collapse
but not in the other. The answer lies in the fact that wavefunction
collapse only happens in the orthodox interpretation when
macroscopic measuring devices are involved. It is only when the
wavefunction can be written as the sum of macroscopically different
pieces that some of them are dropped in the process of reduction.
Now the difference between keeping all the pieces, as in the
Bohm-de Broglie theory, and dropping some of them, as in the
orthodox theory, is only significant experimentally if they can be
made to interfere. However, such interference can only occur if the
pieces can be made identical, which as we have seen (03.6 and
Appendix 6) is so unlikely for macroscopic objects as to be effectively
impossible. The two theories are experimentally
indistinguishable because macroscopic processes are not reversible.
Nevertheless we should emphasis that, where interference can in
principle occur, it is indeed observed. There is no positive evidence
that wavefunction reduction actually happens, so, especially in
view of the problems of Chapter Three, theories that do not require
it have a real advantage.
Given this fact it is perhaps rather remarkable that hiddenvariable
theories are not held in high regard by the general
community of quantum physicists. Why is this so? More importantly,
are there any good reasons why we should be reluctant to
accept them?
We have already hinted at some of the possible answers to the
first question. The many successes of quantum theory created an
atmosphere in which it became increasingly unfashionable to question
it; the argument between (principally) Bohr and Einstein on
whether an experiment to violate the uncertainty principle could be
designed was convincingly won by Bohr (as the debate moved into
other areas the outcome, as we shall see, was less clear); the
elegance, simplicity and economy of quantum theory contrasted
sharply with the contrived nature of a hidden-variable theory which
gave no new predictions in return for its increased complexity; the
whole hidden-variable enterprise was easily dismissed as arising
from a desire, in the minds of those too conservative to accept
change, to return to the determinism of classical physics; the
significance of not requiring wavefunction reduction could only be
appreciated when the problems associated with it had been accepted and, for most physicists, they were not, being lost in the mumbojumbo
of the ‘Copenhagen’ interpretation; this interpretation, due
mainly to Bohr, acquired the status of a dogma. It appeared to say
that certain questions were not allowed so, dutifully, few people
as ked them.
With regard to the second of the questions raised above (namely,
are there any good reasons for rejecting the hidden-variable
approach?), it has to be said that the picture of reality presented
by the Bohm-de Broglie theory is very strange. The quantum force
has to mimic the effects of interference so, although a particle
follows a definite trajectory, it is affected by what is happening
elsewhere. The reflected particle in figure 2 somehow ‘knows about’
the left-hand mirror, though its path does not touch it; similarly,
the particle that goes through the upper slit in the double slit experiment
shown in figure 13 ‘knows’ whether the lower slit is open or
not. This ‘knowledge’arises through the quantum force which can
apparently operate over arbitrarily large distances. To show in
detail the effect of this force we reproduce in figure 19 the particle
trajectories for the double-slit experiment as calculated by Philippidis
et a1 (I/ Nuovo Cimento 52B 15, 1979). We remind ourselves
that, if we are to accept the Bohm theory, then we must believe the
particles really do follow these peculiar paths. Particles have
become real again, exactly as in classical physics, the uncertainty
has gone, but the price we have paid is that the particles behave
very strangely!
Another, perhaps mainly aesthetic, objection to hidden-variable
theories of this type is that, without wavefunction reduction, we
have something similar to the many-worlds situation, i.e. the
wavefunction contains all possibilities. Unlike the many-worlds
case, these are not realised, since the particles all follow definite,
unique, trajectories, but they are nevertheless present in the
wavefunction-waiting, perhaps, one day to interfere with what
we think is the truth! Thus, in our example discussed in Appendix
2, both scenarios act out their complete time development in the
wavefunction. It is all there. The real, existing wavefunction of the
universe is an incredibly complicated object. Most of it, however,
is irrelevant to the world of particles, which are the things that we
actually observe.
The unease we feel about such apparent redundancy can be made
more explicit by expressing the problem in the following way: the pilot wave affects the particle trajectories, but the trajectories have
no effect on the pilot wave. Thus, in the potential barrier experiment,
the reflected and transmitted waves exist and propagate in
the normal way, totally independent of whether the actual particle
is reflected or transmitted. This is a consequence of the fact that the
wavefunction is calculated from the Schrodinger equation which
does not mention the hidden variables. It is a situation totally contrary
to that normally encountered in physics, where, since the time
of Newton, we have become accustomed to action and reaction
occurring together.