The Bohm model

Perhaps the most significant recent development in the Bohm hiddenvariable
model (see $5.2) is that physicists outside of Bohm’s own
students (and John Bell) have begun to take the model seriously. One
group (D. Diirr, S. Goldstein and N. Zhangi, Physics Letters 172A 6,
1992) have invented the rather evocative name ‘Bohmia? mechanics’ to
describe it. This group have considered the requirement that the initial
distribution of positions should be consistent with the quantum theory
probability law, which, as we noted in $5.2, is necessary for the Bohm
model to agree with quantum theory. In particular, they have shown
that the the requirement is expected to be satisfied for any ‘typical’
initial conditions.
Although, given that the above initial requirement holds, the Bohm
model is guaranteed by construction to agree with the statistical
predictions of quantum theory for particle positions (and hence with all
known experiments), there has been a widespread reluctance to accept
this fact, presumably because of a variety of ‘impossibility theorems’
on the lines of that due to von Neumann mentioned in 55.1. One such
theorem is often known as the Kochen-Specker-Bell theorem, which
is a strange irony because John Bell actually gave his simplified proof
of the theorem (Reviews of Modern Physics 38 447, 1966) in order to
show why it was not relevant to the Bohm model! The essence of these theorems is very similar to the non-locality arguments discussed
in 55.4 and Appendix 9. For example, in Appendix 9 we seemed to
show that the performers could not carry cards containing the answers.
Since these answers are the analogues of the hidden variables, this at
first sight means that such things are forbidden if we wish to maintain
agreement with quantum theory. The ‘error’ in this argument is that
it requires the answers to be fixed, whereas in the Bohm model they
are dynamical things which change with time, and which change in a
way that can depend upon what question is being asked of the other
performer (which is where the non-locality enters). The situation here
is sometimes described by saying that measurements are ‘contextual’,
a fancy way of saying that quantum systems in general cannot be
separated into independent parts, and that the answer you get depends
upon the question (i.e., the result depends on the apparatus).
It should be emphasised that the Bohm model looks after all this
automatically. In fact, on re-reading the remarks I wrote at the end
of 55.1, I think I was being unfair to the Bohm model in saying
that it was ‘contrived’. This suggests that much effort was required
in order to devise something that would work, whereas, in fact,
trajectories are defined by one simple property, namely that if we
have many identical systems with identical wavefunctions, and with
particle positions distributed according to the quantum probability law
at a particular time to, then this fact will remain true at other times.
Actually this does not quite define the trajectory uniquely-the Bohm
model is just the simplest possibility.
1 shall now describe a very idealised experiment which shows how
all this works in practice. First, it is necessary to note that in most
versions of the Bohm model trajectories only exist for ‘matter’ particles,
in particular, for the electrons and nucleons that are the constituents of
matter. All these particles have spin equal to !j. Particles of spin zero
or one, e.g., the photon, do not have trajectories-so, in this sense, we
should say that the Bohm model does not have photons. Why then do
we apparently see ‘photons’? Specifically, refemng to the experiment
described in 31.4, why do detectors appear to say that a photon either
goes through the barrier of 51.4 or is reflected, when we know that the
wave does both? We shall see how the existence of matter trajectories
answers this question.
In order to make the calculations as simple as possible, we take
as the measuring device a single particle, moving in one dimension,
initially in a stationary, localised, wave-packet, and suppose that a photon wave-packet interacts with this to give it a momentum. The
details of this interaction are not important. If the detector is placed in,
say, the path of the transmitted wave and if the barrier is removed so
that there is only a transmitted wave, then it is easy to calculate that the
detector particle, initially at rest, will acquire a velocity. Observation
of this velocity will correspond to the photon having been detected.
Thus we have a detector that works properly: a photon wave comes
along and is detected through the motion of the detector particle, i.e.,
the movement of a pointer.
Now let us restore the barrier, so that the photon wave is a
superposition of transmitted and reflected parts (see figure 28). Again
it is possible to calculate what happens to the detector, and it turns out
that, for some initial positions of the detector particle, it moves, and
for others it does not. As indicated in figure 28, the important thing
here is the position of the detector particle, i.e. the hidden-variable,
relative to the position of the detector wave-packet, which of course
is what we refer to as the position of the detector. Thus, whether or
not the detector detects the photon depends on the initial position of its
particle. If it does, we would say that the photon has been transmitted;
if it does not we would say that the photon has been reflected. (Note
that, as in the collapse models discussed in the previous section, these
statements are really statements about the detector, rather than about
the photon). To be more explicit we consider, for simplicity, the case
where transmission and reflection are equally likely (so that PR = PT
in the equations of §4.5), and take a symmetrical initial wave-packet for
the detector. Then those initial starting positions that are on the near
side (relative to the incident photon) will not detect the photon; those
that are on the far side will. This actually follows simply from the
fact that trajectories cannot cross. Provided the distribution of initial
positions, in many repeats of the same experiment, are in accordance
with quantum theory (and hence in this case symmetrical between the
two sides), it follows that the photon will be detected in half of the
experiments, i.e., it will be transmitted with 50 per cent probability as
required. Symbolically, with suitable conventions, this means:

xo > 0 + transmission
and
xo .c 0 + reflection

where xo is the initial position of the particle in the detector and we
have taken the detector to be centred at the origin, x = 0 (see figure
28).
Clearly, very similar considerations hold if we put a detector instead
in the path of the reflected beam. Then we find the analogous results:
yo > 0 -+ reflection
and
yo < 0 -+ transmission
where here yo is the initial position of the particle in the ‘reflection’
detector, which is centred at y = 0.
Next we consider what happens if we have both detectors, one in
the path of the transmitted beam, and the other in the path of the
reflected beam, as shown in figure 29. If these detectors behaved
independently, i.e., acted as if the other were not present, then there
would be the possibility of violating the experimental results (and
also the predictions of quantum theory). For example, if the starting
positions of the detector particles happened to satisfy xg > 0 and yo > 0
then, according to what we saw above, both detectors would record the
photon, which would then appear to have been both transmitted and
reflected! In fact, however, this is where the contextuality becomes

evident. It is straightforward to calculate that the first detector records
the photon, which is therefore transmitted, if
XO - yo > 0.
Otherwise, the second detector records the photon, corresponding
to its being reflected. In general, it is the relative position of the
particles in the two detectors that determines whether a particular event
is observed as a transmitted or reflected photon.
We emphasise again that in this experiment, because we have
assumed there are no photon trajectories, it is the properties of the
detectors that give rise to the apparent existence of ‘photons’ which
appear in specific places. When we say, for example, that the photon
is transmitted we mean no more than that an appropriate detector has,
or has not, recorded a photon. The model is designed to agree with
the predictions of orthodox quantum theory at the level of the output
of detectors, because it is these that correspond to observations. This
last point is particularly significant if we consider experiments where
particles that do have trajectories are used to trigger detectors. In
certain rather special cases it can be shown that the detector records
the particle even though the particle trajectory did not pass through it,
and conversely. One can most easily regard this as being due to nonclassical
effects of the quantum potential (see B. Englert, M.O. Scully,
G. Sussman and H. Walther Z Nutulforsch. 47a 1175 (1992) and C.
Dewdney, L. Hardy and E.J. Squires Physics Letters 184A 6 (1993) for
further details).
Two books covering all aspects of the Bohm model have recently
been published. The Quantum Theory of Motion (Cambridge University
Press, 1993) by Peter Holland, an ex-student of David Bohm, gives
an extremely thorough and detailed treatment of the model and its
applications. The book by David Bohm and Basil Hiley, The Undivided
Universe (Routledge, London, 1993), which was completed just before
Bohm’s death, contains fewer details of calculations in the Bohm
model but more on the general problem of the interpretation of
quantum theory, and comparison with other suggested solutions of the
measurement problem.