Intro.
Ch. 1
Ch. 2
Ch. 3
Ch. 4
Ch. 5
Ch. 6
Ch. 7
Ch. 8
Ch. 9
Ch.10
Ch.11
Ch.12
App.1
App.2
App.3
Biblio.
Index

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Quantum Physics: The Nodal Theory

Hector C. Parr

Chapter 9: The EPR Problem
9.01 The paper
published in 1935 by Einstein, Podolsky and Rosen, Can
Quantum-mechanical description of Physical Reality be considered
complete? (Physical Review 41, 777) has probably prompted more books
and articles, and promoted more discussion and argument, that any other
aspect of Quantum physics. We shall in this chapter attempt our own
analysis of the problem, and try to clear away some misunderstandings,
before showing what contribution the Nodal theory can make to its
interpretation.

9.02 The authors of the original EPR paper described a
thought-experiment in which an elementary particle is made to split into
two equal parts, which than fly apart. Because of the law of momentum
conservation, the two parts must travel with equal and opposite
velocities. Now the complementarity laws of Bohr’s quantum theory decree
that if we know accurately the position of a particle we cannot at the
same time know its momentum. Furthermore, Bohr maintained that this lack
of precision was not the result of clumsy measurement; we can talk about
the momentum of a particle only in relation to a particular arrangement of
measuring apparatus, which could not exist alongside apparatus for
measuring the position of the particle at the same time. Bohr seems to say
that the momentum has no real existence except in relation to the
apparatus used to measure it. But the EPR paper pointed out that if we
measure the momentum of one of the pair of particles we immediately
know the momentum of the other, and the authors maintained that
this shows the latter momentum to have a real existence, whether or not it
is measured. They wrote:

If, without in any way disturbing a system, we
can predict with certainty (i.e., with probability equal to unity) the
value of a physical quantity, then there exists an element of physical
reality corresponding to this physical quantity. ... [It follows that]
quantum mechanics does not provide a complete description of physical
reality.

Bohr defended the opposite point view with
arguments such as the following:

The apparent contradiction in fact discloses
only an inadequacy of the customary viewpoint of natural philosophy for
a rational account of physical phenomena of the type with which we are
concerned in quantum mechanics. Indeed the finite interaction between
object and measuring agencies, conditioned by the very existence of the
quantum of action, entails ... the necessity of a final renunciation of
the classical ideal of causality, and a radical revision of our attitude
towards the problem of physical reality. ... From our point of view we
now see that the wording of the above-mentioned criterion of physical
reality proposed by Einstein, Podolsky and Rosen contains an ambiguity
as regards the meaning of the expression ‘without in any way disturbing
a system’. Of course there is in a case like that just considered no
question of a mechanical disturbance of the system under investigation
during the last critical stage of the measuring procedure. But even at
this stage there is essentially the question of an influence on the
very conditions which define the possible types of predictions regarding
the future behaviour of the system. Since these conditions
constitute an inherent element of the description of any phenomenon to
which the term ‘physical reality’ can be properly attached, we see that
the argumentation of the mentioned authors does not justify their
conclusion that quantum-mechanical description is essentially
incomplete.
The two men continued to discuss their
differences for several years, and had come no closer to agreement at the
time of Einstein’s death in 1955.

9.03 The actual thought-experiment described in the EPR paper
was never actually performed. But in 1951 David Bohm shifted the
discussion sideways by suggesting an alternative experiment which might be
easier to perform in practice, and which also widened considerably the
philosophical challenges which it posed. It is experiments of the type
Bohm invented which have subsequently come to be accepted as the
authoritative version. The pair of identical particles which these
experiments envisage are to be either protons, whose spins must be
oppositely directed in order to conserve angular momentum, or photons
which, if linearly polarised, must share the same axis of polarisation.
The measurements which are to be performed on the two particles are
measurements of spin, which in the case of photons are easily carried out
by attempting to pass them through polarising filters. If a photon is
transmitted by such a filter, conventional quantum theory maintains that
its axis must then become oriented with that of the filter, whereupon
conservation laws decree that the other particle of the pair must at the
same moment be affected in the same way, however remote it is. Two
conclusions follow immediately. If the two filters have their axes
parallel to each other, then whenever one photon is transmitted by
the filter it encounters, the other photon must be transmitted also, and
whenever a photon is stopped by its filter, then so must the other. If,
alternatively, the filters are set up with their axes perpendicular
to each other, then one photon of the pair must always be transmitted, and
the other absorbed.

9.04 There seemed little doubt that this effect would be
observed if the experiment could actually be performed, and many writers
then assumed this would show the two particles were communicating with
each other in some way after they have separated, or were somehow “aware”
of the setting of the polarisers before they encountered them, for how
otherwise could they display the degree of co-operation observed? Many
books written by physicists and mathematicians during the last thirty
years describe this effect, and several of their authors declare,
erroneously as we shall show, that the results prove the existence of such
non-local influences. The EPR effect does present us with some
strange contradictions, but they are not displayed in the simple case we
described above, where the two polarisers have their axes parallel or
perpendicular. Paul Davies, dealing with the case in which the axes of the
two polarising filters are parallel, writes:

The truly mind-boggling implications of [the
experiment] are apparent if we use two parallely orientated polarizers,
one of which is stationed in the path of each of the two correlated
photons. Because the polarizations are forced to be parallel, whatever
we measure for the photon polarization of one we are obliged to find the
same for the other, but as there are only really two polarization states
that are measurable (i.e. parallel and perpendicular) the 'yes-no'
decision of one polarizer must be identical to that of the other. ...
The mystery is, how does the second polariser know that the first
one has let a photon pass, so that it too may do the same? The
experiments could be carried out simultaneously, in which case we can be
sure, on the basis of the theory of relativity, that no message can
travel faster than the photons themselves between the polarizers to say
'let this one pass'. In fact by stationing the polarizers at different
distances from the decaying atom we could arrange for either experiment
to be performed before the other, thereby ruling out any question of one
polarizer signalling the other, or causing it to accept or reject a
photon. (Other Worlds, Penguin, 1980)

9.05 There is no need for Davies' astonishment. We may agree
that information could not be transmitted from one photon to the other
fast enough to ensure their cooperation, but the two photons
themselves can carry information, and may well have “agreed” a
common policy before they separated. We will show a simple method which
photons could employ to meet these requirements without having to pass any
information to each other after the moment when they separate. Suppose
their (common) spin axis is chosen randomly within the range 0o
to 180o, and that a photon always passes through a polariser if
its spin axis makes an angle of less than 45o with the optical
axis of the polariser, and is always stopped by the polariser
otherwise.

In the diagram, we are supposing that a photon is
transmitted by a polariser whenever the axis of the polariser lies within
the red sectors, but is absorbed if the axis lies in the white sectors. It
is easy to see that, whenever a random photon meets a polariser, the
probability of transmission is 50%, as it should be, and in the experiment
we are considering, with the optical axes of the two polarisers parallel,
the two photons are always either both transmitted or both absorbed. It is
equally easy to see that, in the alternative case where the polariser axes
are perpendicular to each other, one photon must be transmitted and
the other is not. We are not suggesting that this is indeed the
explanation of how the EPR results are obtained, but the picture does show
that the results quoted by Davies are not as remarkable as he imagines.
The true EPR paradox is much more subtle.

9.06 So far we have considered only cases in which the two
polarising filters have their axes parallel or perpendicular. If the angle
between the axes has any other value, say qo, then
quantum theory predicts, and no-one doubts, that the results of our
experiment would be as follows: the probability that both photons are
transmitted or both are absorbed will be cos2q, while
the probability that one is transmitted and the other absorbed will be
sin2q. It was in 1964 that John Bell considered in
detail these probabilities, and discovered a remarkable result, which has
come to be known as “Bell’s Inequality”. He envisaged a series of
experiments like the one described above, except that we are able to
rotate the polarisers so that their axes may be set at any angle.
Then, making a number of very reasonable assumptions, he showed that the
results must obey this inequality, unless they were influenced by some
strange effect such as the instantaneous transference of information from
one photon to the other. He showed further that the results predicted by
quantum theory did not obey the inequality, which would seem to
confirm that such an extraordinary effect must be at work. The special
cases considered earlier, with q equal to 0o or
90o, do not display these effects; they become evident only if
we use a number of carefully chosen different values for q

. 9.07 The final episode in this saga was written in the 1980’s,
when improved laboratory techniques made it possible for the first time to
perform some of the experiments which previously had existed only in
imagination. As expected, the results do confirm the standard formalism of
quantum theory; Bell’s inequality is indeed sometimes infringed by the
results of these experiments. So what is Bell’s theorem. and what
conclusions can we draw from the experiments which violate it? A
mathematical statement of the inequality can be found in many textbooks,
but the difficulty of fully understanding it and its implications is
increased by the fact that several different forms exist, which at first
sight are not the same. We shall not present any of these forms here, but
instead simply show the contradiction that is revealed by one simple
series of experiments using the apparatus we have described above.

9.08 Suppose the optical axis of the polariser on the left can
take any of the positions A, B, C or D, these being at 30 degrees to each
other as shown, and those on the right, A', B', C' and D', are
respectively parallel to them. For our first experimental run, set the
axis of the left detector to direction A and the right one to B'. The
angle between them is 30 degrees and so, as the above formula suggests,
the probability of agreement is cos230o, or 3/4;
with a sufficiently long series of readings, about 3/4 of the results at
A, whether “yes” or “no”, will be the same as those at B'. Only 1/4 will
be different, which means that the list of A results would need only 1/4
of them changing to give the B' results.

If instead of setting the left polariser to A it had been at C, while
the right hand polariser remained at B', this in itself could not affect
the run of results at B', which would be the same as before; we are
assuming no non-local effects occur. And as the angle between the
detectors is still 30 degrees, we would again find only 1/4 of the right
hand results to differ from the left.

Thirdly, turn the right polariser to D', leaving the left one at C.
Once again only 1/4 of the D' results will differ from the C results.

9.09 To summarise, only 1/4 of the A results differ from the B'
results, only 1/4 of the B' results differ from the C results, and only
1/4 of the C results differ from those at D'. It follows that no more than
3/4 of the A results can differ from those at D'. But this is not what we
observe; D' is perpendicular to A, and so all its results should
differ. This contradiction is, in effect, an example of the sort displayed
by infringements of Bell's Inequality.

9.10 What exactly is the significance of this infringement? The
usual analysis goes something like this. Bell made two assumptions in
deriving his inequality. The first was that no causal influences could be
transmitted at a speed greater than that of light; we call this the
locality assumption. Bell's second assumption was that the
quantities we measure in quantum mechanics, in this case the orientation
of the photons' axes, really do exist whether or not we attempt to make a
measurement, the so-called reality assumption. So the fact that
Bell's Inequality is broken, so the argument goes, implies that one or
both of these assumptions, locality or reality, must be
false. It is because the abandonment of either of these “obvious” precepts
is so strongly opposed to intuition, that so much effort has been devoted
to testing the EPR results experimentally, and to analysing the real
implications of the results.

9.11 But there is a damaging flaw in the argument we presented
above. We omitted an implicit stage in the discussion. We wrote, in
effect, “With the polarisers at A and B' the probability that the readings
differ is 1/4. So after a long run of trials, about 1/4 of the B' results
will differ from the A results. Similarly about 1/4 of the C results will
differ from the B' results." But we omitted to say, "Therefore no more
than 1/2 of the A results can differ from the C results.” Now this is
meaningless, for there is a fundamental impossibility in obtaining
simultaneously a run of A results and a run of C results. We cannot
combine into one sample space the sample spaces corresponding to the A and
B' settings and the C and B' settings. The former has possible values as
follows:

A transmits B' transmits (prob = 3/8)
A transmits B' absorbs (prob = 1/8)
A absorbs B' transmits (prob = 1/8)
A absorbs B' absorbs (prob = 3/8)

The latter has possible values as follows:
C transmits B' transmits (prob = 3/8)
C transmits B' absorbs (prob = 1/8)
C absorbs B' transmits (prob = 1/8)
C absorbs B' absorbs (prob = 3/8)

If we could design an experiment with all eight of the above results
as alternatives, we could combine them into a single sample space, and
find the probabilities of each of the eight possible outcomes. But A and C
are not possible alternatives. No experiment could determine both the A
results and those at C, for any determination of the photon's state by the
A polarising filter will destroy the possibility of determining how it
would have responded to the C filter. No combined sample space is
possible, and the results at A are not related in any rational way with
the results (in a different experiment) at C.

9.12 We can express this differently. We claimed above that “if
instead of setting the left polariser to A it had been at C, with the
right hand polariser remaining at B', the run of results at B' would be
the same as before”. This is a meaningless conterfactual. Philosophers
have much difficulty in explaining the real meaning of counterfactuals,
and the standard work on the subject, David Lewis’ book
Counterfactuals (1973), leaves many questions unanswered, as I try
to show in a publication on the internet (Conditionals and
Counterfactuals, 1997). I believe any counterfactual in which the
antecedent is not only false, but is essentially impossible, to be
meaningless. This treatment of the EPR paradox provides a good example of
such a meaningless statement. If the left-hand polariser is at A then it
is fundamentally impossible to determine what the B' results would have
been if the left polariser had not been at A. Under these
circumstances I maintain that no meaning can be attached to the statement
that "the run of results at B' would be the same as before", and once
again we find that no conclusion can be drawn concerning the relationship
of the A results to the C results.

9.13 So far as we are aware, the only writer who has realised
fully what the observed infringements of Bell’s inequality do and do not
imply is Thomas Brody (The Philosophy behind Physics,
Springer-Verlag, 1993). He shows that Bell implicitly made a third
assumption in deriving his inequality, in addition to assumptions of
realism and locality. He calls this the Joint Measurability
Assumption. He writes:

The joint measurability assumption refers to
the possibility of measuring two (or more) physical quantities without
mutual interference; this last expression is to be understood in the
sense that neither measurement affects the value obtained by the
other... In the case of spin projections, the JMA is
violated.
He is referring here, of course, to the fact
that a photon having passed through a polarising filter is supposed to be
put into a new state, so that its original axis is no longer measurable.
This is saying, in effect, what our own two statements above have said,
that two measurements on the same photon cannot figure in the same sample
space, or alternatively, that it is meaningless to discuss what result
such an alternative measurement would have given if it could have
been performed.

9.14 So the failure of Bell’s inequality in describing the
results of EPR experiments tells us only that there must be an error in
one (or more) of Bell’s three assumptions, reality, locality
or joint measurability. As the last of these is false, we can make
no deduction concerning the truth of the other two. The questions of the
reality of measurable quantities, or the locality of quantum
causality or influences, remain open, and we must look elsewhere to
resolve them.

9.15 Although we are not forced to abandon either the principle
of locality or of reality by the EPR experiments, because
Brody presents us with a third alternative, it does seem that the
probability which we associate with a particular photon being
transmitted by a particular filter is inflenced in a non-local way by the
orientation of the other filter, and by whether or not the twin photon is
transmitted. Perhaps the two photons are governed by just one wave
function rather than two, but some influence does appear to be transferred
at superluminal speed. This can not be described as a causal
influence, and it is insufficient to transfer information, and so infringe
Special Relativity. The non-local influence on probabilities has no
observable effect on any single pair of particles, for we cannot deduce
the probability of an event merely by observing one instance of it; only
when we observe a large number of particles do the effects of probability
become evident. Indeed, no finite run of successive experiments, however
long, can prove that the probabilities are influenced non-locally,
any more than a run of "heads" when a coin is tossed repeatedly can prove
that the coin is biased. But there is no doubt this influence exists, and
it remains a strange phenomenon, with no parallel in everyday life.

9.16 We feel some discomfort when contemplating such
non-local effects, even if they apply only to the probabilities of
events happening rather than the occurrence of individual events
themselves, but this discomfort is relieved to some extent by the Nodal
theory. The most unpalatable aspect of the picture we painted is of the
photon which has not yet encountered its polariser suddenly changing its
spin axis because the other photon has encountered its polariser.
Nodal theory teaches us, of course, that neither photon actually exists at
this time, and all that is demanded is that, when the (imagined) photons
reach their next nodes, the information fed into each node takes account
of the orientation of the two polarisers, a picture which seems a little
more credible.

9.17 Furthermore, as we have already shown, such
non-local effects are an essential requirement of the nodal theory;
indeed any particle in “moving” from one node to the next must “feel out”
the whole of the surrounding region of space-time in order to “choose” its
next node, and to determine what information is to be transferred to it.
The present discussion extends this concept to cover the rather special
circumstances encountered when a pair of particles become entangled, as
they are in EPR experiments, and the behaviour of each is
influenced by the region around both of them. So accepting the
Nodal hypothesis means that our credulity is stretched no further by the
EPR results than by other more common observations.

***

(c) Hector C. Parr (2002)

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