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).

## Monday, July 2, 2007

### The Bohm model

Perhaps the most significant recent development in the Bohm hiddenvariable

model (see $5.2) is that physicists outside of Bohm’s own

students (and John Bell) have begun to take the model seriously. One

group (D. Diirr, S. Goldstein and N. Zhangi, Physics Letters 172A 6,

1992) have invented the rather evocative name ‘Bohmia? mechanics’ to

describe it. This group have considered the requirement that the initial

distribution of positions should be consistent with the quantum theory

probability law, which, as we noted in $5.2, is necessary for the Bohm

model to agree with quantum theory. In particular, they have shown

that the the requirement is expected to be satisfied for any ‘typical’

initial conditions.

Although, given that the above initial requirement holds, the Bohm

model is guaranteed by construction to agree with the statistical

predictions of quantum theory for particle positions (and hence with all

known experiments), there has been a widespread reluctance to accept

this fact, presumably because of a variety of ‘impossibility theorems’

on the lines of that due to von Neumann mentioned in 55.1. One such

theorem is often known as the Kochen-Specker-Bell theorem, which

is a strange irony because John Bell actually gave his simplified proof

of the theorem (Reviews of Modern Physics 38 447, 1966) in order to

show why it was not relevant to the Bohm model! The essence of these theorems is very similar to the non-locality arguments discussed

in 55.4 and Appendix 9. For example, in Appendix 9 we seemed to

show that the performers could not carry cards containing the answers.

Since these answers are the analogues of the hidden variables, this at

first sight means that such things are forbidden if we wish to maintain

agreement with quantum theory. The ‘error’ in this argument is that

it requires the answers to be fixed, whereas in the Bohm model they

are dynamical things which change with time, and which change in a

way that can depend upon what question is being asked of the other

performer (which is where the non-locality enters). The situation here

is sometimes described by saying that measurements are ‘contextual’,

a fancy way of saying that quantum systems in general cannot be

separated into independent parts, and that the answer you get depends

upon the question (i.e., the result depends on the apparatus).

It should be emphasised that the Bohm model looks after all this

automatically. In fact, on re-reading the remarks I wrote at the end

of 55.1, I think I was being unfair to the Bohm model in saying

that it was ‘contrived’. This suggests that much effort was required

in order to devise something that would work, whereas, in fact,

trajectories are defined by one simple property, namely that if we

have many identical systems with identical wavefunctions, and with

particle positions distributed according to the quantum probability law

at a particular time to, then this fact will remain true at other times.

Actually this does not quite define the trajectory uniquely-the Bohm

model is just the simplest possibility.

1 shall now describe a very idealised experiment which shows how

all this works in practice. First, it is necessary to note that in most

versions of the Bohm model trajectories only exist for ‘matter’ particles,

in particular, for the electrons and nucleons that are the constituents of

matter. All these particles have spin equal to !j. Particles of spin zero

or one, e.g., the photon, do not have trajectories-so, in this sense, we

should say that the Bohm model does not have photons. Why then do

we apparently see ‘photons’? Specifically, refemng to the experiment

described in 31.4, why do detectors appear to say that a photon either

goes through the barrier of 51.4 or is reflected, when we know that the

wave does both? We shall see how the existence of matter trajectories

answers this question.

In order to make the calculations as simple as possible, we take

as the measuring device a single particle, moving in one dimension,

initially in a stationary, localised, wave-packet, and suppose that a photon wave-packet interacts with this to give it a momentum. The

details of this interaction are not important. If the detector is placed in,

say, the path of the transmitted wave and if the barrier is removed so

that there is only a transmitted wave, then it is easy to calculate that the

detector particle, initially at rest, will acquire a velocity. Observation

of this velocity will correspond to the photon having been detected.

Thus we have a detector that works properly: a photon wave comes

along and is detected through the motion of the detector particle, i.e.,

the movement of a pointer.

Now let us restore the barrier, so that the photon wave is a

superposition of transmitted and reflected parts (see figure 28). Again

it is possible to calculate what happens to the detector, and it turns out

that, for some initial positions of the detector particle, it moves, and

for others it does not. As indicated in figure 28, the important thing

here is the position of the detector particle, i.e. the hidden-variable,

relative to the position of the detector wave-packet, which of course

is what we refer to as the position of the detector. Thus, whether or

not the detector detects the photon depends on the initial position of its

particle. If it does, we would say that the photon has been transmitted;

if it does not we would say that the photon has been reflected. (Note

that, as in the collapse models discussed in the previous section, these

statements are really statements about the detector, rather than about

the photon). To be more explicit we consider, for simplicity, the case

where transmission and reflection are equally likely (so that PR = PT

in the equations of §4.5), and take a symmetrical initial wave-packet for

the detector. Then those initial starting positions that are on the near

side (relative to the incident photon) will not detect the photon; those

that are on the far side will. This actually follows simply from the

fact that trajectories cannot cross. Provided the distribution of initial

positions, in many repeats of the same experiment, are in accordance

with quantum theory (and hence in this case symmetrical between the

two sides), it follows that the photon will be detected in half of the

experiments, i.e., it will be transmitted with 50 per cent probability as

required. Symbolically, with suitable conventions, this means:

xo > 0 + transmission

and

xo .c 0 + reflection

where xo is the initial position of the particle in the detector and we

have taken the detector to be centred at the origin, x = 0 (see figure

28).

Clearly, very similar considerations hold if we put a detector instead

in the path of the reflected beam. Then we find the analogous results:

yo > 0 -+ reflection

and

yo < 0 -+ transmission

where here yo is the initial position of the particle in the ‘reflection’

detector, which is centred at y = 0.

Next we consider what happens if we have both detectors, one in

the path of the transmitted beam, and the other in the path of the

reflected beam, as shown in figure 29. If these detectors behaved

independently, i.e., acted as if the other were not present, then there

would be the possibility of violating the experimental results (and

also the predictions of quantum theory). For example, if the starting

positions of the detector particles happened to satisfy xg > 0 and yo > 0

then, according to what we saw above, both detectors would record the

photon, which would then appear to have been both transmitted and

reflected! In fact, however, this is where the contextuality becomes

evident. It is straightforward to calculate that the first detector records

the photon, which is therefore transmitted, if

XO - yo > 0.

Otherwise, the second detector records the photon, corresponding

to its being reflected. In general, it is the relative position of the

particles in the two detectors that determines whether a particular event

is observed as a transmitted or reflected photon.

We emphasise again that in this experiment, because we have

assumed there are no photon trajectories, it is the properties of the

detectors that give rise to the apparent existence of ‘photons’ which

appear in specific places. When we say, for example, that the photon

is transmitted we mean no more than that an appropriate detector has,

or has not, recorded a photon. The model is designed to agree with

the predictions of orthodox quantum theory at the level of the output

of detectors, because it is these that correspond to observations. This

last point is particularly significant if we consider experiments where

particles that do have trajectories are used to trigger detectors. In

certain rather special cases it can be shown that the detector records

the particle even though the particle trajectory did not pass through it,

and conversely. One can most easily regard this as being due to nonclassical

effects of the quantum potential (see B. Englert, M.O. Scully,

G. Sussman and H. Walther Z Nutulforsch. 47a 1175 (1992) and C.

Dewdney, L. Hardy and E.J. Squires Physics Letters 184A 6 (1993) for

further details).

Two books covering all aspects of the Bohm model have recently

been published. The Quantum Theory of Motion (Cambridge University

Press, 1993) by Peter Holland, an ex-student of David Bohm, gives

an extremely thorough and detailed treatment of the model and its

applications. The book by David Bohm and Basil Hiley, The Undivided

Universe (Routledge, London, 1993), which was completed just before

Bohm’s death, contains fewer details of calculations in the Bohm

model but more on the general problem of the interpretation of

quantum theory, and comparison with other suggested solutions of the

measurement problem.

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

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.

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

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!

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

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.

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

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).

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

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.

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.

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