Monday, July 2, 2007

A peculiarly quantum measurement

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

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
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
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
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

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
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

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
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

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
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

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
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
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

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.