The Quantum Mystery

A peculiarly quantum measurement

2007-07-02T01:43:00.001-07:00

It is often said that quantum theory introduces an inevitable, minimum,
disturbance into any measurement. This is true, but here I want to
describe something which at first sight appears to show exactly the
opposite effect, namely, how quantum theory enables us to make a
totally non-disturbing measurement of a type that is impossible in
classical physics.
We consider a two-state system which, in order to have a simple
picture, we regard as a box that can be either EMP?"Y (not contain a
particle) or FULL (contain a particle). From a large sample of such
boxes we are given the task of selecting one that we know is FULL.
The way to do this is to 'look' and see if the box contains a particle.
However, it turns out that one photon falling on the box will either pass
right through, if the box is EMPTY, or be absorbed and destroy the
particle, if the box is FULL. Since we require to use at least one photon
in order to look at the box it follows that, after we have looked, we
either confirm that the box is EMPTY, or we know that it was FULL,
but is so no longer. Clearly, it seems, we cannot select a box that
is certainly FULL. The act of verifying that it is Fuu would simply
destroy the particle.
Here, amazingly, quantum mechanics provides a way to accomplish
our task. We first construct a photon interferometer, as shown in
figure 30. The photons enter at A and reach a beam-splitter (halfsilvered
mirror) at B, where the wave separates into two parts of equal
magnitude travelling on the paths denoted by 1 and 2. They recombine
at a second beam splitter, C, where, by suitable choice of path lengths,
it is arranged that the two contributions to the output towards the D detector destructively interfere, so that D never records a photon. In
other words, the photons always take the E path. Next we suppose
that at a certain place on, say, path 1 we can place one of our boxes
in such a way that if it is FULL the photon will be absorbed, and the
particle in the box destroyed, whereas if it is EMPTY it will have no
effect. We then place each box in turn in the interferometer, and send
in one photon. If the photon does not appear in the detector D then we
discard the box and choose another. When we have a box for which
the detector does record a photon, then we know that we have a box
that is FULL.
It is easy to see why: if the box had been EMPTY, then it would
have no effect, and by construction of the interferometer, the photon
could not go to the detector at D. Thus if a photon is seen at D, the
box is necessarily FULL. Note, also, that a FULL box just acts as
another detector, so with beam splitters having equal probabilities of
transmission and reflection, half of the experiments with a FULL box
will result in the photon destroying the particle in the box. In the
other half, the photon will reach the second beam-splitter, at C, and
one-half of the time will pass through and reach the D detector. Thus
one-quarter of the FULL boxes will lead to a photon being seen at D,
and therefore will actually be selected as FULL. What we have here is is a perfect ‘non-disturbing’ measurement, because we can see that the
photon has actually gone on the other path (path 2); nevertheless, if it
appears at the detector, it has verified that the box is FULL.
The basic ideas behind the arguments of this section are due
to A.C.Elitzw and L.Vaidman in an unpublished article from the
University of Tel Aviv (1991). Other applications of similar ideas
are given by L. Hardy Physics Letters 167A 11 (1992) and Physical
Review Letters 68 2981 (1992).

The Bohm model

2007-07-02T01:42:00.001-07:00

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.

Recent Developments of Quantum

2007-07-02T01:41:00.000-07:00

Models with explicit collapse
In 53.7, and Appendix 7, we considered how the measurement problem
of quantum mechanics could be solved by changing the theory so that a
wavefunction would evolve with time to become a state corresponding
to a unique value of the observable that was being measured. Two
difficulties with this approach were noted, namely, it seemed to require
prior knowledge of what was to be measured (since a state cannot in
general correspond to a unique value of several observables), and also
it had to happen very quickly in circumstances involving observation,
but at most very slowly in the many situations where the Schrijdinger
equation is known to work very well.
An explicit model, in which both these difficulties were overcome,
was proposed by three Italians, GianCarlo Ghirardi, Albert0 Rimini
and Tullio Weber (now universally known as GRW), in a remarkable
article published in 1986 (Physical Review D 34 470). They noted, first,
that all measurements ultimately involve the position of a macroscopic
object. (The special role of position is already used implicitly in the de
Broglie-Bohm model, as was noted in 55.2). Thus the measurement
problem can be solved if wavefunctions evolve so as to ensure
that macroscopic objects quickly have well-defined positions. By a
macroscopic object we here mean something that can seen by the naked
eye, say, something with a mass greater than about gm. Similarly,
a well-defined position requires the spread of the wavefunction to be
less than an observable separation, say, less than about cm.
In order to achieve this end, GRW postulated that all particles suffer
(infrequent) random ‘hits’ by something that destroys (makes zero)
all their wavefunction, except that within a distance less than about lO-%m from some fixed position. This position is chosen randomly
with a probability weight proportional to the square magnitude of the
particle’s wavefunction, i.e., to the probability of its being found at that
position if its position were measured (see 82.2).
GRW assumed that the typical time between hits was of the order
of lo%, which ensures that the effects of the hits in the microscopic
world are negligible, and do not disturb the well established agreement
between quantum theory and experiment. However, even the small
macroscopic object referred to above, with mass gm, contains
about 10l8 electrons and nucleons, so typically about one hundred of
these will be hit every second. Although it might at first sight seem
that hitting a few particles out of so many would have a negligible
effect, it turns out that, in a measurement situation, just one hit is
enough to collapse the whole state: when one goes, they all go! This
is the real magic of the GRW proposal. To see how it comes about we
imagine that the macroscopic object represents some sort of detector
(a ‘pointer’) which tells us whether a particle has, or has not, passed
through a barrier (see Chapter 1). Explicitly, suppose the pointer is
in position 1, with wavefunction D’, if the particle has been reflected,
and in position 2, with wavefunction D2, if it has not. Note that, for
example, D’ corresponds to all the particles of the object being close
to position 1. We assume that, in a proper measurement, the separation
between the two positions is greater than both the size of the object
and the GRW size parameter lo4 cm. The wavefunction describing
this situation has the form (cf 54.5):
Now we suppose that one of the particles is hit. The centre of the
hit will most likely occur where the wavefunction is big, i.e., in the
neighbourhood of either position 1 or position 2 (with probabilities
IPR12,l P~lre~sp ectively). Suppose the random selection chooses the
former. Then the whole wavefunction given above will be multiplied
by a function which is zero everywhere except in the neighbourhood
of position 1. Since the second term in the above state is zero except
when all the particles are near position 2, it will effectively be removed
by this hit (there are no values for the position of the hit particle
for which both factors, the hitting function and the wavefunction D2,
simultaneously differ from zero). In other words the wavefunction
will have collapsed to the state in which the particle was reflected. Notice that it is something that happens in the detector that establishes
whether or not the particle is transmitted; without a detector no such
determination is made (except within a time of around 10l6 s, the
average collapse time for a single particle).
Since, as we have seen, even for a small detector the typical time
between the collapses is of the order of s, which is less than the
time it takes for a person to respond to an observation, it is clear that the
GRW model has the desired effect of giving outcomes to measurements.
As a working, realistic, model of quantum theory it is important. It
provides insight into the theory; it raises fascinating questions relating
to when a conscious observation has actually occurred, particularly
because the disappearance of the unwanted terms is only approximate
and so-called ‘tails’ always remain; it also gives a structure in which
questions like the relation with relativity can be discussed. Whether
it is true is another question. It seems very unnatural, although more
satisfying versions in which the hitting is replaced by a continuous
process (similar to that discussed in Appendix 7) have been developed
by GRW, Philip Pearle and others. A review of this work, and further
references, is given in the articles by Ghirardi and Pearle published in
Proceedings of the Philosophy of Science Foundation 2 pp 19 and 35
(row).
The predictions of collapse models do not agree exactly with those
of orthodox quantum theory; for example, they give a violation of
energy conservation. It is this that puts limits on the parameters-the
process must not happen too quickly. Any bound system, initially in
its stable, lowest energy state, will have a certain probability of being
excited to a higher energy state if one of the constituents is ‘hit’. Thus,
for example, hydrogen atoms will spontaneously emit photons. Philip
Pearle and I have recently shown that the best upper limit on the rate
(i.e., lower limit on T), probably comes from the fact that protons are
known to be stable up to something like years. These protons are
in fact bound states of three quarks, and every time a quark is ‘hit’
there is a very small probability that the proton will go to an excited
state which will spontaneously decay. The fact that such decays have
not been observed puts severe restrictions on GRW-type models (and
may even rule out some simple versions).
In one sense it is an advantage for a model that it gives clear,
distinctive predictions, because this allows the possibility that it might
be verified. On the other hand, in the absence of any positive evidence
for the unconventional effects, the fact that the free parameters of the model have to be chosen rather carefully-to make the process
happen fast enough in a measurement situation, but not too fast to give
unobserved effects elsewhere-is a negative feature; why should nature
have apparently conspired so carefully to hide something from us?
A partial answer to this last question might lie in the possibility that
the parameters of the collapse are not in fact independent of the other
constants of the physical world, but arise in particular from gravity, as
suggested in Appendix 7. In his wide-ranging book, The Emperor’s
New Mind (Oxford University Press, 1989), Roger Penrose gives other
reasons for believing that gravity might be associated with collapse.
He also develops the idea that the human mind’s ability to go beyond
the limits of ‘algorithmic computation’, i.e., the use of a closed set of
rules, shows that it can only be explained by really new physics, and
that such new physics, which would be ‘non-computable’, might well
be associated with the collapse of the wavefunction.

Early history and the Copenhagen interpretation

2007-07-02T01:40:00.000-07:00

We have not, in this book, been greatly concerned with the
historical development of quantum theory. When an idea is new
many mistakes are made, blind alleys followed, and the really
significant features can sometimes be missed. Thus history is unlikely to be a good teacher. Nevertheless, it is of interest to look
back briefly on how the people who introduced quantum theory
into physics interpreted what they were doing.
Already we have noted that Einstein, surely the premier scientist
of this century, was always unhappy with quantum theory, which
he considered to be, in some way, incomplete. Initially his objections
seemed to be to the lack of causality implied by the theory,
and to the restrictions imposed by the uncertainty principle. He had
a long running controversy with Bohr on these issues, a controversy
which it is fair to say he lost. In addition, however, Einstein was
one of the first to realise the deeper conceptual problems. These he
was not able to resolve. Many years after the time when he was the
first to teach the world about photons, the particles of light, he admitted
that he still did not understand what they were.
Even more remarkable, perhaps, was the attitude of Schrodinger .
We recall that it was he who introduced the equation that bears his
name, and which is the practical expression of quantum theory,
with solutions that contain a large proportion of all science. In
1926, while on a visit to Copenhagen for discussions with Bohr and
Heisenberg, he remarked: ‘If all this damned quantum jumping
were really to stay, I should be sorry I ever got involved with quantum
theory.’ (This quote, which is of course a translation from the
original German, is taken from the book by Jammer, The
Philosophy of Quantum Mechanics, p 57). The ‘jumping’
presumably refers to wavefunction reduction, a phenomenon
Schrodinger realised was unexplained within the theory, which he,
like Einstein, therefore regarded as incomplete. To illustrate the
problem in a picturesque way he invented, in 1935, the
‘Schrodinger cat’ story, which we have already discussed in §4.4.
He considered it naive to believe that the cat was in an uncertain,
dead or alive, state until observed by a conscious observer, and
therefore concluded that the quantum theory could not be a proper
description of reality.
Next we mention de Broglie, who, it will be recalled, was the first
to suggest a wave nature for electrons. He was also unhappy with
the way quantum theory developed, and took the attitude that it
was wrong to abandon the classical idea that particles followed
trajectories. He believed that the role of the wavefunction was to
act as a pilot wave to guide these trajectories, an idea which paved
the way for hidden-variable theories. Thus, of the four people (Planck, Einstein, Schrodinger, de
Broglie) who probably played the leading roles in starting quantum
theory, three became, and remained, dissatisfied with the way it
developed and with its accepted ‘orthodoxy’. This orthodoxy is
primarily due to the other three major figures in the early development
of the theory, Bohr and, to a lesser extent, Heisenberg and
Born. It has become known as the ‘Copenhagen’ interpretation.
A precise account of what the Copenhagen interpretation actually
is does not exist. Quotations from Bohr’s articles do not always
seem to be consistent (which is not surprising in view of the fact
that the ideas were being developed as the articles were being
written). Almost certainly, two present-day physicists, who both
believe that they subscribe to the orthodox (Copenhagen) interpretation,
would give different accounts of what it actually means.
Nevertheless there are several key features which, with varying
degrees of emphasis, would be likely to be present. We shall
endeavour to describe these.
(i) Bohr made much use of the notion of ‘complementarity’:
particle and wave descriptions complement each other; one is
suitable for one set of experiments, the other for different
experiments. Thus, since the two descriptions are relevant to
different experiments, it does not make sense to ask whether they
are consistent with each other. Neither should be used outside its
own domain of applicability.
(ii) The interpretation problems of quantum theory rest on
classical ways of thinking which are wrong and should be abandoned.
If we abandon them then we will have no problems. Thus
questions which can only be asked using classical concepts are not
permitted. Classical physics enters only through the so-called ‘correspondence’
principle, which says that the results of quantum
theory must agree with those of classical mechanics in the region
of the parameters where classical mechanics is expected to work.
This idea, originally used by Planck, played an important role in
the discovery of the correct form of quantum theory.
(iii) The underlying philosophy was strongly ‘anti-realist’ in tone.
To Bohr: ‘There is no quantum world. There is only an abstract
quantum physical description. It is wrong to think that the task of
physics is to find out how nature is. Physics concerns what we can say about nature.’ Thus the Copenhagen interpretation and the
prevailing fashion in philosophy, which inclined to logical
positivism, were mutually supportive. The only things that we are
allowed to discuss are the results of experiments. We are not
allowed to ask, for example, which way a particle goes in the interference
experiment of 61.4, The only way to make this a sensible
question would be to consider measuring the route taken by the
particle. This would give us a different experiment for which there
would not be any interference. Similarly, Bohr’s reply to the alleged
demonstration of the incompleteness of quantum theory, based on
the EPR experiment, was that it was meaningless to speak of the
state of the two particles prior to their being measured. (It should
be noted that Einstein himself had made remarks which were in this
spirit. Indeed Heisenberg, a convinced advocate of the Copenhagen
interpretation, was apparently helped along this line by one such
remark: ‘It is the theory which decides what we can observe.’)
(iv) All this leaves aside the question of what constitutes a
‘measurement’ or an ‘observation’. It is possible that somewhere in
the back of everyone’s mind there lurked the idea of apparatuses
that were ‘classical’, i.e. that did not obey the rules of quantum
theory. In the early days the universality of quantum theory was
not appreciated, so it was more reasonable to divide the world into,
on the one hand, observed systems which obeyed the rules of
quantum mechanics, and, on the other, measuring devices, which
were classical.
These, then, are the ingredients of the Copenhagen interpretation.
It is very vague and answers few of the questions; anybody
who thinks about the subject today would be unlikely to find it
satisfactory: yet it became the accepted orthodoxy. We have
already, in $5.2, suggested reasons why this should be so. The
theory was a glorious success, nobody had any better answers to the
questions, so all relaxed in the comfortable glow of the fact that
Bohr had either answered them or told us that they should not be
asked.
1 was a research student in Manchester in the 1950s. Rosenfeld
was the head of the department and the Copenhagen interpretation
reigned unquestioned. One particular Christmas, the department
visited the theoretical physics department in Birmingham to sing
carols (that, at least, was the excuse). Some of the carols were parodied. In particular, I remember the words we used for the carol
that normally begins ‘The boar’s head in hand bear 1’. They were:
At Bohr’s feet I lay me down,
For I have no theories of my own
His principles perplex my mind,
But he is oh so very kind.
Correspondence is my cry, I don’t know why,
I don’t know why.
But we were all afraid to ask!

More quantum mystery

2007-07-02T01:39:00.000-07:00

Quantum theory has been the basis of almost all the theoretical
physics of this century. It has progressed steadily, indeed gloriously.
The early years established the idea of quanta, particularly
for light, then came the applications to electrons which led to all
the developments in atomic physics and to the solution of
chemistry, so that already in 1929 Dirac could write that ‘The
underlying physical laws necessary for the mathematical theory of
a large part of physics and the whole of chemistry are thus com pletely known.. .’ (Proceedings of the Royal Society A123 714).
The struggle to combine quantum theory with special relativity,
discussed in the preceding section, occupied the period from the
1930s to the present, and its successes have ranged from quantum
electrodynamics to QCD, the theory of strong interactions. We are
now at the stage where much is understood and there is confidence
to tackle the remaining problems, like that of producing a quantum
theory of gravity.
The interpretation problem has been known since the earliest
days of the subject (recall Einstein’s remark mentioned in 0 1. l),
but here progress has been less rapid. The ‘Copenhagen’ interpretation,
discussed in the next section, convinced many people that the
problems were either solved or else were insoluble. The first really
new development came in 1935 with the EPR paper, which, as we
have seen, purported to show that quantum theory was incomplete.
We must then wait until the 1950s for Bell’s demolition of the von
Neumann argument regarding the impossibility of hidden-variable
theories, and, later, for his theorem about possible results of local
theories in the EPR experiment. Throughout the whole period there
were also steady developments leading to satisfactory hiddenvariable
theories. At present, attempts are being made to see if
these are, or if they can be made, compatible with the requirements
of special relativity.
What progress can we expect in the future? In the very nature of
the case, new insights and exciting developments are unlikely to be
predictable. We can, however, suggest a few areas where they
might occur.
Let us consider, first, possible experiments. There is much interest
at present in checking the accuracy of simple predictions of
quantum theory, in order, for example, to see whether there is any
indication of non-linear effects. No such indications have been seen
at the present time, but continuing checks, to better accuracy and
in different circumstances, will continue to be made.
Another area where there is active work being done is in the
possibility of measuring interference effects with macroscopic
objects, or at least with objects that have many more degrees of
freedom than electrons or photons. The best hope for progress here
lies in the use of SQUIDS (superconducting quantum interference
devices). These are superconducting rings, with radii of several
centimetres, in which it is hoped that interference phenomena, as predicted by quantum theory, between currents in the rings can be
observed. Such observations will verify (or otherwise) the predictions
of quantum theory for genuinely macroscopic objects. In
particular, it should be possible to see interference between states
that are macroscopically different, and thereby verify that a system
can be in a quantum mechanical superposition of two such states
(cf the discussion of Schrodinger’s cat, etc, in 44.3).
The success of quantum theory, combined with its interpretation
problems, should always provide an incentive to experimentalists to
find some result which it cannot predict. Many people would
probably say that they are unlikely to find such a result, but the
rewards for so doing would be great. If something could be shown
to be wrong with the experimental predictions of orthodox quantum
theory then we would, at last, perhaps have a real clue to
understanding it.
It must be admitted that the likelihood of there being any practical
applications arising from possible discoveries in this area is
extremely low. There are many precedents, however, that should
prevent us from totally excluding them. We have already noted in
$5.6 that genuine observation of wavefunctions, were it ever to be
possible, might lead to the possibility of instantaneous transmission
of signals. To allow ourselves an even more bizarre (some would
say ridiculous) speculation, we recall that, as long as the wavefunction
is not reduced, then all parts of it evolve with time according
to the Schrodinger equation. Thus, for example, the quantum
world contains the complete story of what happens at all subsequent
times to both the transmitted and reflected parts of the wavefunction
in a barrier experiment. Suppose then that a computer is
programmed by a non-reduced wavefunction which contains many
different programs. In principle this is possible; different input keys
could be pressed according to the results (‘unobserved’, of course)
of a selection of barrier type experiments, or, more easily, according
to the spin projections of particles along some axis. As long as
the wavefunction is not reduced, the computer performs all the
programs simultaneously. This is the ultimate in parallel processing!
If we observe the output answer by normal means we select one
set of results of the experiments, and hence one program giving a
single answer. The unreduced output wavefunction, however, contains
the answers to all the programs. It is unlikely that we will
ever be able to read this information, but . . . On the theoretical side, we have already mentioned the passibility
that the difficulties with making a quantum theory of gravity
just might be related to the defects of quantum theory. Maybe
some of our difficulties with non-locality suggest that our notions
of time and space are incomplete. If, for example, our three dimensions
of space are really embedded in a space of more dimensions
then we might imagine that points of space which seem to us to be
far separated are in reality close together (just as the points on a
ball of string are all close, except to an observer who, for some
reason, can only travel along the string).
Bearing in mind the issue of causality, we might ask why we
expect this to exist in the first place, in particular, why we believe
that the past causes the present. Indeed we could wonder why there
is such a difference between the past, which we remember, and the
future, which we don’t! In case we are tempted to think these things
are just obvious, we should note that the fundamental laws of
physics are completely neutral with regard to the direction of time,
i.e. they are unchanged if we change the sign of the time variable.
In this respect time is just like a space variable, for which it is clear
that one direction is not in any fundamental respect different from
any other. Concepts like ‘past’ and ‘present’, separated by a ‘now’,
do not have a natural place in the laws of physics. Presumably this
is why Einstein was able to write to a friend that the distinction
between past and present was only a ‘stubbornly persistent
illusion’.
It may well be that, in order to understand quantum theory, we
need totally new ways of thinking, ways that somehow go beyond
these illusions. Whether we will find them, or whether we are so
conditioned that they are for ever outside our scope is not at
present decidable.

Quantum theory and relativity

2007-07-02T01:38:00.000-07:00

This is a difficult section, from which we shall learn little that has
obvious relevance to our theme. Nevertheless, the section must be
included since its subject is very important and is an extremely
successful part of theoretical physics. There is also the possibility,
or the hope, that it could one day provide the answers to our
problems.
The mysteries that we met in Chapter One arose from certain
experimental facts. We have learned that quantum theory predicts
the facts but does not explain the mysteries. Now we must learn
that quantum theory also meets another separate problem, namely
that it is not compatible with special relativity.
The reason for this'is that special relativity requires that the laws
of physics be the same for all observers regardless of their velocity
(provided this is uniform). This requirement implies that only relative velocities are significant, or, in other words, that there is no
meaning to absolute velocity. In practice this fact makes little
difference to physics at low velocity; it is only when velocities
become of the order of the velocity of light (3 x lo8 m s-’) that the
new effects of special relativity are noticed.
Quantum theory, as originally developed, did not have this
property of being independent of the velocity of the observer, and
is thus inconsistent with special relativity. Although the practical
effects of this inconsistency are very tiny for the experiments we
have discussed, there are situations where they are important, and
it is natural to ask whether quantum theory can be modified to take
account of special relativity, and even to ask whether such
modifications might provide some insight into our interpretation
problems. The answer to the first of these questions is a qualified
‘yes’; to the second it is a tentative ‘no’.
The relativistic form of quantum mechanics is known as relativistic
quantum field theory. It makes use of a procedure known
as second quantisation. To appreciate what this means we recall
that, in the transition from classical to quantum mechanics, variables
like position changed from being definite to being uncertain,
with a probability distribution given by a wavefunction, i.e. a
(complex) number depending upon position. In relativistic
quantum field theory we have a similar process taken one stage
further: the wavefunctions are no longer definite but are uncertain,
with a probability given by a ‘wavefunctional’. This wavefunctional
is again a (complex) number, but it depends upon the
wavefunction, or, in the case where we wish to talk about several
different types of particle, upon several wavefunctions, one for
each type of particle. Thus we have the correspondence:
First quantisation:
Second quantisation:
x, y, . . .
W(x), V(x ), . . .
replaced by W(x, y, . . . )
replaced by Z( W(x), U(x). . .),
The analogue of the Schrodinger equation now tells us how the
wavefunctional changes with time.
An important practical aspect of relativistic quantum field theory
is that the total number of particles of a given type is not a fixed
number. Thus the theory permits creation and annihilation of
particles to occur, in agreement with observation.
For further details of relativistic quantum field theory we must refer to other books. (Most of these are difficult and mathematical.
An attempt to present some of the features in a simple way
is made in my book To Acknowledge the Wonder: The story of
fundamental physics, referred to in the bibliography.) There is no
doubt that the theory has been enormously successful in explaining
observed phenomena, and has indeed been a continuation of the
success story of ‘non-relativistic’ quantum theory which we outlined
in 82.5. In particular, it incorporates the extremely accurate predictions
of quantum electrodynamics, has provided a partially unified
theory of these interactions with the so-called weak interactions,
and has provided us with a good theory of nuclear forces. In spite
of these successes there are formal difficulties in the theory. Certain
‘infinities’ have to be removed and the only way of obtaining results
is to use approximation methods, which, while they appear to
work, are hard to justify with any degree of rigour.
Do we learn anything in all this which might help us with the
nature of reality? Apparently not. If, in our previous, nonrelativistic,
discussion, we regarded the wavefunction as a part of
reality, we now have to replace this by the wavefunctional, which
is even further removed from the things we actually observe. The
wavefunctions have become part of the observer-created world, i.e.
things that become real only when measured.
We must now consider the problem of making quantum theory
consistent with general relativity. Since general relativity is the
theory of gravity, this problem is equivalent to that of constructing
a quantum theory of gravity. Much effort has been devoted to this
end, but a satisfactory solution does not yet exist. Maybe the lack
of success achieved so far suggests that something is wrong with
quantum theory at this level and that, if we knew how to put it
right, we would have some clues to help with our interpretation
problem. This is perhaps a wildly optimistic hope but there are a
few positive indications. Gravity is negligible for small objects, i.e.
those for which quantum interference has been tested, but it might
become important for macroscopic objects, where, it appears,
wavefunction reduction occurs. Could gravity somehow be the
small effect responsible for wavefunction reduction, as discussed in
$3.7?
Probably the correct answer is that it cannot, but if we want
encouragement to pursue the idea we could note that the magnitudes
involved are about right. The ratio of the electric force (which is responsible for the effects seen in macroscopic laboratory
physics) to the gravitational force, between two protons, is about
For larger objects the gravitational force increases (in fact it
is proportional to the product of the two masses), whereas this
tends not to happen with the electric force because most objects are
approximately electrically neutral, with the positive charge on
protons being cancelled by the negative charge on electrons.
Consider, then, the forces between two massive objects, each of
which has charge equal to the charge on a proton. The electric force
will be equal to the gravitational charge if the objects weigh about
10-6g. Thus we can see that gravitational forces become of the
same order as electrical forces only when the objects are enormously
bigger than the particles used in interference effects, but
that they are certainly of the same order by the time we reach
genuine macroscopic objects. (See also the remarks at the end of
Appendix 7.)
We end this section by noting a few other points. General
relativity is all about time and space, about the fact that our
apparently ‘flat’ space is only an approximation, about the
possibility that there are singular times of creation, and/or extinction,
about the existence of black holes with their strange effects.
Some of these facts could be relevant, but at the present time all
must be speculation. As an example of such speculation we mention
the suggestion of Penrose that there might be some sort of
trade-off between the creation of black holes and the reduction of
wave packets (see the acticle by Penrose, ‘Gravity and State Vector
Reduction’ in Quantum Concepts in Space and Time, ed C J Isham
and R Penrose [Oxford: Oxford University Press 19851).

The Mysteries of the Quantum World

2007-07-02T01:37:00.001-07:00

Readers who have read this far are probably confused. Normally
this is not a good situation to be in at the start of the last chapter
of a book. Here, however, it could mean that we have at least
learned something: the quantum world is very strange. Certain
experimentally observed phenomena contradict any simple picture
of an external reality. Although such phenomena are correctly
predicted by quantum theory, this theory does not explain how they
occur, nor does it resolve the contradictions.
What else ought we to have learned? We have seen, again on the
basis of experiment, that a local picture of reality is false. In other
words, the assumption that what happens in a given region of space
is not affected by what happens in another, sufficiently distant,
region is contrary to observation.
Nothing else is certain. We have met questions which appear to
have several possible answers. None of these answers, however, are
convincing. Indeed, it is probably closer to the truth to say that all
are, to our minds, equally implausible. The quantum world teaches
us that our present ways of thinking are inadequate.
I have tried to give a quick survey of the questions and their
possible answers in tables 6.1 and 6.2 The first of these tables
presents the problem purely in terms of the potential barrier experiment
introduced in 81.3. No reference is made here to quantum
theory or its concepts.

Can signals travel faster than light?

2007-07-02T01:36:00.000-07:00

According to the special theory of relativity, the velocity of light
(or, more generally, of electromagnetic radiation) in vacuum is a
fundamental property of time and space. The rules for combining
velocities, and the laws of mechanics, etc, ensure that nothing can
move with a velocity that exceeds this.
It would take us too far outside the scope of this book to explain
special relativity; we can, however, assert with confidence that it is
now firmly based on experimental observation and that it is a vital
ingredient of the structure of contemporary theoretical physics.
That its effects are not immediately obvious in our everyday
experience is due to the large size of the velocity of light,
c = 3 x 10' ms-'.
How then do we understand the fact that, according to quantum
theory, wavefunction reduction happens instantaneously over
arbitrarily large distances and, further, that such behaviour is
apparently confirmed by experiment?

The first thing to notice here is that we cannot actually use this
type of wavefunction reduction to transmit real messages from one
macroscopic object to another. To help us appreciate what is meant
by this statement we should distinguish the transmission of a
message between two observers from what happens when the two
observers both receive a message. For example, two people, one on
Earth and one on Mars, could make an agreement that they will
meet at a particular time either on Earth or on Mars. In order to
determine which, they might agree to measure spins, in a prearranged
direction, of electrons emitted in a particular EPR experiment.
If they obtained + 1/2 they would wait on their own planet,
whereas if they obtained -1/2 they would travel to the other’s
planet. The correlation between the results of their measurements,
noted in $5.4, would ensure that the meeting would take place. It
would be possible for them to make their measurements at the same
time, so they would receive the message telling them the place of
the meeting simultaneously. However this message would not have
been sent from one to the other.
We contrast this with the situation where the prior agreement is
that the person on Earth will decide the venue and then try to communicate
this to the person on Mars. How car, he use the EPR type
of experiment to transmit this message? The only option he has is
either to make a measurement of the spin of the electron or not to
make the measurement. A code could have been agreed: the
measurement of the spin of A along a previously decided direction
would mean that the meeting is to be on Earth, whereas no such
measurement would mean that Mars would be the venue. Thus, at
a particular time, he decides on his answer-he either makes the
measurement or he does not. Immediately the wavefunction of B
‘knows’ this answer; in particular, if it is Earth then B will have a
definite spin along the chosen direction, otherwise it will not.
The person on Mars, however, although he can observe the
particle B, cannot ‘read’ this information because he is not able to
measure a wavefunction. There is no procedure that the observer
could use that would allow him to know whether or not the spin
of B was definite or not.
The same conclusion is reached if we use, instead of a single
experiment, an ensemble of identical experiments. In this case,
if we decide on the venue Earth, then we would measure the spins
of all the A particles in the specified direction. This would

immediately mean that all the B particles had a definite spin in that
direction. Now, if these were all the same, e.g. if they were all
+ 1/2, then we could verify this by simply measuring them.
However, they would not all be the same, half would be + 1/2 and
half would be - 1/2, which is exactly the same distribution we
would have obtained if the spins were not definite, i.e. if the venue
had been Mars and no measurements of A had been made.
The situation could be very different if the quantum theory
description is incomplete and there are hidden variables. If these
could, by some as yet unknown means, be measured, then, since
measurements at A inevitably change these variables at B, the
possibility of sending messages at an infinite velocity would seem
to exist, in violation of the theory of special relativity. Such a violation
can be seen explicitly in some types of hidden-variable theories
where a quantum force is required to act instantaneously over
arbitrarily large distances. This contrasts with the known forces,
which in fact are due to exchange of particles and whose influence
therefore cannot travel faster than the velocity of light.
We here have another very unpleasant feature of hidden-variable
theories. It is not, however, possible to use this argument to rule
them out entirely. Special relativity has only been tested in experiments
that do not measure hidden variables; if we ever find ways
of measuring them then the theory might be shown to be wronggeneralising
results from one set of experiments to an entirely
different set has often led to mistakes.
Even within normal quantum mechanics the question of how a
wavefunction can reduce instantaneously, consistently with special
relativity, is one that requires an answer. To discuss it would take
us into relativistic quantum field theory, which is the method by
which quantum theory and special relativity are combined.
Although this theory has had many successes, it is certainly not
fully understood and at the present time does not appear to have
anything conclusive to say.

Experimental verification of the non-local predictions of quantum theory

2007-07-02T01:35:00.000-07:00

As we discussed in 82.5, quantum theory has been successfully
applied to a truly enormous variety of problems, and its status as
a key part of modern theoretical physics, with applications ranging
from the behaviour of the early universe and the substructure of
quarks to practical matters regarding such things as chemical binding,
lasers and microchips, is unquestioned. New tests of such a
theory might therefore be seen as adding very little to our
knowledge. The reason why, in spite of this, the experiments which
we describe here have attracted so much attention is that they test
certain simple predictions of the theory which violate conditions (in
particular the Bell inequalities) that very general criteria of
localisability would lead us to expect.
Following the publication of the first of the Bell inequalities, in
1965, there have been a succession of attempts to test them against
real experiments. These experiments are, in fact, quite difficult to
do with sufficient accuracy and early attempts, although they
generally supported quantum theory, with one exception, were
rather inconclusive. We shall therefore confine our discussion to
the recent series of experiments which have been performed in
France by Aspect, Dalibard, Grangier and Roger.
In all these experiments a particle emits successively two photons
in such a way that their total spin is zero. We recall that photons
are the particles associated with electromagnetic radiation, e.g.
light. They are spin one particles, in contrast to the spin 1/2
particles which we have previously used in our discussion. It is
convenient to measure the ‘polarisation’ of the photons rather than
their spin projections. These are related in a way that need not concern us. The only difference we need to note is that in the
predicted expression for ( E ) the angle between the various directions
has to be doubled, i.e. we find
(E(a, b)) = -cos 2(a - b). (5.9)
In the first experiment the spin measurements were carried out in
such a way that a particle with spin + 1 in a chosen direction was
deflected into the detector and counted, whereas a particle with spin
- 1 in the same direction was deflected away from the detector and
not counted. The experiment then measured the number of coincident
counts, i.e. counts at both sides. Because of imperfections
in the detectors it could not be assumed that no count meant that
the particle had spin - 1, it could have had spin + 1 and just been
'missed'. To take this into account it was necessary to run the
experiment with one or both of the spin detectors removed, and
then to use a modified form of the Bell inequality. We refer to the
experimental papers, listed in the bibliography, for details.
The important quantity that is measured is a suitably normalised
coincidence counting rate, which is predicted by quantum theory to
be given by
(5.10)
The factor 0.984, rather than unity, arises from imperfections in
the detectors (some particles are missed). If this prediction holds
throughout the whole range of angles then the Bell inequality is
violated. In figure 26 we show the results. The agreement with
quantum theory is perfect.
To demonstrate how effectively these results violate the Bell
inequality, and hence forever rule out the possibility of a local
realistic description of the world, the authors measured explicitly
at the angles where the violation was maximum, namely with the
configuration shown in figure 27, i.e. with a - b = b - a' =
a' - b' = 22.5", and a - b' = 67.5". A particular quantity S which
according to the Bell inequality has to be negative, but which
according to quantum theory has to be 0.1 18 2 0.005, is measured
to be 0.126 2 0.014. It is very clear that quantum theory and not
locality wins.
In the next set of experiments both spin directions were explicitly
detected, so the set-up was closer to that envisaged in the proof of
the original Bell inequality. From the measurements, the value
of(F(a, a ’ , b, b’)), defined in the previous section, was calculated
as
Fexp=t 2.697 i 0.015 (5.11)
for the orientation given by figure 27. This exceeds the bound given
in the inequality by more than 40 times the uncertainty. On the
other hand it agrees perfectly with the prediction of quantum
theory which, again allowing for the finite size of the detectors, is
calculated to be
Fq‘ = 2.70 & 0.05 (5.12)
instead of 2,2, which is the result with perfect detectors.

The third experiment was designed to investigate the following
question. Quantum theory suggests that measurement at A, say,
causes an instantaneous change at B, and this seems to be confirmed
by experiment. It appears therefore that ‘messages’ are sent with
infinite velocity (see the next section for further discussion of this).
Such a requirement would, however, not be needed if it were
assumed that the spin detecting instruments somehow communi cate their orientations to each other prior to the emission of the
photons, rather than when a photon actually reaches a detector. In
order to eliminate this possibility it is necessary to arrange that the
orientations are ‘chosen’ after the photons have been emitted.
Clearly the time involved is too small to allow the rotation of
mechanical measuring devices, so the experiment had two spin
detectors at each side, with pre-set orientations, and used switching
devices to deflect the photons into one or the other detector. The
switches were independently controlled at random. Thus, when the
photons were emitted, the orientations that were to be used had not
been decided. We refer to the original paper for further details of
this experiment and here record only the result, which was again in
complete agreement with quantum theory, and in violation of the
Bell inequality. Of course, it could be that nothing is really random
and that the devices that controlled the switching themselves communicated
with each other prior to the start of the experiment.
Such bizarre possibilities are hard to rule out (though if we were
sufficiently clever we could arrange that the signals which switch the
detectors originate from distant, different, galaxies that, according
to present ideas of the evolution of the universe, can never
previously have been in any sort of communication).

In this series of experiments it was also possible to vary the
distance between the two detectors and so test whether the wavefunction
showed any sign of ‘reducing’ as a function of time, as it
would according to the type of theory discussed in 03.4. Even when the separation was such that the time of travel of the photons was
greater than the lifetime of the decaying states that produced them
(which might conceivably be expected to be the time scale involved
in such an effect), there was no evidence that this was happening.
Thus it appears that, once again, quantum theory has been
gloriously successful. Maybe most of the people who regularly use
it are not surprised by this; they have learned to live with its strange
non-locality. The experiments we have described confirm this
feature of the quantum world; no longer can we forget about it by
pretending that it is simply a defect of our theoretical framework.
We close this section by noting the interesting irony in the history
of the developments following the EPR paper. Einstein believed in
reality (as we do); quantum theory seemed to deny such a belief and
was therefore considered by Einstein to be incomplete. The EPR
thought experiment was put forward as an argument, in which the
idea of locality was implicitly used, to support this view. We now
realise, however, that the experiment actually demonstrates the
impossibility of there being a theory which is both complete and
local.

Bell’s theorem

2007-07-02T01:34:00.000-07:00

This theorem, published in 1964 (Physics 1 195), expresses one of
the most remarkable results of twentieth century theoretical
physics. It exposes, in a clear quantitative manner, the real nature
of the conflict between ‘common sense’ and quantum theory which
exists in the EPR type of experiment. As we shall show, the
theorem is easy to prove (once one has seen it), but the fact that
nobody at the time of the early controversy following the publication
of the EPR paper realised that such a result could be found is
the real measure of the magnitude of John Bell’s achievement.
In order to appreciate properly the meaning of the theorem we
must first emphasise an important distinction; one which we have
indeed already met. The EPR experiment suggests that
measurements on one object (A) alter what we can predict for
subsequent measurements on another object (B), regardless of how
far apart the objects may be at the time of the measurements. There
are two completely different ways of explaining this, namely:
(i) it could be that measurements on A actually have an effect on
B, or alternatively,
(ii) it could be that measurements on A only affect our knowledge
of the state of B, i.e. they tell us something about B which was in
fact already true before the measurement.
The first of these possibilities, which Bell’s theorem shows is the
case in quantum theory, is totally contrary to the idea of locality.
The second, on the other hand, is an everyday occurrence and has
no great significance.
As a trivial example illustrating the last remark, we imagine that
a box is known to contain two billiard balls, one of which is black
and the other white. We then remove one ball, in the dark, and put
it on a rocket which flies off into space. At this stage all that we
know about the colour of this ball is that there is a 50% chance of
its being white, and a 50% chance of its being black (just like a spin
in a given direction might have a 50% chance of being either + 1/2
or - 1/2). We then look at the ball remaining in the box and if it
is black (white) we immediately know that the other ball is white
(black). Again this is superficially rather like our experiment with
two spin 1/2 particles. However we know that in no sense do we do
anything to the distant ball by looking in the box. It already was
either white or black. Because of our lack of knowledge, our
previous description of it was incomplete. A complete description
did however exist, and with such a complete description, the observation
of the colour of the remaining ball would clearly have no
effect.
The question now is whether such a complete description can
exist in the EPR spin experiment, i.e. is it possible that there is a
way of specifying the state of particle B such that measurements on
A have no effect on B? Bell’s theorem allows us to give a negative
answer to this question both on the basis of quantum theory, and
of experiment (see next section).
It is instructive to see exactly what is involved in the theorem, in
particular, how little, so we shall give the proof even though it
again involves a small amount of mathematical symbolism. (A
simpler form of the theorem, described in terms of the behaviour
of people rather than particles, can be found in Appendix 9.)
To begin, we suppose that the spin-measuring apparatus, at each
side, is connected to a machine that records the results of the
measurements. We arrange that these machines record + 1 for a
spin measurement of + 1/2 and - 1 for a measurement of - 1/2.
Let M and N be the values recorded for the A and B particles
respectively. In fact we shall be concerned only with the product of
M and N, which we denote by E. The appropriate experimental
arrangement is depicted in figure 23. (Note that throughout this section we are making the natural assumption that a measurement
gives only one result. Thus we are ignoring the many-worlds
possibility. For further discussion of this point see the article of
Stapp, ‘Bell’s theorem and the foundations of quantum physics’,
American Journal of Physics 53 306 1985.) Not surprisingly, in view of the statistical nature of quantum
theoretical predictions, the argument requires us to consider not
just one event but many, i.e. the decay of a large number of
identical spin zero particles. For each such event we can record a
value of E (always + or - l), and we then calculate the average
over all events. This will depend upon the orientation of the two
spindetectors, which are given by the angles a and b, so we write
it as ( E ( a , b ) ) . Thus,
@(a, b)) = Average value of E
= Average value of M - N. (5.1)
Clearly this number lies between + 1 and - 1.
Next we introduce the variable H which is supposed to give the
required complete description of the two spin 1/2 particles. It is not
important for our purpose whether H consists of a single number
or several numbers. However, for convenience we shall refer to it
as though it were just a single number. When we know the value
of H we know everything that can be known about the system.
Each event will be associated with a certain value of H and in a number of such events there will be a certain probability for any
particular value occurring. If the hidden-variable theory is deterministic
(a restriction we shall later drop), then the values of Mand
Nin a given event, and for given angles a and b, are uniquely determined
by the value of H.
Now we introduce the assumption of locality which is here
expressed by the assertion that the value of M does not depend on
b and the value of N does not depend on a. In other words, the
value we measure for the spin of the particle A cannot depend on
what we choose to measure about particle B, and vice versa. It
follows that M depends only on H and a, whilst N depends only
on W and b. We express these dependences by writing the values
obtained as M(H, a) and N(H, b) respectively. The resulting value
of E is then given by
E(H, a, b) = M(H, a)N(H, b). (5.2)
For a particular value of H this is a fixed number. Different values
of H can occur when we repeat the experiment many times, and the
average value of E that is measured will equal the average of
E(H, a, b) over these values of H, i.e. the hidden-variable theory
predicts
(E@,b ) )= Average over H of E(H,a , b). (5.3)
At this stage we do not appear to have got very far. Since we do
not know anything about the variation of M or N with H, or about
the distribution of the values of H, all that we can say about the
predicted value of (E(a,b)) is that it lies between + 1 and - 1. This
of course we already knew.
Now comes the clever part. We consider two different orientations
for each of the spin measuring devices. Let these be denoted
by the angles a and a’ for measurements on A and by b and b’ for
measurements on B. For a ked value of H, there are now two
values of Mand two values of N, i.e. four numbers, each of which
is either + 1 or - 1. In table 5.1 we show all possible sets of values
for these four numbers, We also show the corresponding values for
the quantity F(H, a, a ’ , b, b‘ ), defined by
F(H,a,a’b,, b ’ ) = E ( H , a , b )+ E ( H , a ’ , b ’ )
+E(H,a‘,b)-E(H,a,b’). (5.4)

In all cases this quantity is + or -2, from which it follows that
its average value over the (unknown) distribution of H lies between
- 2 and + 2. Hence our local hidden-variable theory predicts that
the particular combination of results defined by
(F(a,a',b,b'))= (E(a,b))+ ( E ( a ' , b ' ) )
+ (E@', b)) - (E@, b ' ) ) (5.5)
(5.6)
This is one form of the Bell inequality.
It is important to realise that locality rather than determinism is
the key ingredient of this proof. In order to demonstrate this, we
relax the assumption that H determines the values of M and N
uniquely, and suppose instead that each value of H determines a
probability distribution for M and N. The locality assumption is
now a little more subtle. It is that the probability distribution for
M does not depend on the value measured for N, and vice versa.
To appreciate why this is so we recall that measurement of N
cannot tell us anything further about particle A, since H is intended
to be the complete description of the state of the system; equally,
because of locality, it cannot change the state of A. Hence the
probability of obtaining any given value of M does not depend on
the value measured for N.
We can then define independent averages of M and N, for each
value of H. We denote these by M"'(H, a) and N"'(H, b). Because
of the assumption of independence, the average value of the
product of M and N, which we write as E"', is equal to the product
of the average values, i.e.
satisfies:
-2 < (F(a,a',b,b') ) < +2.
Eav(HU, , b)= M"'(H, a)N"'(H, b). (5.7) It is now possible to prepare a table similar to that above with
Ma’ replacing M, etc. Instead of taking values of + or - 1, these
quantities lie somewhere between these limits. It is then quite easy
to show that the particular combination defining F, which we now
denote by Fa’, always takes a value that lies between -2 and +2.
When we then average Fa’ over all values of H we again obtain the
Bell inequality.
For a more complete discussion of the circumstances in which the
inequality, or various alternative versions of it, can be proved we
refer to the review article of Clauser and Shimony listed in the
bibliography (86.5)
The significance of the Bell inequality lies in the fact that,
unlike the inequality we found for E, it does not have to be true
if the locality assumption is dropped. Indeed it turns out that the
inequality is violated by the predictions of quantum theory.
Before we discuss these predictions it is interesting to see why
quantum theory fails to satisfy the assumptions of the theorem. In
quantum theory the full specification of the state is the wavefunction,
so this plays the role of the quantity H. We can then define,
as above, the averages of M and N over many measurements.
However, these averages are not independent; the distribution of
values of Mdepends on what has been measured for N. As a simple
illustration of this we note that, with our wavefunction corresponding
to total spin zero, the averatge value of M or N measured
independently is zero (regardless of the angles a, b). However, in
the special case of a = b, then we know that M is always opposite
to N, so the product is always - 1. Thus the average value of MN
is - 1, which is not the product of the separate averages of M and
N.
In general, the quantum theoretical prediction for (E(a, b ) )
depends on the difference between the angles a and b. As we show
in Appendix 8 it is given by
(E(a, b)) = -cos(a - b). (5.8)
This function is drawn in figure 24. As expected it lies between - 1
and + 1. However, and this is the reason why it leads to a conflict
with the Bell inequality, it cannot be factorised into a product of
a function of a and a function of b.
The resulting prediction for (F(a, a‘, b, b ’ ) ) is now easily
found. A particularly simple case is when the angles are chosen as in figure 25.. Here the violation of the inequality is maximised; each
term in Fcontributing the same amount, (@)/2, to the sum. Hence,
F= 2 3 2: 2.83
which is in clear violation of the Bell inequality.

Thus the Bell inequality shows that any theory which is local
must contradict some of the predictions of quantum theory. The
world can either be in agreement with quantum theory or it can
permit the existence of a local theory; both possibilities are not
allowed. The choice lies with experiment; the experiments have
been done and, as we explain in the next chapter, the answer is
clear.

The Einstein-Podolsky-Rosen thought experiment

2007-07-02T01:32:00.002-07:00

In 1935, Einstein, Podolsky and Rosen published a paper entitled
‘Can Quantum-Mechanical Description of Physical Reality Be
Considered Complete?’ (Physical Review 47 777), which has had,
and continues to have, an enormous influence on the interpretation
problem of quantum theory. In this paper, they proposed a simple
thought experiment and analysed the implications of the quantum
theory predictions for the outcome of the experiment. These made
explicit the essentially non-local nature of quantum theory and,
according to the authors, proved that the theory must be
incomplete, i.e. that a more complete (hidden-variable) theory
exists and might one day be discovered. Much later, as we discuss
in the next section, John Bell carried the analysis considerably
further and showed that no local hidden-variable theory could
reproduce all the predictions of quantum theory. Naturally this
work prompted experimentalists to turn the thought experiments
into real experiments, and so check whether these predictions are
correct, or whether the actual results deviated from them in such
a way as to permit the existence of a satisfactory local theory. These
experiments, which we discuss in 05.5, beautifully confirm
quantum theory.
We shall refer to the general class of experiments with the same
essential features as that proposed by Einstein, Podolsky and
Rosen as EPR experiments. The orginal work is sometimes called
the EPR paradox, or the EPR theorem.
The particular EPR experiment that we shall describe is
somewhat different from the original, but is more suited to our
later discussion. We consider the situation shown in figure 20, in
which a particle with zero spin at rest in the laboratory decays spontaneously
into two, identical, particles, each with spin 112. These
particles, which we call A and B respectively, will move apart with
velocities that are equal in magnitude and opposite in direction.
(This ensures that their momenta add to zero so that the total
momentum, which was initially zero, is conserved.)
The experiment now consists of measuring the spin components
of the two particles in any particular directions-in fact, for
simplicity, we consider only directions perpendicular to the direction
of motion. Thus we have an apparatus that will measure the
spin component of particle A in a direction we can specify by the
angle a. Similarly we have an apparatus to measure the spin component
of B in a direction specified by some angle b. The full
experiment is illustrated in figure 21. The form of the apparatus
used to measure the spin is irrelevant for our purpose, but in order
to demonstrate that the measurement is possible we could consider
the case in which the spin 1/2 particles are charged, e.g. electrons.
In that case the particles would have a magnetic moment which
would be in the same direction as the spin. Then to measure the spin along a specific direction we could have a varying magnetic
field in that direction which would deflect the electron, up or down
according to the value of the spin component In order to discuss the form of the results we must digress a little
to think about spin. We first recall, from the earlier discussion of
spin in 53.7 (also Appendix 8), that a measurement of a spin component
of a spin 1/2 particle in any given direction will always give
a value either + 1/2 or - 1/2, i.e. the spin is always either exactly
along the chosen direction or exactly contrary to it. Suppose, for
example, that we know the particle has a spin component + 1/2 in
a particular direction (see figure 22). Whereas according to classical
mechanics we would obtain some value in between + 1/2 and - 1/2
for this second measurement, in fact, according to quantum
theory, we will obtain either of the two extremes, each with a
calculable probability. This probability will depend on the angle
between the two directions, and will be such that the average value
agrees with that given by classical mechanics. Within quantum
theory it will not be possible to predict which value we will obtain
in a given measurement; the situation in fact will be very analogous
to the choice of reflection or transmission in the barrier experiment
of 51.3. Further details of all this are given in Appendix 8. For the following
discussion the important fact we shall need to remember is that,
in quantum theory, the spin of a particle can have a definite value
in only one direction. We are free to choose this direction, but once
we have chosen it and determined a value for the spin in that direction,
the spin in any other direction will be uncertain. The fact that
when we measure the spin in this new direction we automatically
obtain a precise value implies that the measurement does something
to the particle, i.e. it forces it into one or the other spin values along
the new line. This of course is an example of wavefunction reduction
about which we have already written much.
The next thing that we need to learn is that the total spin, in any
given direction, for an isolated system, remains constant, Readers who know about such things will recognise this as being related to
the law of conservation of angular momentum. It is true in quantum
mechanics, as well as in classical mechanics; in particular, it
is true for individual events and not just for averages, a fact which
has been experimentally confirmed.

The pilot wave

2007-07-02T01:32:00.001-07:00

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.

Hidden Variables and Non-locality

2007-07-02T01:31:00.000-07:00

Review of hidden-variable theories
In $1.3 we saw that it is possible to repeat an experiment several
times, under apparently exactly the same conditions, and yet obtain
different results. In particular, for example, we could direct identical
particles, all with the same velocities, at identical potential
barriers, and some would be reflected and some transmitted. The
initial conditions would not uniquely determine the outcome.
Quantum theory, as explained in Chapter Two, accepts this lack
of determinism; knowledge of the initial wavefunction only
permits probabilistic statements regarding the outcome of future
measurements.
Hidden-variable theories have as their primary motivation the
removal of this randomness. To this end they regard the
‘apparently’ identical initial states as being, in reality, different;
distinguished by having different values of certain new variables,
not normally specified (and therefore referred to as ‘hidden’). The
states defined in quantum theory would not correspond to precise
values of these variables, but rather to certain specific averages over
them. In principle, however, other states, which do have precise
values for these variables, could be defined and with such initial
states the outcome of any experiment would be uniquely
determined. Thus determinism, as understood in classical physics,
would apply to all physics. Particles would then have, at all times,
precise positions and momenta, etc. The wavefunction would not
be the complete description of the system and there would be the
possibility of solving the problems with wavefunction reduction
which we met in Chapter Three. This latter fact is, to me at least,
a more powerful motivation than the desire for restoration of
determinism.
Any satisfactory hidden-variable theory must, of course, agree
with experimental observations and therefore, in particular, with
all the verified predictions of quantum theory. Whether it should
agree exactly with quantum theory, or whether it might deviate
from it to a small degree, while still remaining consistent with
experiment, is an open question, The normal practice seems to have
been to seek hidden-variable theories for which the agreement is
exact. A hidden-variable theory will, of course, tell us more than
quantum theory tells us-for example, it tells us which particles will
pass through a given barrier. What we require is that it gives the
same, or very nearly the same, results for those quantities that
quantum theory can predict.
There have been, and still are, many physicists who would regard
the question of the possiblity of such a hidden-variable theory,
agreeing in all measurable respects with quantum theory, as being
an unimportant issue. Readers who are still with us, however, are
presumably convinced that the quest for reality is meaningful, so
they will take a different view. The question is interesting and
worthy of our attention. Indeed, there are even pragmatic grounds
for pursuing it: different explanations of a set of phenomena, even
though they agree for all presently conceivable experiments, may
ultimately themselves suggest experiments by which they could be
distinguished. There is also the hope that better understanding of
quantum theory might help in suggesting solutions to some of the
other unsolved problems of fundamental physics.
The subject of hidden-variable theories was for many years
dominated by an alleged ‘proof’, given by von Neumann in 1932
(in his book Mathematische Grundlagen der Quantenmechanik
[Berlin : Springer] , English translation published by Princeton
University Press, 1955), that such theories were impossible, i.e. that
no hidden-variable, deterministic, theory could agree with all the
predictions of quantum theory. The proof was simple and elegant;
its mathematics, though subject to much scrutiny, could not be
challenged. However, the mathematical theorem did not really
have any relevance to the physical point at issue. The reason for
this lay in one of the assumptions used to prove the theorem. We
shall give a brief account of this assumption in the following paragraph. Since this account is rather technical and not used in the
subequent discussion, some readers may prefer to omit it.
Let us suppose that two quantities, call them X and Y, can be
separately measured on a particular system, and that it is also
possible to measure the sum of the two quantities, X + Y, directly.
Then the assumption was that the average value of X + Y, over any
collection of identical systems, i.e. any ensemble, was equal to the
average value of X plus the average value of Y. Since, in general,
the variable X+ Y is of a different kind, measured by a different
apparatus, from either X or Y, there is no reason why such an
equality should hold. Von Neumann was led to assume it because
it happens to be true in quantum theory, i.e. for those ensembles
specified by a given wavefunction. In a hidden-variable theory,
however, other states, defined by particular values of the hidden
variables, can, at least in principle, exist, and for such states the
assumption does not have to be true. Although several people
seemed vaguely to have realised this problem with von Neumann’s
theorem, it was not until 1964 that John Bell finally clarified the
issue, and removed this theoretical obstacle to hidden-variable
theories. The article was published in Reviews of Modern Physics
38 447 (1966).
At this stage we should emphasise that, although hidden variable
theories are possible, they are, in comparison to quantum theory,
extremely complicated and messy. We know the answers from
quantum theory and then we construct a hidden-variable, deterministic,
theory specifically to give these answers. The resulting
theory appears contrived and unnatural. It must, for example, tell
us whether a given particle will pass through a potential barrier for
all velocities and all shapes and sizes of the barrier. It must also tell
us ihe results for any type of experiment; not only for the
reflection/transmission barrier experiment of 0 1.3, but also for the
experiment with the mirrors. In the latter case, there can now be
no question of interference being the real explanation of what is
happening, because a given particle is certainly either reflected or
transmitted by the barrier and hence can only follow one path to
the detectors. Nevertheless, although it reaches only one of the
mirrors, which reflects it to the detectors, the path it follows must
be influenced by the other mirror. This is brought about by the
introduction of a new ‘quantum force’ which can act over
arbitrarily large distances. This quantum force is constructed in
order to give the required results. For details of all the various hidden-variable theories that are
available we refer to the excellent book by Belinfante, A survey of
hidden-variable theories [Oxford: Pergamon 19731. Here, we shall
only discuss a particular class of such theories; they appear to be
the most plausible and are the topic of our next section.

The many-worlds interpretation

2007-07-02T01:30:00.000-07:00

In 1957 H Everett I11 wrote an article entitled “‘Relative State”
Formulation of Quantum Mechanics’ (Reviews of Modern Physics
29 454) which introduced what has become known as the ‘manyworlds’
interpretation of quantum theory. He began by noting that
the orthodox theory requires wavefunctions to change in two
distinct ways; first, through the deterministic Schrodinger equation
and, secondly, through measurement, which causes the reduction
of the wavefunction to a new wavefunction which is not uniquely
determined. It is this second type of change that causes problems;
what is a ‘measurement’?, what are the non-quantum forces that
cause it?, how can it occur instantaneously over large distances?,
etc. Everett was in fact motivated in his work by yet another
problem: he was interested in applying quantum theory to the
whole universe, but how could he then have an ‘external’ observer
to measure anything?
The solution that Everett proposed to the problems of wavefunction
reduction was to say simply that it does not happen. Any
isolated system can be described by a wavefunction that changes
only as prescribed by the Schrodinger equation. If this system is
observed by an external observer then, in order to discuss what
happens, it is necessary to incorporate the observer into the system,
which then becomes a new isolated system. The new wavefunction,
which now describes the previous system plus the observer, is again
determined for all times by the Schrodinger equation.
To help us understand what this means we shall put it into
symbolic form. To this end we return to the barrier experiment, in
particular as this was discussed in $3.5. We write the wavefunction,
after interaction with the barrier, in the form:
PR( w ’ D ~ ~ +D P~T(W ~‘D~RO )FFD ~ON ) ,

This is not really as complicated as it might appear. The Ws
describe the particle, with the arrows indicating the direction, and
the Ds the two detectors. The first bracket is then the wavefunction
of the reflected wave and the second that of the transmitted wave.
Each of these wavefunctions is taken to be 'normalised' so that it
corresponds to one particle. Then the PR and PT are the parameters
that give the magnitudes of the two wavefunctions. The squares of
these numbers give the probability for reflection and transmission
respectively. We notice that this wavefunction correctly describes
the correlations between the states of the detectors and those of the
particle, e.g. that if the right-hand detector is ON then the particle
has been reflected, etc. This correlation exists because, as noted in
$3.4, the wavefunction is not simply a product (in fact in this case
it is the sum of two products).
According to the orthodox interpretation of quantum theory
such a wavefunction reduces, on being observed, to
w'DON OFF with probability Pi
R DL
or to
w-DOFF ON with probability Pf. R DL
(See figure 15.)
In the interpretation due to Everett, however, this reduction does
not occur. The true reality is always expressed by the full wavefunction
containing both terms. This is all very well, we are saying,
but did we not convince ourselves previously that the reduction had
to occur; that deterministic theories are not adequate to describe
observation? We certainly did, so we must examine the argument.
It relied on the fact that we, or more properly I, do not see both
pieces of the wavefunction. To me, either reflection or transmission
has occurred, not both. Clearly then, in order to understand what
is happening, it is necessary to introduce ME into the experiment
and to include ME in the wavefunction. Although my wavefunction
is very complicated the only relevant part for our purpose here is
whether I am aware of reflection or transmission. We denote these
two states of myself by ME"' and ME"^^ respectively. Thus the
complete wavefunction, according to Everett, is:
P~(WDD)-MEI~+* Pr(W DD)+ME"~"~
where we have simplified the notation in an obvious way. Notice that again the wavefunction contains the correct correlations: if the
particle is transmitted then I have observed transmission, etc.
Previously we argued (e.g. in 54.1) that, since we are only aware
of one possibility, one of the terms in the above expression must
be eliminated. Everett would argue instead that there are two MES,
both conscious but unaware of each other. Thus, through my
observation of what happens in the barrier experiment, I have split
the world into two worlds, each containing one possible outcome of
the observation.
Similar considerations apply to other types of observation. In all
cases the Everett interpretation requires that all possible outcomes
exist. Whenever a measurement is made we can think of the world
as separating into a collection of worlds, one for each possible
result of the measurement. It is through this way of thinking that
the name ‘many worlds’ has arisen. Such a name was not, however,
in the original Everett paper, and in some ways it is misleading. The
key point of this way of interpreting quantum theory is that
measurements are not different from other interactions; nothing
special, like wavefunction reduction, happens when a measurement
is made; everything is still described, in a deterministic way, by the
Schrodinger equation.
How can we reconcile this with our previous belief that
measurements were special? The previous argument was basically
as follows:
I am only aware of one outcome of a measurement, therefore there
is only one outcome.
Now we would argue differently:
I am only aware of one outcome of a measurement because the ME
that makes this statement, is the ME associated with one particular
outcome. There are other MES, which are associated with different
terms in the wavefunction, and which are aware of different outcomes.
The wavefunction given above for the barrier experiment
illustrates this: both of the terms exist, there are two MES but they
are not aware of each other.
It will be seen that, from the point of view of the many-worlds
interpretation, the ‘error’ we made earlier was that we inserted a
tacit assumption that our minds were able to look at the world from outside, and hence to conclude from our certainty of a particular
result that the other results had not occurred.
The ‘branching’ of the world into many worlds is therefore an
illusion of the conscious mind. The reality is a wavefunction which
always contains all possible results. A conscious mind is capable of
demanding a particular result (this is what we mean by making an
observation) and thereby it must select one branch in which it
exists. Since, however, all branches are equivalent, the conscious
mind must split into several conscious minds, one for each possibie
branch.
Is this then the answer to the problem of reality in the quantum
world? At first sight it appears more satisfactory than our previous
ideas where consciousness seemed to have to affect wavefunctions;
now this is not required. Nevertheless the general view of the
theoretical physics community has been to reject the many-worlds
interpretation. This of course is not in itself a strong argument
against it, particularly when we realise that many writers have
rejected it on grounds that suggest they have failed to understand
it. Here I should admit that the above discussion was an attempt
to describe what I think is the most plausible form of the Everett
interpretation. The original paper, and others mentioned in the
bibliography, contain mainly the formalism of orthodox quantum
theory with little comment on the interpretation.
It is probably fair to say that much of the ‘unease’ that most of
us feel with the Everett interpretation comes from our belief, which
we hold without any evidence, that our future will be unique. What
I will be like at a later time may not be predetermined or calculable
(even if all the initial information were available), but at least I will
still be one ‘1’. The many-worlds interpretation denies this. For an
example to illustrate this lack of uniqueness (some would say rather
to show how silly it is) we might return to the barrier experiment
and suppose that the right-hand detector is attached to a gun which
shoots, and kills, me if it records a particle. Then after one particle
has passed through the experiment, the wavefunction would contain
a piece with me alive and a piece with me dead. One ‘I’ would
certainiy be alive, so we appear to have a sort of Russian roulette,
in which we cannot really lose! Indeed, since all ‘aging’ or ‘decaying’
processes are presumably quantum mechanical in nature, there
is always a small part of the wavefunction in which they will not
have occurred. Thus, to be completely fanciful, immortality is guaranteed-Z will always be alive in the only part of the wavefunction
of which Z am aware!
It is important to realise that the fact that another observer does
not see two '1's is not an argument against this interpretation. As
soon as YOU, say, interact with me so that you can discover
whether I am alive or dead, you become two Yous, for one of
which I am dead and the other I am alive. In wavefunction
language, using the previous notation, we would have:
PR(WDD)*ME~~~YOU' + WDD)~ME"~~~YOU'.
Neither of the two YOUS is aware that there are two MEs.
Two final remarks in favour of the many-worlds interpretation
should be made here. It has long been known that, for many
reasons, the existence of 'life' in the universe seems to be an incredible
accident, i.e. if many of the parameters of physics had
been only a tiny bit different from their present values then life
would not have been possible. Even within the framework of
'design' it is hard to see how everything could have been correct.
However, it is possible that most of the parameters of physics were
fixed at some early stage of the universe by quantum processes, so
that in principle many values were possible. In a many-worlds
approach, anything that is possible happens, so we only need to be
sure that, for some part of the wavefunction, the parameters are
correct for life to form. It is irrelevant how improbable this is,
since, clearly, we live in the part of the wavefunction where life is
possible. We do not see the other parts. Thinking along these lines
is referred to as using the anthropicprinciple; for further discussion
we refer to articles listed in the bibliography.
The other remark concerns the origin of the observed difference
between past and future, i.e. the question of why the world exhibits
an asymmetry under a change in the direction of time when all the
known fundamental laws of physics are invariant under such a
change. One aspect of this asymmetry is psychological: we
remember the past but not the future. (Note that it is because of
this clear psychological distinction between past and future that we
sometimes find it hard to realise that there is a problem here, e.g.
it is possible to fool ourselves that we have derived asymmetric
laws, like that concerning the increase of entropy, from laws that
are symmetric.) The many-worlds interpretation gives an obvious
explanation of this psychological effect: my conscious mind has a unique past, but many different futures. Each time I make an
observation my consciousness will split into ‘as many branches as
there are possible results of the observation. Some readers may
wish to note that this might allow vague, shadowy, probabilistic,
‘glimpses’ into the future-thus, a prophecy is likely to be fulfilled,
but only for one of the future MES.

Does wavefunction reduction require conscious observers?

2007-07-02T01:29:00.000-07:00

Does the human body deviate from the laws of physics, as
gleaned from the study of inanimate nature? The traditional answer
to this question is ‘No’: the body influences the mind but the mind
does not influence the body. Yet at least two reasons can be given
to support the opposite thesis.?
Here we shall examine more closely the possibility that external
reality consists of a wavefunction and that this wavefunction only
reduces when an observation is made by a conscious observer. It is
the existence of consciousness that introduces the probabilistic
aspects into the quantum world.
One reason for at least considering such a suggestion is that
wavefunction reduction (which is an inexplicable phenomenon
associated with quantum theory) and decision making (which is an
inexplicable phenomenon associated with the conscious mind)
share the common feature of producing an increase of information;
in both something previously ‘unknown’ becomes ‘known’. More
specific motivation arise, as we have seen, from the following facts.
‘Simple’ systems apparently do not reduce wavefunctions; they are
also not conscious (we ignore here the possibility (c) of the previous
section). At some stage of ‘complexity’, and for reasons that may
be solely due to complexity or may involve something totally new,
there are systems that do reduce wavefunctions, and there are
systems that are conscious. In both cases our certainty here is due
to our own experience. We experience effects which appear to
correspond to reduced wavefunctions and we are certainly
conscious. It is therefore reasonable that we should try to relate the
two phenomena, an idea that has been argued most convincingly,
in recent times, by the eminent theoretical physicist, Eugene
Wigner. (It should be noted, however, that Wigner’s latest work on
this topic shows a move to the more conventional position that
other complex, yet not conscious, systems can also cause wavefunction
reduction. We refer, for example, to his article in
Quantum Optics, Experimental Gravity, and Measurement Theory
ed P Meystre and MO Scully [New York: Plenum 19831 .)
We must now explain carefully what is involved here. We
consider an isolated system, which may be as complicated as we
desire, but which must not contain any conscious mind. According
to our assumption, such a system is described by a wavefunction
which changes with time according to the rules of quantum theory.
A conscious observer now makes a measurement of some property
of this system, e.g. of the position of a particle. When the result
of this measurement enters the mind of the observer, then the
wavefunction reduces to the form corresponding to the particular
value of the measured quantity. Notice that it is not enough for the
conscious observer simply to be aware of only the part of the
wavefunction corresponding to the observed value. If this was all
that happened then we could not be sure that a different observer
would see the same value of the observed quantity. We require that
the act of conscious observation actually changes the wavefunction.
Thus, in our potential barrier experiment (see figures 15
and 16), if an observer sees the right-hand detector as being ON
then it must be ON, and not in the state of part ON and part OFF.
Then another observer will also see that it is ON, and both will
agree that the particle has been reflected. In figure 18 we illustrate
this difference between the observed and unobserved systems.
The last paragraph shows the first of the reasons noted by
Wigner, in the quotation at the start of this section, for believing
that mind affects physical things. The second reason he gives is that
in all other parts of physics, action and reaction occur together, i.e.
if A affects B then B affects A. Thus, since the physical world
clearly affects the conscious mind, we expect the converse to apply. To be quite fair, the assertion we have made here, that conscious
minds change the wavefunction, is not absolutely necessary. It
would, presumably, be possible that the act of observation does not
change the system, but that conscious observers somehow communicate
with each other so that they all ‘see’ the same thing. Such
an interpretation of quantum theory is possible and we shall discuss
it further in 54.5. We return to our assumption that only conscious observers can
reduce wavefunctions, and must now comment on how utterly
outrageous such a statement really is. To see this we might suppose
that the detectors in the potential barrier experiment are
photographic plates. What we are saying is that they are in a state
of ‘perhaps blackened but perhaps not’ until they have been
observed by a conscious mind, which may, of course, be years after
the event. Indeed, nothing ever really happens, e.g. no particle ever
decays, except through the intervention of a conscious mind. We
are not quite in the situation of denying external reality-which
possibility we considered and rejected in 0 1.2-but we are denying
that the external world possesses the properties we observe, until we
actually observe them. This is a picture of reality that we find hard
to accept.
The paradox of ‘Schrodinger’s cat’ is an example of the sort of
problem we can get into here. We suppose, for example, that the
right-hand detector in our potential barrier experiment is a trigger
that fires a gun and kills a cat as soon as a particle reaches it. After
one particle has passed through the apparatus the wavefunction
thus contains a piece in which the cat is dead and a piece in which
the cat is alive. Only if the cat is conscious can we say that one of
these represents the truth. What however could we say if the cat
were asleep? If, on the other hand, a cat is not conscious, or if we
used instead a being or a thing that is not conscious, then it remains
in a state of being part-dead/part-alive until some conscious
observer forces the wavefunction to go to one state or the other.
Like Schrodinger himself we probably consider this an unlikely
picture of reality.
The assumption we are considering appears even more weird
when we realise that throughout much of the universe, and indeed
throughout all of it at early times, there were presumably no conscious
observers. Thus the wavefunction did not reduce, and all the
possibilities inherent in the development of the wavefunction since
the beginning of time would have persisted until the first conscious
observers appeared. Even worse are the problems we meet if we
accept the modern ideas on the early universe in which quantum
decays (of the ‘vacuum’, but this need not trouble us here) were
necessary in order to obtain the conditions in which conscious
observers could exist. Who, or what, did the observations necessary
to create the observers? The only possibility here seems to be that observation, indeed
conscious observation, can be made by ‘minds’ outside the physical
universe. Such is one of the traditional roles of God and/or gods.
This is the realm of theology; a realm into which we shall, with
trepidation, enter briefly in the next section.
Before we close this section, however, there is one obvious
question we must ask. Since we have suggested that consciousness
might offer a possible, even if unlikely, solution to a problem of
physics, can physics help with the problem of the nature of consciousness?
Again the answer may well be that it cannot, but the
issue is certainly being discussed. The fact that quantum theory
frees physics from the rigid causality of classical mechanics is an
obviously immediately relevant fact. There just seems to be more
room for ideas like free will in a quantum world than in a classical
one. Already quantum tunnelling, as described in 81.3, has been
used to explain certain processes in the nervous system-see, for
example, Walker, International Journal of Quantum Chemistry 11
103 (1977). (We should, however, be cautious here. There is a big
difference between the idea of freedom to choose, where the choice
is presumably made by rational thought, and the apparent
randomness of quantum theory, so a relation between the two,
though possible, is not obvious,) It is also natural to try to associate
the very non-local nature of wavefunctions with the similar lack of
locality of ‘thoughts’, etc. For some discussion along these lines,
and for other references, we refer to the article by Stapp,
‘Consciousness and Values in the Quantum Universe’, Foundations
of Physics 15 35 (1985).

What is a conscious observer?

2007-07-02T01:28:00.001-07:00

Consciousness or ‘awareness’ is something we, as people, possess.
We talk about it; we have a vague understanding of what it is;
through it we experience many emotions, happiness, sorrow,
jealousy, love, etc; we develop concepts like free will and purpose
which really have no meaning without it; we can even refer to its
absence, e.g. to ‘unconscious’ decisions, etc; but we do not have any way of defining it. It cannot be expressed in terms of other
things or even be likened to other properties. It is something unique
and totally different from anything else.
To discuss it, we should therefore begin with what we know. Or,
rather, I should begin with what Z know.
I am conscious. This fact I can express in an alternative way by
saying that I have a conscious mind, or that my consciousness
exists, However expressed, and regardless of the fact that we do not
really know precisely what the statements mean, the truth they
convey cannot be denied, Even if I wish to deny reality to
everything else, which, as we saw in 0 1.2, is logically possible even
if rather pointless, I cannot deny the reality of my own thoughts.
As a natural extrapolation of my experience, it is reasonable that
I should assume that you, my readers, are conscious, and then to
extend this to all other people. Already, however, there are those
who would question this. A Princeton University psychologist,
Janes, has written a book in which he claims that consciousness is
a comparatively recent feature of the human race. (The book is
called The Origin of Consciousness in the Breakdown of the
Bicameral Mind [ Harmondsworth: Penguin 19801 . Though I am
fairly convinced that I do not believe the claim, it is expertly, and
interestingly, argued.)
Having agreed that we possess consciousness, do we know what
it is? It is a private ‘space’ in which each of us rules alone, and into
which we can introduce whatever we desire of real things, i.e. those
we believe exist elsewhere, or abstract things which are purely our
creation. But does such a vague description allow us to say where
else it might exist, who, or what, might possess it? People? Yes, by
extension of ourselves. But dogs? worms? amoebae?
I hope my readers will allow me a personal note here. I
remember, as a schoolboy, sitting by a riverside listening to a
skylark. I think I should have been revising for examinations but,
instead, I doodled some verses of poetry. Though I can just about
remember them, they are inferior to the precedent I was following,
so I will not expose them to public view. I mention them because
in the first verse I asked whether the bird was singing because it was
happy, in the second I wondered whether it was instead singing in
response to feelings of sadness, but then I wondered whether it was
neither, whether in fact the skylark was capable of feeling either
happy or sad or whether it possessed any awareness of anything:

‘Is’t only nature’s law that makes thee want to sing?’ I was asking
myself, perhaps for the first time, the question about consciousness
that I have asked many times since. I still do not know the answer,
and I have no idea how to go about finding it.
The nature of the problem here can be demonstrated by the
following thought experiment (which could, with a little expense,
even be a real experiment). Suppose we devised a series of tests for
consciousness. A conscious being would, for example, be expected
to show pleasure in some suitable way when it was praised, it would
back away from any object that hit it, or otherwise showed
threatening behaviour, it would seek ‘food’, i.e. whatever it
required to sustain its activity, when needed, and would express the
need urgently if the search proved unsuccessful. The list could easily
be extended. Whatever property of this type we included, however,
it is easy to see that we could design a computer-robot to make all
the correct responses. Such a machine would pass our tests for
consciousness. I believe, though I am not sure why, that it would
nevertheless not be conscious. Somehow ‘physical’ systems, even
when designed to have the attributes of consciousness, do not seem
to us to be conscious. Thus, although it is easy to simulate the
effects of consciousness, we should avoid making the mistake of
believing that in so doing we have created consciousness.
Conversely, it would be possible, by careful analysis of what
happens in the human brain, to correlate the various feelings like
joy, sadness, anger, etc, which we associate with consciousness,
with particular chemical or physical processes in the body, the
release of various hormones, and such like. But surely joy is not a
chemical compound, or a particular pattern of electrical currents,
Or is it? Or is it just caused by particular physical processes
occurring in the right place?Alternatively, is the truth nearer to the
statement that the thoughts of the conscious mind cause the
appropriate currents to flow? Are the emotions, or their material
effects, primary?
Certainly conscious thoughts appear to have physical effects. I
have just made a conscious decision to write these particular words
in my word processor. The fact that you are reading them is
evidence that my thoughts had real effect in the physical world. In
one sense, of course, this could be an illusion (whatever that might
mean in this context). The process of my writing these words could
be entirely a consequence of all the particles that make up my hand, brain, etc, moving inexorably according to the laws of motion.
Somewhere along the series of events in my body that leads to the
typing, particular things happen that make me think I am
‘deciding’ what to write. But what is cause, and what merely effect?
The problems we are discussing here are more than simply a
question of the language that we use to describe things. There are
of course ‘language’ issues. For example, we could describe a
pocket calculator as a machine that ‘allows particular currents to
flow’, or, alternatively, as a machine that ‘does arithmetic’. These
are different sets of words describing the same thing. Our concern
is more with the question of whether the calcualator knows that it
is doing anything at all. That real issues are involved can be seen
from the fact that our behaviour to some extent depends on how
we answer these questions. Part of the reason for the concern we
sometimes (too rarely) feel for people, animals, . . . is that we
believe these creatures are conscious.
It is outside the range of this particular book, and beyond the
ability of its author, to take this discussion any further. Much has
been written on the subject. In this sense it is rather like the interpretation
problem of quantum theory. I have the impression that
the two topics are similar in another sense-very little is understood
of either!
We close this section by offering three possible ‘answers’ to the
question of what makes an object conscious.

The relevance of conscious observers

2007-07-02T01:27:00.000-07:00

In $3.5 we proved that an instrument governed by the laws of
quantum theory is not capable of making a proper measurement,
that is, it cannot cause the wavefunction of a system to change to
a state corresponding to a particular value of the quantity to be
measured. As an example we saw that, in the potential barrier
experiment, even after the attempted measurement of transmission
or reflection, the wavefunction still contained pieces corresponding
to both possibilities.
We must not, of course, conclude from this that true
measurements are impossible. We know that they occur. We can
observe which of the two detectors flashes and hence deduce
whether or not a particle has passed through the barrier. Our brain
certainly does not permit both possibilities. Thus, although a
simple, microscopic, instrument, obeying the laws of quantum
theory, does not reduce wavefunctions, they are certainly reduced
by the time the information reaches our brain.
What, then, is responsible for the reduction and what are the
characteristics of ‘instruments’ that are able to cause it? We do not
know the answers to these questions. It could be that, with increasing
complexity and size, correction terms in the equations of quantum
mechanics become more significant, so that any macroscopic
apparatus can do the reduction. On the other hand, it could be that
something totally new is required and that some things possess it whereas others do not. In either case it is an obvious question to
ask whether there are other features of wavefunction-reducing
systems that distinguish them from simpler systems. One obvious
possibility that arises here is to go to the extreme end of the chain
of observation and consider the possibility that the reduction does
not occur until we know that it must, i.e. that it only occurs when
conscious observers are involved.
Such a wild suggestion tends to horrify the austere minds of
most physicists. We fear that it takes our subject, beloved for its
high standards of objectivity, rigour, precision and experimental
support, into a realm where nothing can be properly defined, where
feelings and personality replace detached measurement, even, perhaps,
to put it on a par with astrology and the reading of tea leaves!
From another point of view, however, it should perhaps be seen as
an exciting new development. Maybe it allows the possibility that
the enormously successful methodology of physics might enter a
totally new field of investigation. This would be a revolution that
would, in its significance, dwarf those to which we referred in 4 1.1.
Although it is probably fair to say that such a revolution is
unlikely, we should, before dismissing it entirely, remember that
J C Maxwell, the creator of the theory of electromagnetism and
undoubtedly one of the greatest physicists of all time, once
expressed the view that the study of atoms would be forever outside
the scope of physics! Such a precedent will guard us from making
similar rash statements about consciousness.
If we are to consider seriously the relevance of consciousness in
the collapse of wavefunctions we must ask, and at least try to
answer, the question of what it is. To this topic we turn in our next
section.

Can quantum mechanics be changed so that it will reduce wavefunctions?

2007-07-02T01:26:00.002-07:00

In the quantum theoretical decription of the potential barrier
experiment, which we discussed in @2.3 and 2.4, the wavefunction
split into two pieces, one travelling to the left and one to the right.
This behaviour was good because both pieces were needed to
explain the interference effects. However, it is possible to modify
the Schrodinger equation so that, after a certain time, the form of
the wavefunction changes: one of the peaks grows and the other
falls to zero. If, for example, it is the right-hand peak that remains,
then the equation will have predicted that the particle is reflected.
In this way, reduction of the wavefunction becomes a consequence
of the modified equation. To obtain the probabilistic element which
is vital for agreement with observation it is necessary that the
modification contains some randomly chosen contribution. Then it
is possible to arrange things so that either one of the peaks remains,
with a probability proportional to its original area. In this way we
obtain complete agreement with observation, and we have
automatic reduction of the wavefunction.
Actually, what we have described here for one particular
example can be done in general. Suitable additional terms can be
added to the Schrodinger equation, so that the wavefunction
automatically reduces to the form associated with a particular value
for some measured quantity, always with the correct probability
distribution. These extra terms must contain a random input. There
is also a free constant which can be used to fix the overall
magnitude of the new effects; this determines how long it takes for
the reduction to occur. Some further details of the very pretty
mathematics involved are given in Appendix 7.
At first sight all this appears to be just what we require for a
theory of wavefunction reduction. On closer examination, however,
it is clearly seen to be very unsatisfactory. The first reason for this concerns the time scale which is required for the reduction to
occur. As noted above, this can be adjusted to any desired value
by suitable choice of the magnitude of the extra terms in the
equation. However, no choice can satisfy the experimental constraints,
because these are mutually contradictory. On the one
hand, it is sometimes observed that reduction takes place very
rapidly, whereas, on the other hand, the observation of interference
effects from radio waves that have travelled distances of the
order of the size of the galaxy requires that the reduction time must
be very long. No time scale for automatic reduction of the
wavefunction is compatible with all observations.
The second reason why these ideas are unsatisfactory is that the
wavefunction has to reduce to a form appropriate for any type of
measurement. Hence the particular terms that have to be put into
the Schrodinger equation depend upon what is going to be
measured. In our example we have always thought in terms of
position measurements, but we could instead decide to measure
velocities. This would require a very different type of wavefunction
reduction.
It is worth introducing here another type of experiment, totally
different from anything we have met before, which illustrates this
last point very well and which will also be of use later. Many
particles have a ‘spin’, which always has a constant magnitude. For
example, we shall consider electrons, where the magnitude of the
spin, measured in suitable units, is always 1/2. (Appendix 8 gives
some further details.) The only variable associated with the spin is
its direction. (It is convenient to think of this as the direction of the
axis of a spinning top.) In order to ascertain this direction we
measure the spin along any line in space. It is a consequence of
quantum theory that, in such a measurement, we will always find
one of two values, + 1/2, corresponding to the spin being along the
chosen line, and - 1/2, corresponding to its being in the opposite
direction (see figure 17). Thus, when we make a measurement, the
wavefunction will reduce to the form corresponding to plus or
minus 112 along the line chosen. As we have stated above, it is
possible for this wavefunction reduction to happen automatically if
quantum theory is suitably modified. However, the final form of
the reduced wavefunction, and therefore the modification required,
will depend upon which particular line in space is chosen for the
measurement. There cannot be one equation which describes the future evolution of the electron wavefunction, regardless of what
we choose to measure,

It is clear that both these objections to the type of theory
involving automatic reduction of the wavefunction can be met if
the modifications to the standard quantum theory ‘know about’
what is to be measured and when. In other words, the new
Schrodinger equation must depend upon the form of all the
apparatus involved, including the measuring instruments and, for
example, whatever (or whoever) decides on the direction for a spin
determination. The work that we have outlined above suggests that
theories of this type might be possible, but much work remains to
be done and there is a danger that what emerges will look more like
an arbitrary prescription to obtain the results than like a proper
theory. Certainly it is hard to see how it can look at all natural.
There are three other points which might be relevant to this
section and which certainly should be mentioned. First, all real
measuring instruments are macroscopic. To appreciate how different
such an object is from a single electron, say, we should
realise that an object with a mass of one kilogram contains about
lo2’ particles. It is, therefore, not hard to imagine that effects which are utterly negligible for single particles might build up to
something important for macroscopic objects. Two particular ways
in which the mass of an object might appear in the formulae for
reduction are suggested at the end of Appendix 7.
Secondly, as we have seen in the previous section, all tests of
interference effects refer to particles. It is just not possible to test
whether they would also occur for macroscopic objects where a
very large number of degrees of freedom are involved. The
difference between whether they really do occur, as predicted by
quantum theory, or whether they do not, has no obvious
measurable consequences. This is unfortunate, because the question
has enormous relevance to the issues we are discussing.
Finally, if it is true that really new effects arise for large,
complex, systems, then we should ask whether there are other
manifestations of these. Is it even possible that one such effect
could be consciousness, which might also be expected to occur only
for large systems? Maybe, somewhere here, there is a link between
this section and the subject of our next chapter.

Interference and macroscopic objects

2007-07-02T01:26:00.001-07:00

We have stressed that a wavefunction which contains a sum of
several terms (a pure state) is genuinely different from a wavefunction
which is either one term or another (a mixed state).
The reason for the difference is that, in principle, it is possible
to arrange that two terms in a sum interfere when a particular probability is calculated, and this interference can be observed.
Indeed, as we have seen, e.g. in $2.5, there are many experiments
where this interference has been measured and found to agree
perfectly with the predictions of quantum theory. However, in
practice, in many situations, and in all cases where macroscopic
apparatus is involved, it is not possible to design a suitable experiment
to observe the interference, so the two wavefunctions are
effectively indistinguishable.
To understand why this is so, let us suppose we want to check
that the wavefunction for the barrier-plus-two-detectors experiment
really does contain the sum of two pieces, e.g. as in figure
15(b). To this end we would like to arrange that the two pieces are
allowed to interfere. Thus, in effect, we need to do both the potential
barrier experiments of Chapter One in the same experiment.
However, even when the waves corresponding to the reflected and
transmitted particles are brought together by mirrors they will not
interfere because, unlike the situation in 01.4, they now contain
different states of the detectors (ON/OFF or OFF/ON respectively). In
order to have interference it is necessary that the detectors be
brought to the same state. At first sight this might appear to be
easy; they can be switched to the OFF position, say. However, in
order to have interference the states must be identical, and for
macroscopic objects that is not possible. To reverse exactly the
process whereby one of the detectors was switched to ON is, by
many orders of magnitude, outside the range of any conceivable
experimental technique; there is, for example, no conceivable
mechanical interaction between macroscopic objects that does not
remove a few atoms, slightly change the temperature of the object,
alter its shape, etc. This is the reason why interference between
macroscopic objects cannot be experimentally verified.
For a proper treatment of this topic we would need to use the
mathematical formalism of quantum mechanics. Some of the ideas
are discussed further in Appendix 6. Here we shall be content with
the above rather sketchy outline of the argument. The key to it, to
which we shall return, is the inherently irreversible nature of
macroscopic changes.
It is clear that there is a continuum of scales ranging from the
micro- to the macroscopic, so we naturally ask how far towards the
latter we can go with interference experiments. At present, it seems
as though the answer is not very far: all experiments so far performed deal with ‘elementary’ particles, or, more precisely, with
systems that, for the purpose of the experiment considered, can be
regarded as having very few degrees of freedom. Some ideas for
doing interference experiments with larger systems are being
explored at the present time

Measurement in quantum theory

2007-07-02T01:25:00.000-07:00

As we have seen, it is not normally correct to say that a particle,
described by quantum theory, is at a particular position. Rather,
the particle has a wavefunction which tells us the probability of
finding it at any given position when a measurement of position is
made. Similarly, the wavefunction tells us the probability of
obtaining a given value for the velocity if we make a velocity
measurement. Thus measurements play a more positive role in
quantum theory than in classical physics because they are not
merely observations of something already present, they actually
help to produce it.
A measuring instrument can be defined as something that
enables us to make a measurement of the above type. That such
instruments exist follows from the fact that we do actually make
such measurements. We would, of course, like to believe that the
apparatuses can be described by physics, i.e. that they too satisfy
the rules of quantum theory. It is, however, very easy to show that
this is impossible. An instrument that is able to make a measurement,
in the above sense, cannot be completely described by
quantum theory.
To illustrate this fact we shall consider again the potential barrier
experiment with the two detectors in position. Recall that the lefthand
detector records the passage of a transmitted particle and the
right-hand detector the passage of a reflected particle. We suppose
that each detector is a simple quantum mechanical system that can exist in one of two states OFF and ON, and that the transition
between these is caused by the passage of a particle through the
detector.
The complete experiment is now described by a wavefunction
which contains information about both detectors as well as about
the particle. Thus, for example, it would tell us the probability of finding the particle at a given position, with one detector in the OFF
position and the other in the ON position, etc. We know the initial
form of this wavefunction; it describes the particle as being incident
from the right and both detectors being in the OFF position. A
pictorial representation of this is given in figure 15(a).
The system now evolves with time according to the Schrodinger
equation. This equation is more complicated than before because
it must include the interaction between the detectors and the
particle. We are assuming that this interaction only occurs when the
particle is in the neighbourhood of a detector, and that its effect is
to change the detector from OFF to ON as the particle passes
through. The precise details here are not important. We can then
go to a later time when the particle will certainly have passed
through one detector, i.e. the two parts of the wavefunction shown
in figure 11 have passed beyond the positions of the detectors. The
wavefunction will now be the sum of two pieces (compare the
discussion given earlier). The first piece describes a peak travelling
to the right, with the right-hand detector ON and the left-hand
detector OFF. The second describes a peak travelling to the left,
with the right-hand detector OFF and the left-hand detector ON.
Figure 15(b) gives a picture of this wavefunction.
We notice, first, that our measuring instruments are doing their
job properly in the classical sense, that is they correctly correlate
the ON/OFF positions of the detectors with the reflection/transmission
of the particle. However they have not selected one or the
other; the wavefunction still contains both possibilities and has not
been reduced. Thus we have not succeeded in making a proper
measurement in the quantum theoretical sense as we described it at
the beginning of this section. Such a measurement would have left
us with a final state expressible as either the left-hand detector ON
and the right-hand detector OFF, or the other way round (with a
certain probability) and not as the sum of both. Pictorially, the
wavefunction would have had the form of figure 16 rather than
figure 15(b). Readers who wish to see the difference expressed in
terms of mathematical symbols should consult §4.5.
It is important now to realise that the difference between these
two forms of wavefunction is not just ‘words’ (or even, in 84.5,
‘symbols’). They are different. The difference can be seen from the
fact that, at least in principle (see next section), the two parts of the
sum can be brought together and made to interfere. Such interference is not possible if the wavefunction has become just one of
the two pieces.

The result we have obtained, that quantum theory does not allow
the reduction of the wavefunction, is extremely important. We
have obtained it in a very specialised and idealised situation, but in
fact it is a completely general result. A wavefunction that can be
expressed as a sum of several terms, like that of figure 15(b), is called
a pure state. One that is expressed as a selection of alternative
possibilities, like that of figure 16, is called a mixed state. From the
laws of quantum theory it is possible to prove that a pure state cannot
change into a mixed state. Thus the wavefunction can never be
reduced. An easy way to understand why this is so is to recall that wavefunctions change with time in a deterministic way, as long as
they are described by quantum theory, hence they can never give
the probabilistic aspects associated with measurements.
Note that we cannot solve our problem by saying, in the potential
barrier example, that all we need to do is look at the detectors to
see whether they are ON or OFF. This is equivalent to saying that we
measure the state of the detectors. We then have to repeat the
process and describe the new measuring apparatus, e.g. our eyes,
by quantum theory. The resulting wavefunction now contains
information describing this additional apparatus. It will remain a
pure state.
Quantum theory, therefore, when applied to individual systems,
contains an internal contradiction. It cannot describe instruments
suitable for making measurements.
Faced with this situation, and bearing in mind the enormous success
of quantum theory, it is natural that we should seek to modify
it in such a way as to leave its successful predictions unchanged and
yet to allow wavefunction reduction in appropriate circumstances.
Attempts along these lines will be described in 43.7, and we shall
see that there are formidable problems.
Are there any alternatives? Well, if quantum theory says that
wavefunctions do not reduce we should look again at why we need
them to reduce in the first place. Why must measurements choose?
How do we know that a detector will tell us that a particle either
passed through or not? The obvious answer is that we are conscious
of seeing only one result. Our conscious minds do not contain both
parts of the wavefunction. Maybe, then, in order to understand
what is happening, we need to examine this answer more closely
and to consider the concept of consciousness.

The wavefunction as part of external reality

2007-07-02T01:24:00.002-07:00

We now want to consider the possibility that the wavefunction
should be treated rather more seriously than in the preceding two
sections, so that we can use it to tell us something about the
external reality. We shall try to regard the wavefunction not as just
a description of a statistical ensemble, as in 53.1, or as a catalogue
of our information about a system, as in 53.2, but as something
that really exists, something that is, indeed, part of the external
reality which we observe.
There are at least three good reasons why we should want to consider
this assumption. First, since the classical picture of a single
particle, always having a precise position and following a specific
path, is not compatible with the observations described in 01.4, we
do not have any other object available for our representation of
reality. Secondly, the evidence that wavefunctions can interfere
strongly suggests that they are real, e.g. just like ripples on the surface
of a pond. In order to understand the third reason we need to
know about certain symmetry properties that have to be imposed
on wavefunctions describing more than one particle. If we have two
identical particles, e.g. two electrons, then in classical mechanics
we could distinguish them, for example, by their positions. In
quantum theory, on the other hand, they are described by a
wavefunction which tells us the probability of finding an electron
at one place and an electron at another place; in no way are the two
electrons distinguished. This means that the wavefunction must be
symmetrical in the two electrons, i.e. it must not change if we interchange
them. Actually, the truth is a little different from this
because in some particular cases the wavefunction has to change its
sign. Such a change, however, does not alter any of the physics,
which is determined by the square of the magnitude of the
wavefunction. A more detailed discussion of this is given in
Appendix 4. Here we merely note that the symmetry properties give rise to important, testable, predictions, which have been verified
and which would be very hard to understand without the assumption
that wavefunctions have a real existence.
Our tentative picture of the potential barrier experiment is
therefore that of a wavefunction which has a value that varies with
the point of space being considered. We are familiar with quantities
of this type, e.g. the temperature of the air at different points of
a room, or the number of flies per unit volume in a field of cattle.
Actually the wavefunction is a little different since, as we recall, it
is a line or, alternatively, two numbers at each point of space. This
fact, however, does not affect the present discussion, so we shall
continue to refer simply to the value of the wavefunction.
As is illustrated for example in figure 11, the wavefunction'is in
general not constant but changes with time. Again this is a concept
with which we are familiar; the temperatures at various points in
a room, for example, will similarly change with time, e.g. when the
heating has been switched off. We therefore have a simple picture
of reality, with the wavefunction describing something that actually
happens.
There are, however, two difficulties associated with this picture.
The first of these is due to the fact that the world does not consist
of just one particle. We remember that the wavefunction we have
used so far was specifically designed to treat only one particle. How
do we generalise this to accommodate additional particles?
Consider a world of two particles, which we shall call A and B.
As a first guess we might try having a wavefunction for particle A
and a separate and independent one for particle B. Then the
probability of finding A at some point would not depend on the
position of B. This is reasonable for particles that are genuinely
independent, i.e. not interacting. It is, however, quite unreasonable,
and is indeed false, for particles that are interacting. In this
case the wavefunction must depend on rwo positions. It will then
tell us the probability for finding particle A at one position and partide
B at the other. (Some further details are given in Appendix 4.)
One can express this by saying that the wavefunction does not exist
in the usual space of three dimensions but in a space of two-timesthree
dimensions. It is no longer true to say that at a particular
point of space the wavefunction has a particular value. Rather we
have to say that, associated with every two points of space (or, if we prefer to express it this way, with every point of a sixdimensional
space) there is a particular value for the wavefunction.
Of course, we cannot stop at two particles and must go on to
include 3,4, etc, with the wavefunction depending on the corresponding
number of points, 9, 12, etc, in space. At this stage the
wavefunction starts to look more like a mathematical device than
something that is part of the real world. Certainly it is not now of
the form of the familiar quantities mentioned earlier. These are
local, i.e. at a single point of space there is a number which is the
temperature. The wavefunction, on the contrary, is non-local; in
order to establish its value we need to give many positions in space.
We shall find this non-locality occurring again in our discussion.
It should be noted here that the two-particle wavefunction is
not, in general, simply a product of two one-particle wavefunctions.
To understand this distinction we recall that the square of the
magnitude of the wavefunction gives the probability of finding
a particle at each of the two points. If the particles are quite
independent, and not in any way correlated in position, then the
probability of finding a particle at a point P will not depend on the
position of the other. In such a case the wavefunction will be a
simple product of two wavefunctions, each depending upon one
position. In most real situations, however, particles interact and
therefore their positions are correlated. The wavefunction is then
not of the product type but is, rather, one function with an explicit
dependence upon two positions. Again we refer to Appendix 4 for
further details.
The second difficulty that arises when we regard the wavefunction
as part of reality is one to which we have already referred,
the process of reduction of the wavefunction. As we saw in $2.3,
the wavefunction changes when a measurement is made. This
change appears to be sudden and discontinuous. It is also very nonlocal
in the sense that measurements at one point of space can
change the wavefunction at other points, regardless of how far
away these might be. The measurement by means of a detector on
the right-hand side of the potential barrier provides a good example
of this. If this flashes it means that the particle has been reflected,
so the piece of the wavefunction on the left (e.g. in figure 11) immediately
becomes zero. This, at least, appears to be what is happening.
Whenever a measurement is made on a system described by a wavefunction, then one of the possible values consistent with the
probability distribution is obtained. The measurement somehow
selects part of the wavefunction. We cannot be content, however,
with merely postulating that this happens. We must ask how it
happens. In particular, we have claimed that quantum mechanics
is a universal theory and applies to everything. It should therefore
apply to the apparatus which we use to make a ‘measurement’, and
should, therefore, contain the answer to our question-that is,
quantum mechanics should be able to explain how the wavefunction
reduces. In fact, however, it says very clearly that the
wavefunction cannot reduce! Such a startling fact deserves another
section.

The wavefunction as a measure of our knowledge

2007-07-02T01:24:00.001-07:00

In 52.2 we tended to regard the wavefunction as describing a
particular system (in fact, just a single particle). Suppose, however,
we take the view that the wavefunction instead describes our
knowledge of the system. By implication the system might then be
thought of to have other properties of which, at the time, we are
ignorant. For example, the particle may actually be at a specific
position, whereas we know only that it has a certain probability of
being in some region. This, at first sight, appears to be a very
reasonable view. It is indeed the situation that occurs whenever
probability aspects arise in non-quantum situations.
To have a trivial example of this, let us suppose that I am in a
room with 10oO other people. Assume also that I know the lo00 is
made up of 498 French men, 2 French girls, 200 Norwegian men
and 300 Norwegian girls. With this information I would know that
the probability of the person immediately behind me being French
was one in two. Now suppose that I looked at the person behind
me and saw that she was female. The probability of the person
being French would immediately change to one in fifty.
If the situation in quantum theory is of a similar nature then the
issue of the reduction of the wavefunction, raised in 52.3, immediately
goes away. When the wavefunction is just an expression
of our knowledge of the truth, then it is not surprising, and is even
expected, that is should suddenly change to something else when a
measurement is made. A measurement has simply changed our
knowledge (this of course is normally the purpose of making
measurements).
Superficially attractive though this view of the wavefunction may
be, it is in one very important respect inadequate. It cannot explain
the phenomenon of interference. We remind ourselves here that
there is abundant experimental evidence for interference effects and, contrary to what appears to happen in some discussions.of the
interpretation problems of quantum theory, they cannot be ignored.
Wavefunctions which merely represent our knowledge of a system
cannot interfere. We can see this immediately in the case of the
potential barrier experiment. There we require that ‘something’
follows both routes to the detector. That ‘something’ cannot be our
knowledge, which, if it is anywhere, is in our brain. If the particle
really has followed one route then we are back with the problem
as to how its motion can be influenced by the presence of the other
mirror. It is not an answer to this to say that we know about the
other mirror; the behaviour of the particles surely cannot depend
upon the information contained in the brains of particular
individuals.
We can therefore be sure that, if interference actually occurs, this
interpretation of the wavefunction must be wrong. However, the
form of the qualification used here is important. What we know is
that the results of our observation can be predicted from the
calculation of the interference effect. It looks as though interference
is actually happening but it is possible that this is not so,
but that, instead, the calculation just ‘happens’ to give the right
answer. A simple analogy might help here. An umpire at a cricket
match counts the number of balls that have been bowled by placing
pebbles in his pocket, one for each ball. When six pebbles are in
the pocket he calls ‘over’ and play changes ends. Now the reason
for this change is not directly anything to do with pebbles in the
umpire’s pocket, it is because six balls have been bowled and the
rules say that play changes ends every six balls. The pebbles can be
used by the umpire to make the calculation because of the rules of
arithmetic which ensure that the right answer will be obtained. It
could be that a similar thing is happening with the interference
calculation; it gives the right answer but the real reason for the
experimental facts lies elsewhere.
Where? Clearly we must look at the hidden information-at the
properties not contained in our knowledge of the system, and
therefore not in the wavefunction. We are then in the domain of
hidden variable theories which we discuss in detail in Chapter Five.
However, to complete this section we should look ahead and note
that such theories do not in fact eliminate the need for an interfering
wavefunction. Indeed, it is inconceivable that any theory could
successfully reproduce all the correct effects of interference unless the interference actually happens. Thus, although it was important
to mention the reservations of the previous paragraph, I believe
they can now be forgotten.

Quantum Theory and External Reality

2007-07-02T01:23:00.000-07:00

As we have seen, quantum theory deals with a wavefunction, which
it states is causally determined from some initial conditions. The
passage from this wavefunction to experimental observation uses
the assumption that the wavefunction gives probabilities for
measurements to yield particular values. In order to test the predictions
of the theory it is necessary to prepare a large number of
identical systems and perform the same measurement on each. We
recall that we used this procedure to define the probabilities of
transmission and reflection in $1.3. Of course, the word identical
now must refer to the wavefunction, i.e. ‘identical systems’ are
defined to be systems with the same initial wavefunction (and
therefore the same wavefunction for all future times).
The large number of identical systems is referred to as an
ensemble. For any such ensemble the predictions of quantum
theory are precise and deterministic. For example, quantum theory
tells us what percentage of a given (large) number of particles will
pass through a potential barrier. What it cannot tell us, of course,
is whether any particular one of the particles will pass through.
Some writers on this topic have therefore adopted the view that
quantum theory is a theory of ensembles and as such tells us
anything about individual systems. This is a perfectly reasonable
view and it may be the correct one to take. There are then no
further difficulties in the ‘interpretation of quantum theory’, and
the subject does not cause any philosophical problems. We must
not, however, go on from this to claim that we have solved the
problems met in the first chapter. We have merely ignored them.
We do not only have experimental results for ensembles. Individual
systems exist and the problems arise when we observe them. It is
possible to argue that quantum theory says nothing about such individual systems but, even if this is true, the problems do not go
away.
We shall, in this chapter, adopt a more positive view and
continue to hope that the theory which predicts our results might
also help to explain them.

Other applications of quantum theory

2007-07-02T01:22:00.000-07:00

In this section we shall outline some of the most important applications
of quantum theory to various areas of physics, applications which ensured that, in spite of its problems, it rapidly gained acceptance.
Nothing in the remainder of our discussion will depend on
this section, so it may be omitted by readers who are in a hurry.
The section is also somewhat more demanding with regard to
background knowledge of physics than most.
The understanding of electricity and magnetism, besides being
the prerequisite for the scientific and technological revolutions of
this century, was the great culminating triumph of nineteenth century,
classical, physics. By combining simple experimental laws,
deduced from laboratory experiments, into a mathematically consistent
scheme, Maxwell unified electric and magnetic phenomena
in his equations of electromagnetism. These equations predicted
the existence of electromagnetic waves capable of travelling
through space with a calculable velocity. Visible light, radio waves,
ultraviolet light, heat radiation, x-rays, etc, are all examples, differing
only in frequency and wavelength, of such waves.
The first hint of any inadequacy within this scheme of classical
physics came with the calculation of the way in which the intensity
of electromagnetic radiation emitted by a ‘black body’ (i.e. a body
that absorbs all the radiation falling upon it at a particular
temperature) varies with the frequency of the radiation. The
assumptions which went into the calculation were of a very general
nature and were part of the accepted wisdom of classical physics;
the results, however, were clearly incompatible with experiment. In
particular, although there was agreement at low frequency, the
calculated distribution increased continuously at high frequency
rather than decreasing to zero as required.
Max Planck, in 1900, realised that one simple modification to the
assumptions would put everything right, namely, that emission and
absorption of radiation by a body can only occur in finite sized
‘packets’ of energy equal to h times the frequency. The constant of
proportionality introduced here, and denoted by h, is the original
Planck’s constant. For various reasons it is usual now to work
instead with the quantity h, which we quoted in equation (2.2), and
which is equal to h divided by 27r.
The packets of energy, introduced by Planck, are the ‘quanta’
which gave rise to the name quantum theory. Each such quantum
is now known to be a photon, i.e. a particle of electromagnetic
radiation, but such a concept was a heresy at the time of Planck’s
original suggestion; electromagnetic radiation (e.g. light, radio waves, etc) was known to be waves! The quantisation was therefore
assumed to be simply something to do with the processes of
emission and absorption.
Such a view was shown to be untenable by the observation of
the photoelectric effect, in which electrons are knocked out of
atoms by electromagnetic radiation. If we assume that the energy
in a uniform beam of light, incident upon a plate, is distributed
uniformly across the plate, then it is possible to calculate the time
required for sufficient energy to fall on one atom to knock out an
electron. This is normally of the order of several seconds, in
contrast to the observation that the effect starts immediately.
Further, the energy of the emitted electrons is, apart from a
constant, proportional to the frequency of the radiation. Einstein,
in 1905, showed that all the observations were in perfect agreement
with the assumption that the radiation travelled as photons, each
carrying the energy E appropriate to its frequency according to the
relation previously used by Planck:
E = hf (2.3)
where f is the frequency.
The final confirmation of the idea of photons came from the
observation, in 1922, of the Compton effect, in which radiation was
seen to decrease in frequency when it was scattered by electrons.
This can be explained very simply as being due to the loss of energy
in the photon-electron collision, a loss that can be exactly calculated
from the laws of conservation of energy and momentum.
Although quantum theory began with its application to radiation,
the ideas were soon applied to particles. In 191 l , de Broglie
suggested that, if waves can have particle properties, then it is
reasonable to expect particles to have wave properties. He
introduced the relation:
I = h/mv (2.4)
between the wavelength I , the velocity v, and the mass m of a
particle. The major achievements of quantum mechanics have
been, following this relation, in its application to matter, in
particular to the structure of atoms.
The experimental work of Rutherford, early this century,
showed that an atom consists of a small, positively charged,
nucleus, which contains most of the mass of the atom, surrounded by a number of negatively charged electrons which are bound to the
nucleus by the attractive electric force. Each atom was therefore
like a miniature solar system, with the electrons playing the role of
planets, orbiting the nuclear ‘sun’. Prior to the advent of quantum
theory there were, however, serious problems with this picture: why
did the orbiting electrons not radiate electromagnetic waves,
thereby losing energy so that they would fall into the nucleus? Why
were the energies available to a given atom only a set of discrete
numbers, rather than a continuum as would be expected from
classical mechanics?
Quantum theory provides a complete answer to these questions.
All the energy levels of atoms can be calculated from. the
Schrodinger equation, in perfect agreement with experiment. The
interactions between atoms, as observed in molecules, chemical
processes and atomic scattering experiments can also be understood
from this equation. As we mentioned in 81.1, quantum theory
successfully brought a whole new range of phenomena into the
domain of calculable physics.
The details of all this are outside the scope of this particular
book, but it is worthwhile to give a simple picture of why the wave
nature of the electron helps us to understand the quantum answers
to the problems mentioned above with the classical picture of the
atom. If we consider a wave on a string with fixed end points,
then only certain wavelengths are allowed, because an integral
multiple of the wavelength must fit exactly into the string. A consequence
is that the string can only vibrate with a particular set of
frequencies; a fact which is crucial to many musical instruments.
The frequencies which occur can be altered by changing either the
length or the tension of the string. In an atom the situation is
similar, except that, instead of having a wave on a string with fixed
end points, we have a wave on a circle (the orbit), which must join
smoothly on to itself. Thus the circumference of the circle has to be
an exact integral multiple of the wavelength. As we show in
Appendix 5 , this condition yields the energy levels of the simplest
atom.
The transition from one energy level in an atom to another, by
the emission of a photon, i.e. by electromagnetic radiation, is an
example of an important class of very typically quantum
phenomena, in which one particle spontaneously ‘decays’ into (say)
two others. Calling the first particle A and the others B and C, we can write this as
A+B+C.
If we start with a large number of A particles then, after a given
time, some of them will have decayed. It is usual to define a
‘half-life’ as the time taken for half of the particles in a large initial
sample to have decayed. The half-life depends on the process considered
and values ranging from tiny fractions of a second to times
beyond the age of the universe are known.
Even though the half-life for the decay of a certain type of
particle, e.g. the A particle above, might be known, it will not be
possible to say when a particular A particle will decay. This is
random; like, for example, the choice of transmission or reflection
in the potential barrier experiment. Indeed, one can think of some
types of decays as being rather like a particle bouncing backwards
and forwards between high potential barriers; eventually the
particle passes through a barrier and decay occurs. In general, if we
start with a wavefunction describing only identical A particles, then
it will change into a sum of a wavefunction describing A particles, which will have a magnitude decreasing with time, and one describing
(B + C), which will have an increasing magnitude.
Finally, we mention the recent, very accurate, experiments which
show that neutrons passing through a double slit, as in figure 13,
interfere exactly as predicted by quantum theory. An example is
shown in figure 14. These experiments were carried out in response
to the recent upsurge of interest in checking carefully the validity
of quantum theoretical predictions in as many circumstances as
possible. We shall later mention other such tests. In all cases so far
the theory is completely satisfactory.

Interference

2007-07-02T01:21:00.000-07:00

We shall next consider the quantum theoretical description of the second type of barrier experiment discussed in Chapter One. In
this, we recall, there were mirrors which could bring both the
reflected and the transmitted particles to the same set of detectors.
We begin then with the same initial state as before (figure ll(a))
and follow the wavefunction to the situation shown in figure 1 l ( d ) .
Here, to a good approximation, the wavefunction can be regarded
as a sum of two wavefunctions, one giving the left-hand peak and
the other the right-hand peak. Note that the operation of adding
the two wavefunctions is rather trivial at this stage since, at any
given point of space, at most one of the two wavefunctions which
are added is different from zero. In the subsequent motion each of
the two peaks will change independently; in fact they will move in
a manner closely resembling the classical motion of a free particle.
(It is irrelevant here that the area under each peak is not actually
equal to one.)
Eventually, if the mirrors are present, the peaks will come
together in the neighbourhood of the detectors. At this stage the
addition is no longer trivial since both wavefunctions are different
from zero at the same place. This means that the feature mentioned
at the end of 52.2 becomes relevant, and the probability resulting
from the two wavefunctions is not equal to the sum of the probabilities
associated with the separate wavefunctions.
We have here an example of an extremely important
phenomenon known as ‘interference’. It occurs in a wide range of
physical situations even where quantum effects are not relevant. As
an example, we can think of two pebbles being dropped onto the
surface of a still pond. Ripples will spread out from the points of
impact. At some positions on the pond the ‘ups’ and the ‘downs’
from the two circular wave patterns will always come at the same
time and the wave will therefore be enhanced. At others they will
be ‘out of phase’, i.e. an ‘up’ from one will arrive at the same time
as a ‘down’ from the other, in which case they will cancel each
other and the water will remain still. Figure 12 illustrates this
situation.
In our quantum mechanics problem the situation is rather more
complicated since we are not just adding numbers, which can be
positive or negative, but adding ‘lines’, and we recall that the result
depends on the angle between the lines. On the other hand, if we
think just of the real parts of the wavefunctions, then what happens
is very similar to the case of water waves, The precise forms of the two wavefunctions to be added will depend on the length of the
path to any particular detector (see figure 4, for example). It
follows that the nature of the interference observed will depend on
which detector is considered. Certainly, in general, the probability
resulting from the sum of the two wavefunctions will be different
from the sum of the probabilities coming from each separately. This is in accordance with the observations which we found so
surprising in 41.4.
Detailed calculations yielding precise results are, of course,
possible. Similar calculations can be done for other situations in
which quantum mechanical interference occurs, and where the
results can be verified by experiments. Of particular importance are
experiments where electrons are scattered off crystals. Here the
interference is between parts of the wavefunction scattered off
different sites in the crystal. Comparison of the results with
calculated predictions reveals information on the structure of the
crystal.
A brief historical note is of interest here. The long-standing
conflict between a corpuscular theory of light (favoured by Isaac
Newton) and a wave theory was generally believed to have been
settled in favour of the latter by observation of interference effects
when light was passed through two slits (see figure 13). Interference
implied waves. It was therefore a shock when electrons, long
established as particles, were also found to show interference
effects. This schizophrenic behaviour became known as ‘particlewave
duality’. The same duality applies to electromagnetic
radiation, of which light is an example. The ‘particles’ of light are
called photons. In our potential barrier example, the particle nature
is seen most naturally in the first set of experiments where the
particle is observed either to be transmitted or reflected. The wave
nature is seen in the second set, where there is evidence for
interference effects.
Quantum theory successfully incorporates both features and
enables us to calculate correctly all microscopic phenomena that do
not involve ‘relativistic’ effects. A brief review of some of the
successes of the theory is given in the next section, with which we
conclude this chapter.

The potential barrier according to quantum mechanics

2007-07-02T01:20:00.000-07:00

We require for this problem an initial state which corresponds as
closely as possible to the classical situation, i.e. a particle on the
right of the barrier and moving towards it with a velocity v. To this
end we take a wavefunction with a magnitude that is peaked in the
neighbourhood of the initial position and with an angular variation
such that the average velocity is equal to v. There will of course
be an uncertainty in both the position and the velocity, according
to equation (2.1). A possible form for the square of the magnitude,
which we recall is proportional to the probability, is shown in figure
ll(a). Since we are dealing with one particle the area under this
peak will be equal to one.
The Schrodinger equation now determines the subsequent
behaviour of this wavefunction. We shall not discuss the method
of solving the equation but merely state the results. The peak in the
wavefunction moves towards the barrier with a velocity approximately
v-this is very similar to the classical motion of a particle
where there are no forces. There is, in addition, a small increase in
the width of the peak, so the situation at a later time is shown in
figure 1 l(b). When the peak reaches the barrier, where the effect of
the force begins to be felt, it spreads out more rapidly and then
splits into two peaks, as seen in figure 1 l(c). These two peaks then
move away from the barrier in opposite directions, so a little later we have the situation shown in figure 1 l(d). Our wavefunction has
separated into two peaks, one reflected and one transmitted by the
barrier.
It is a consequence of the Schrodinger equation that, throughout
the motion, the total area under the graph of the square of the length of the wavefunction remains equal to one. In fact we know
that this has to be true for consistency with the probability
interpretation-the particle always has to be somewhere. The probability
that it is on the right of the barrier, i.e. that it has been
reflected, is given by the area under the right-hand peak, whereas
the probability for transmission is given by the area under the peak
on the left. Thus the calculation allows us to predict these
probabilities and to compare with the results of experiments as
discussed in $1.3. In all cases where calculations using the
Schrodinger equation have been compared with experiment the
agreement is perfect. In particular, it is worth mentioning that we
obtain agreement with the classical result for a very high or very
low potential barrier, namely almost 100% reflection or transmission
respectively.
We must now look more closely at what our calculation for the
potential barrier experiment really tells us. After collision with the
barrier the wavefunction, and hence the probability, is the sum of
two pieces. Here we are ignoring the fact that the two parts are in
practice joined because the wavefunction is never quite zero, just
very small, between them. What, then, happens when we make an
observation which tells us whether the particle has been reflected?
Clearly, in some sense, we ‘select’ one of the two peaks in the wavefunction.
In other words, we might say that the wavefunction has
jumped from having two peaks to having only one. This process is
referred to as ‘reduction of the wave packet’. What it means, whether
it happens and, if so, how, are topics to which we shall return.
To close this section we emphasise that the wavefunction is
determined from the initial conditions in a completely deterministic
way. Knowing the initial wavefunction exactly (e.g. figure 1 l ( ~ ) ) ,
we can calculate, without any uncertainty, the wavefunction at all
later times and hence the probability of transmission or reflection.
The non-deterministic, probabilistic, aspects of the potential
barrier experiment arise because we do not observe wavefunctions
but rather particles; in particular, we can observe the position of
an individual particle after it has interacted with the barrier.

The wavefunction

2007-07-02T01:19:00.001-07:00

We consider a system of a single particle acted upon by some
forces. In classical mechanics the state of the system at any time is
specified by the position and velocity of the particle at that time.
The subsequent motion is then uniquely determined for all future
times by solution of Newton’s second law of motion, which tells us
that the acceleration is the force divided by the mass.
In quantum theory the state of the system is specified by a
wavefunction. Instead of Newton’s law we have Schrodinger’s
equation. This plays an analogous role because it allows the
wavefunction to be uniquely determined at all times if it is known at some initial time. Thus quantum mechanics is a deterministic
theory of wavefunctions, just as classical mechanics is of positions.
The wavefunction of a particle exists at all points of space. It
consists of two numbers, whose values, in general, vary with the
point considered. We shall find it convenient later to picture these
two numbers by regarding the wavefunction as a line on a plane,
like that shown in figure 7. The two numbers are then the length
of the line and the angle it makes with some fixed line. We shall
refer to these numbers as the magnitude and the angle of the
wavefunction.

As mentioned in the previous section, the wavefunction at a
given point determines the probability for the particle to be at that
point. In fact, the relation between the wavefunction and the probability
is very simple: the probability is proportional to the square
of the magnitude of the wavefunction. It does not depend in any
way on the angle of the wavefunction.
The classical notion of a particle’s position is therefore related to
the magnitude of the wavefunction. What about the classical
velocity? Not surprisingly, this is related to the angle. In fact, the
velocity is proportional to the rate at which the angle of the
wavefunction varies with the point of space, i.e. with x. The reason for this is discussed in Appendix 4 (but only for readers with the
necessary mathematical knowledge). Note that here we are speaking
of the actual velocity, not the uncertainty in the velocity which,
as discussed earlier, is proportional to the width of the peak in the
probability.
For easier visualisation of what is happening it is useful to
simplify the idea of a wavefunction by thinking about its so-called
real part, which is the projection of the wavefunction along some
fixed line, as shown in figure 7. For example, the real part of the
wavefunction corresponding to the probability distribution of
figure 5 might look like figure 8. The dashed line in this figure is
the magnitude of the wavefunction. The rate of oscillation of the
real part is proportional to the velocity of the particle.
We shall see later that it is necessary to have a method of
‘adding’ wavefunctions. The method we use can be understood by
reference to figure 9. We wish to add the wavefunctions represented
by the lines in figures 9(a) and (b). To do this we join the beginning
of the first line to the end of the second; then the line joining the
beginning of the second to the end of the first is the line that
represents the sum of the two wavefunctions. This is illustrated in
figure 9(c). It is not hard to show that, with this definition, it is
irrelevant which line is called the first and which the second. We
now notice the important fact that this definition is not the same
as using ordinary addition to add the numbers associated with each
wavefunction. In particular, the magnitude of the sum of two
wavefunctions is not the same as the sum of the magnitudes of the
wavefunctions. As an example of this, whereas, since magnitudes
are always positive, the sum of two magnitudes is always greater
than either, this is not necessarily the case for the magnitude of the
sum, as is seen in figure 10. Note, however, that the real parts of
wavefunctions do add just like ordinary numbers.

Quantum Theory

2007-07-02T01:18:00.000-07:00

The description of a particle in
quantum theory
The familiar, classical, description of a particle requires that, at all
times, it exists at a particular position. Indeed, the rules of classical
mechanics involve this position and allow us to calculate how it
varies with time. According to quantum mechanics, however, these
rules are only an approximation to the truth and are replaced by
rules that do not refer explicitly to this position but, instead,
predict the time variation of a quantity from which it is possible to
calculate the probability of the particle being in a particular place.
We shall indicate below the circumstances in which the classical
approximation is likely to be valid.
The probability will be a positive number (any probability has to
be positive) which, in general, will vary with time and with the
spatial point considered. As an example, figure 5 is a graph of such
a probability, and shows how it varies with the distance, denoted
by x, along a straight line from some fixed point 0. This graph
represents a particle which is close to the point labelled P. The
width of the distribution, shown in the figure as U,, gives some idea
of the uncertainty in the true position of the particle. There are
precise methods of defining this uncertainty but these are not
important for our purpose. Clearly a very narrow peak corresponds
to accurate knowledge of the position of the particle and, conversely,
a wide peak to inaccurate knowledge.

At this stage it might be thought that we can always use the
classical approximation, where particles have exact positions, by
working with sufficiently narrow peaks. However, if we do this we
lose something else. It turns out that the width of the peak is also
related to the uncertainty in the velocity of the particle, more
precisely the velocity in the direction of the line between the points
0 and P, only here the relation is the opposite way round: the
narrower the peak, the larger the uncertainty. In consequence,
although there is no limit to the accuracy with which either the
position or the velocity can be fixed, the price we have to pay for
making one more definite is loss of information on the other. This
faa is known as the Heisenberg uncertainty principle.
Quantitatively, this principle states that the product of the
position uncertainty and the velocity uncertainty is at least as large
as a certain fixed number divided by the mass of the particle being
considered. The fixed number is, in fact, the constant +z introduced
earlier. We can then write the uncertainty principle in the form
U,U, > &/m (2.1)
where U, is the uncertainty in the velocity and m is the mass of the
particle.
The quantity +I is Planck’s constant. We quote again its value,
this time in SI units:
4 = 1.05 x kgm2s-’.
This is a very small number! We can now see why quantum effects
are hard to see in the world of normal sized, i.e. ‘macroscopic’,
objects. For example, we consider a particle with a mass of one
gram (about the mass of a paper clip). Suppose we locate this to
an accuracy such that U, is equal to one hundredth of a centimetre
(10-4m). Then, according to equation (2.1), the error in velocity
will be about 10-”m per year. Thus we see that the uncertainty
principle does not put any significant constraint on the position and
velocity determinations of macroscopic objects. This is why
classical mechanics is such a good approximation to the macroscopic
world.
We contrast this situation with that which applies for an electron
inside an atom. The uncertainty in position cannot be larger than
the size of the atom, which is about 10-”m. Since the electron
mass is approximately kg, equation (2.1) then yields a
velocity uncertainty of around lo6 ms-’. This is a very large
velocity, as can be seen, for example, by the fact that it corresponds
to passage across the atom once every 10-l6s. Thus we guess,
correctly, that quantum effects are very important inside atoms.
Nevertheless, readers may be objecting on the grounds that, even
in the microscopic world, it is surely possible to devise experiments
that will measure the position and velocity of a particle to a higher
accuracy than that allowed by equation (2.1), and thereby
demonstrate that the uncertainty principle is not correct. Such
objections were made in the early days of quantum theory and were
shown to be invalid. The crucial reason for this is that the
measuring apparatus is also subject to the limitations of quantum
theory. In consequence we find that measurement of one of the
quantities to a particular accuracy automatically disturbs the other
and so induces an error that satisfies equation (2.1). As a simple
example of this, let us suppose that we wish to use a microscope to measure the position of a particle, as illustrated in figure 6. The
microscope detects light which is reflected from the particle. This
light, however, consists of photons, each of which carries momentum.
Thus the velocity of the particle is continuously being altered
by the light that is used to measure its position. It is not possible
to calculate these changes since they depend on the directions of the
photons after collision. The resulting uncertainty can be shown to
be that given by the uncertainty relation. The caption to figure 6
explains this more fully. Most textbooks of quantum theory, e.g.
those mentioned in the bibliography ($6.5), include a detailed
analysis of this experiment and of other similar ‘thought’
experiments.

The experimental challenge to reality

2007-07-02T01:17:00.002-07:00

We continue with our experiment in which particles are directed at
a potential barrier but now, instead of having detectors to tell us
whether a particle has been reflected or transmitted, we have
‘mirrors’ which deflect both sets of particles towards a common
detector. There are many ways of constructing such mirrors, par ticularly if our particles are charged, e.g. if they are electrons, when
we could use suitable electric fields. For this experiment we must
also allow the particles to follow slightly different paths, which can
easily be arranged if there is some degree of variation in the initial
direction. To be specific, we suppose that the source of particles
gives a uniform distribution over some small angle. Then the final
detector must cover a region of space sufficiently large to see particles
following all possible paths. In fact, we split it into several
detectors, denoted by A, B, C, etc, so that we will be able to
observe how the particles are distributed among them. In figure 2
we give a plan of the experiment.

We now do three separate sets of experiments. For the first set
we only have the right-hand mirror. Thus only the particles that are
reflected by the barrier will be able to reach the detectors. When we
have sent N particles, where N is large, the detectors will have
flashed R times. These R flashes will have some particular distribution
among the various detectors.

Next, we repeat these experiments with the right-hand mirror
removed and the left-hand mirror in place. This time only the
transmitted particles will reach the detectors, so, when we have sent
N particles, we will have T flashes. In figure 3 (b) we show a
possible distribution of these among the same five detectors.
For our third set of experiments we have both mirrors in
position. Thus all particles, whether reflected or transmitted by the
barrier, will be detected. When N particles have been sent, there
will have been N flashes. Can we predict the distribution of these
among the various detectors? Surely, we can. We know what
happens to the transmitted particles, e.g. figure 3 (b), and also to
the reflected particles, e.g. figure 3 (a). We also know that the particles
are sent separately so they cannot collide or otherwise get in
each other’s way. We therefore expect to obtain the sum of the two
previous distributions. This is shown in figure 3 (c) for our
example. The world, however, is not in accord with this expectation.
The distribution seen when both mirrors are present is not the
sum of the distributions seen with the two mirrors separately.
Indeed, it is quite possible for some detectors to receive fewer
particles when both mirrors are present than when either one is
present. A typical possible form showing this effect is given in
figure 3 (d).
Can we understand these results? Can we understand, for
example, why there are paths for particles to reach detector B when
either mirror is present but such paths are not available if both
mirrors are present? The only possibility is that in the latter case
each individual particle ‘knows about’, i.e. is influenced by, both
mirrors. This is not compatible with the view of reality, discussed
in the previous section, in which a particle either passes through or
is reflected. On the contrary, the reality suggested by the experiments
of this section is that each particle somehow splits into two
parts, one of which is reflected by one mirror and one by the other.
Such a picture is, however, not compatible with the results of the
detector experiments in which each individual particle is seen to go
one way or the other and never to split into two particles. Thus the
simple pictures of reality suggested by these two sets of experiments
are mutually contradictory.
Clearly we should not accept this perplexing situation without
examing very carefully the steps that have led to it. The first thing
we would want to check is that the experimental results are valid,
Here I have to make an apology. Contrary to what has been implied
in the above discussion, the experiments that have been described
have not actually been done. For a variety of technical reasons no
real experiment can ever be made quite as simple as a ‘thought’
experiment. The apparent incompatibility we have met does occur
in real experiments, but the discussion there would be much more
complicated and the essential features would be harder to see. The
‘results’ of our simple experiments actually come from theory, in
particular from quantum theory, but the success of that theory in
more complicated, real, situations means that we need have no
doubt about regarding them as valid experimental results.
As another possibility for rescuing the picture of reality given in
the previous section, we might ask whether we abandoned it too
readily in the face of the evidence from the mirror experiments. On
examining the argument we see that a key step lay in the statement
that a reflected particle, for example, could not know about the
left-hand mirror. Behind this statement lay the assumption that
objects sumciently separated in space cannot influence each other.
Is this assumption true and, if so, were our mirrors sufficiently well
separated? With regard to the second question one answer is that,
according to quantum mechanics, which provided our results, the
distance is irrelevant. Perhaps more important, however, is the fact
that the irrelevance of the distance scale seems to be experimentally
supported in other situations. The only hope here, then, is to question
the assumption; maybe the belief that objects can be spatially
separated so that they no longer influence each other is false. If this
is so, then it is already a serious criticism of the normal picture of
reality, in which the idea that objects can be localised plays a
crucial role. We shall return to this topic later.
Are there any other alternatives? Certainly some rather bizarre
pggsibilities exist. The ‘decision’ to put the second mirror in place
was made prior to the experiment with two mirrors being performed.
Maybe this process somehow affected the particles used in
the experiment and hence led to the observed results. Alternatively,
it could in some way have affected the first mirror, so that the two
mirrors ‘knew about’ each other and therefore behaved differently.
Such things could be true, but they seem unlikely. We mention
them here to emphasise how completely the results we have
discussed in this chapter violate our basic concept of reality, and
also because they are, in their complexity, in stark contrast to the
elegant simplicity of the quantum theoretical description of these
experiments. It is this description that forms the topic of the next
chapter.

Hidden variable theories

2007-07-02T01:17:00.001-07:00

Hidden variable theories. In such theories the particles reaching the
barrier are not identical; they possess other variables in addition
to their velocities and, in principle, the values of these variables
determine the fate of each particle as it reaches the barrier; no
breakdown of determinism is required and the probability aspect
only enters through our ignorance of these values, exactly as in
classical physics. At this stage of our discussion readers are probably
thinking that hidden variable theories surely contain the truth,
and that we have not yet given any good reasons for abandoning
determinism. They are right, but this will soon change and we shall
see that hidden variable theories, which are discussed more fully in
Chapter 5, have many difficulties.
Before proceeding we shall look a little more carefully at our
potential barrer experiment. Since we are interested in whether or
not particles pass through the barrier we must have detectors which
record the passage of a particle, e.g. by flashing so that we can see
the flash. We shall assume that our detectors are ‘perfect’, i.e. they
never miss a particle. Then if we have a detector on the left of the
barrier it will flash when a particle is transmitted, whereas one on
the right will flash for a reflected particle. Suppose N particles, all
with the same velocity, are sent and suppose we see R flashes in the
right-hand detector and Tin the left-hand detector. Because every
particle must go somewhere, we will find

Orthodox theories

2007-07-02T01:16:00.001-07:00

Orthodox theories. In such theories it is accepted that the particles
genuinely are identical, so there is nothing available with which to
answer the question except the statement that it is a random choice,
subject only to the requirement that when the same experiment is
repeated many times the correct proportion have been reflected.
Quantum theory, as normally understood, is a theory of this type.
If such theories are correct then determinism, as defined in 0 1.1, is
not a property of our world; probability enters physics in an
intrinsic way and not just through our ignorance. The situation is thus different in nature from that of people passing the Jet d’eau
in Geneva. Herein lies the second revolution of quantum physics to
which we referred in the opening section. The physical world is not
deterministic. It is worth noting here that, although quantum
phenomena are readily seen only on the microscopic scale, this lack
of determinism can easily manifest itself on any macroscopic scale
one might choose.

The potential barrier and the breakdown of determinism

2007-07-02T01:15:00.000-07:00

We now want to describe a set of simple experiments which
demonstrate the crucial features of quantum phenomena. To begin
we suppose that we have a flat table on which there is a smooth ‘hill’, This is illustrated in figure 1. If we roll a small ball, from the
right, towards the hill then, for low initial velocities, the ball will
roll up the hill, slowing down as it does so, until it stops and then
rolls back down again. In this case we say that the ball has been
reflected. For larger velocities, however, the ball will go right over
the hill and will roll down the other side; it will have been
transmitted.

Now we introduce quantum physics. The simple result expressed
by equation (l.l), which we obtained from experiment and which
is in agreement with the laws of classical mechanics, is not in fact
correct. For example, even when v < Vthere is a possiblity that the
particle will pass through the barrier. This phenomenon is sometimes
referred to as quantum tunnelling. The reason why we
would not see it in our simple laboratory experiment is that with
objects of normal sizes (which we shall refer to as ‘macroscopic’
objects), i.e. things we can hold and see, the effect is far too small
to be noticed. Whenever v is measurably smaller than V the
probability of transmission is so small that we can effectively say
it will never happen.

With ‘microscopic’ objects, i.e. those with atomic sizes and
smaller, the situation is very different and equation (1.1) does not
describe the results except for sufficiently small, or sufficiently
large, velocities. For velocities close to V we find, to our surprise,
that the value of v does not tell us whether or not the particle will
be transmitted. If we repeat the experiment several times, always
with a fixed initial velocity (v) we would find that in some cases the
particle is reflected and in some it is transmitted. The value of v
would no longer determine precisely the fate of the particle when
it hits the barrier; rather it would tell us the probability of a particle
of that velocity passing through. For low velocities the probability
would be close to zero, and we would effectively be in the classical
situation; as the velocity rose towards V the probability of
transmission would rise steadily, eventually becoming very close to
unity for v much larger than V, thus again giving the classical
result.

External reality

2007-07-02T01:13:00.000-07:00

As I look around the room where I am now sitting I see various objects. That is, through the lenses in my eyes, through the structure
of the retina, through assorted electrical impulses received in
my brain, etc, I experience sensations of colour and shape which
I interpret as being caused by objects outside myself. These objects
form part of what I call the ‘real world’ or the ‘external reality’.
That such a reality exists, independent from my observation of it,
is an assumption. The only reality that I know is the sensations of
which I am conscious, so I make an assumption when I introduce
the concept that there are real external objects that cause these sensations.
Logically there is no need for me to do this; my conscious
mind could be all that there is. Many philosophers and schools of
philosophy have, indeed, tried to take this point very seriously
either by denying the existence of an external reality, or by claiming
that, since the concept cannot be properly defined, proved to exist,
or proved not to exist, then it is useless and should not be discussed.
Such views, which as philosophic theories are referred to by words
such as ‘idealism’ or ‘positivism’, are logically tenable, but are
surely unacceptable on aesthetic grounds. It is much easier for me
to understand my observations if they refer to a real world, which
exist even when not observed, than if the observations are in
fact everything. Thus, we all have an intuitive feeling that ‘out
there’ a real world exists and that its existence does not depend
upon us. We can observe it, interact with it, even change it, but we
cannot make it go away by not looking at it. Although we
can give no proof, we do not really doubt that ‘full many a flower
is born to blush unseen, and waste its sweetness on the desert
air’.

It is important that we should try to understand why we have this
confidence in the existence of an external reality. Presumably one
reason lies in selective evolution which has built into our genetic
make-up a predisposition towards this view. It is easy to see why
a tendency to think in terms of an external reality is favourable to
survival. The man who sees a tree, and goes on to the idea that
there is a tree, is more likely to avoid running into it, and thereby
killing himself, than the man who merely regards the sensation of
seeing as something wholly contained within his mind. The fact of
the built-in prejudice is evidence that the idea is at least ‘useful’.
However, since we are, to some extent, thinking beings, we should
be able to find rational arguments which justify our belief, and
indeed there are several. These depend on those aspects of our experience which are naturally understood by the existence of an
external reality and which do not have any natural explanation
without it. If, for example, I close my eyes and, for a time, cease
to observe the objects in the room, then, on reopening them, I see,
in general, the same objects. This is exactly what would be expected
on the assumption that the objects exist and are present even when
I do not actually look at them. Of course, some could have moved,
or even been taken away, but in this case I would seek, and
normally find, an explanation of the changes. Alternatively I could
use different methods of ‘observing’, e.g. touch, smell, etc, and I
would find that the same set of objects, existing in an external
world, would explain the new observations. Thirdly, I am aware
through my consciousness of other people. They appear to be
similar to me, and to react in similar ways, so, from the existence
of my conscious mind, I can reasonably infer the existence of real
people, distinct from myself, also with conscious minds. Finally,
these other people can communicate to me their observations, i.e.
the experiences of their conscious minds, and these observations
will in general be compatible with the same reality that explains my
own observations.
In summary, it is the consistency of a vast range of different
types of observation that provides the overwhelming amount of
evidence on which we support our belief in the existence of an
external reality behind those observations. We can contrast this
with the situation that occurs in hallucinations, dreams, etc, where
the lack of such a consistency makes us cautious about assuming
that these refer to a real world.

We turn now to the scientific view of the world. At least prior
to the onset of quantum phenomena this is not only consistent
with, but also implicitly assumes, the existence of an external
reality. Indeed, science can be regarded as the continuation of the
process, discussed above, whereby we explain the experiences of
our senses in terms of the behaviour of external objects. We have
learned how to observe the world, in ever more precise detail, how
to classify and correlate the various observations and then how to
explain them as being caused by a real world behaving according
to certain laws. These laws have been deduced from our experience,
and their ability to predict new phenomena, as evidenced by the
enormous success of science and technology, provides impressive support for their validity and for the picture of reality which they
present.

This beautifully consistent picture is destroyed by quantum
phenomena. Here, we are amazed to find that one item, crucial to
the whole idea of an external reality, appears to fail. It is no longer
true that different methods of observation give results that are consistent
with such a reality, or at least not with a reality of the form
that had previously been assumed. No reconciliation of the results
with an acceptable reality has been found. This is the major revolution
of quantum theory, and, although of no immediate practical
importance, it is one of the most significant discoveries of science
and nobody who studies the nature of reality should ignore it.
It will be asked at this stage why such an important fact is not
immediately evident and well known. (Presumably if it had been
then the idea of creating a picture of an external reality would
not have arisen so readily.) The reason is that, on the scale
of magnitudes to which we are accustomed, the new, quantum
effects are too small to be noticed. We shall see examples of this
later, but the essential point is that the basic parameter of
quantum mechanics, normally denoted by f~ (‘h bar’) has the
value 0.OOO OOO 000 OOO OOO OOO OOO OOO 001 (approximately) when
measured in units such that masses are in grams, lengths in
centimetres and times in seconds. (Within factors of a thousand or
so, either way, these units represent the scale of normal experience.)
There is no doubt that the smallness of this parameter is
partially responsible for our dimculty in understanding quantum
phenomena-our thought processes have been developed in situations
where such phenomena produce effects that are too small to
be noticed, too insignificant for us to have to take them into
account when we describe our experiences.

Reality in the Quantum World

2007-07-02T01:12:00.000-07:00

Quantum mechanics, created early this century in response to
certain experimental facts which were inexplicable according to
previously held ideas (conveniently summarised by the title
‘classical physics’), caused three great revolutions. In the first place
it opened up a completely new range of phenomena to which the
methods of physics could be applied: the properties of atoms and
molecules, the complex world of chemical interactions, previously
regarded as things given from outside science, became calculable in
terms of a few fixed parameters. The effect of this revolution has
continued successfully through the physics of atomic nuclei, of
radioactivity and nuclear reactions, of solid-state properties, to
recent spectacular progress in the study of elementary particles. In
consequence all sciences, from cosmology to biology, are, at their
most fundamental level, branches of physics. Through physics they
can, at least in principle, be understood. Indeed, on contemplating
the success of physics, it is easy to be seduced into the belief that
‘everything’ is physics-a belief that, if it is intended to imply that
everything is understood, is certainly false, since, as we shall see,
the very foundation of contemporary theoretical physics is
mysterious and incomprehensible.

The second revolution was the apparent breakdown of determinism,
which had always been an unquestioned ingredient and an
inescapable prediction of classical physics. Note that we are using the word ‘determinism’ solely with regard to physical systems,
without at this stage worrying about which systems can be so
described; that is, we are not here concerned with such concepts as
free will. In a deterministic theory the future behaviour of an
isolated physical system is uniquely determined by its present state.
If, however, the world is correctly described by quantum theory,
then, even for simple systems, this deterministic property is not
valid. The outcome of any particular experiment is not, even in
principle, predictable, but is chosen at random from a set of
possibilities; all that can be predicted is the probability of particular
results when the experiment is repeated many times. It is important
to realise that the probability aspects that enter here do so for a different
reason than, for example, in the tossing of a coin, or throw
of a dice, or a horse race; in these cases they enter because of our
lack of precise knowledge of the orginal state of the system,
whereas in quantum theory, even if we had complete knowledge of
the initial state, the outcome would still only be given as a
probability.
Naturally, physicists were reluctant to accept this breakdown of
a cherished dogma-Einstein’s objection to the idea of God playing
dice with the universe is the most familiar expression of this
reluctance-and it was suggested that the apparent failure of determinism
in the theory was due to an incompleteness in the description
of the system. Many attempts to remedy this incompleteness,
by introducing what are referred to as ‘hidden variables’, have been
made. These attempts will form an important part of our later
discussion.

We are accustomed to regarding the behaviour, at least of simple
mechanical systems, as being completely deterministic, so if the
breakdown of determinism implied by quantum mechanics is
genuine, it is an important discovery which must affect our view of
the physical world. Nevertheless, our belief in determinism arises
from experience rather than logic, and it is quite possible to conceive
of a certain degree of randomness entering into mechanics; no
obvious violation of ‘common sense’ is involved. Such is not the
case with the third revolution brought about by quantum
mechanics. This challenged the basic belief, implicit in all science
and indeed in almost the whole of human thinking, that there exists
an objective reality, a reality that does not depend for its existence
on its being observed. It is because of this challenge that all who endeavour to study, or even take an interest in, reality, the nature
of ‘what is’, be they philosophers or theologians or scientists,
unless they are content to study a phantom world of their own
creation, should know about this third revolution.

To provide such knowledge, in a form accessible to nonscientists,
is the aim of this book. It is not intended for those who
wish to learn the practical aspects of quantum mechanics. Many
excellent books exist to cover such topics; they convincingly
demonstrate the power and success of the theory to make correct
predictions of a wide range of observed phenomena. Normally
these books make little reference to this third revolution; they omit
to mention that, at its very heart, quantum mechanics is totally
inexplicable. For their purpose this omission is reasonable because
such considerations are not relevant to the success of quantum
mechanics and do not necessarily cast doubt on its validity. In
1912, Einstein wrote to a friend, ‘The more success the quantum
theory has, the sillier it looks.’ [Letter to H Zangger, quoted on
p 399 of the book Subtle is the Lord by A Pais (Oxford: Clarendon
1982).] If it is true that quantum mechanics is ‘silly’, then it is so
because, in the terms with which we are capable of thinking, the
world appears to be silly. Indeed the recent upsurge of interest in
the topic of this book has arisen from the results of recent
experiments; results which, though they beautifully confirm the
predictions of quantum mechanics, are themselves, quite
independent of any specific theory, at variance with what an
apparently convincing, common-sense, argument would predict

We can emphasise the essentially observational nature of the
problem we are discussing by returning to the experimental facts we
mentioned at the start of this section, and which gave birth to quantum
mechanics. Although, by abandoning some of the principles of
classical physics, quantum theory predicted these facts, it did not
explain them. The search for an explanation has continued and we
shall endeavour in this book to outline the various possibilities. All
involve radical departures from our normal ways of thinking about
reality.

On almost all the topics which we shall discuss below there is a
large literature. However, since this book is intended to be a
popular introduction rather than a technical treatise, I have given very few references in the text but have, instead, added a detailed
bibliography. For the same reason various ifs and buts and
qualifying clauses, that experts might have wished to see inserted
at various stages, have been omitted. I hope that these omissions
do not significantly distort the argument.
I have tried to keep the discussion simple and non-technical,
partly because only in this way can the ideas be communicated to
non-experts, but also because of a belief that the basic issues are
simple and that highly elaborate and symbolic treatments only
serve to confuse them, or, even worse, give the impression that
problems have been solved when, in fact, they have merely been
hidden. The appendices, most of which require a little more
knowledge of mathematics and physics than the main text, give
further details of certain interesting topics.

Finally, I conclude this section with a confession. For over thirty
years I have used quantum mechanics in the belief that the problems
discussed in this book were of no great interest and could, in
any case, be sorted out with a few hours careful thought. I think
this attitude is shared by most who learned the subject when I did,
or later. Maybe we were influenced by remarks like that with which
Max Born concluded his marvellous book on modern physics
[Atomic Physics (London: Blackie 1935)] : ‘For what lies within
the limits is knowable, and will become known; it is the world of
experience, wide, rich enough in changing hues and patterns to
allure us to explore it in all directions. What lies beyond, the dry
tracts of metaphysics, we willingly leave to speculative philosophy.’
It was only when, in the course of writing a book on elementary
particles, I found it necessary to do this sorting out, that I
discovered how far from the truth such an attitude really is. The
present book has arisen from my attempts to understand things
that I mistakenly thought I already understood, to venture, if you
like, into ‘speculative philosophy’, and to discover what progress
has been made in the task of incorporating the strange phenomena
of the quantum world into a rational and convincing picture of
reality.