The End of Time: The Next Revolution in Physics

CHAPTER 14

The Greater Mysteries

SCHRÖDINGER’S VAST ARENA

The true heart of quantum mechanics and the way to quantum cosmology is the way in which it describes composite systems – that is, systems consisting of several particles. It is an exciting, indeed extraordinary story, though it is seldom well told. When Schrödinger discovered wave mechanics, he said it could be generalized and ‘touches very deeply the true essence [wahre Wesen] of the quantum prescriptions’. But it was not just the Bohr quantization prescriptions that came into focus: at stake here are the rules of creation. A bold claim, but one I hope to justify as the book goes on. First, we have to see how Schrödinger opened the door onto a vast new arena.

The central concept of this book is Platonia. It is a relative configuration space. The new arena that Schrödinger introduced is something similar, a configuration space (without the ‘relative’). The notion is easily explained. Each possible relative arrangement of three particles is a triangle and corresponds to a single point in the three-dimensional Triangle Land. But now imagine the three particles located in absolute space. Besides the triangle they form, which is specified by three numbers (the lengths of its sides), we now have to consider the location of its centre of mass in absolute space, which requires three more numbers, and also its orientation in absolute space, which also requires three more numbers. Location in Triangle Land needs three numbers, in absolute space it needs nine. Just as each triangle corresponds to one point in three-dimensional Triangle Land, the triangle and its location in absolute space correspond to one point in a nine-dimensional configuration space. The tetrahedron formed by four particles corresponds to one point in six-dimensional Tetrahedron Land and one point in the corresponding twelve-dimensional configuration space. For any Platonia corresponding to the relative arrangements of a certain number of particles, the matching configuration space has six extra dimensions. Schrödinger called such a space a Q, and I shall follow his example. Such a Q is a ‘hybrid Platonia’, since it contains both absolute and relative elements. This hybrid nature is very significant, as will become apparent.

The most important thing about Schrödinger’s wave mechanics is that it is formulated not in space and time, but in a suitably chosen Q and time. This is not apparent for a single particle, for which the configuration space is ordinary space. Since most accounts of quantum mechanics consider only the behaviour of a single particle, many people are unaware that the wave function is defined on configuration space. That is where ψ lives. It makes a huge difference.

An illustration using a plastic ball-and-strut model of molecules may help to bring this home. Imagine that you are holding such a model in some definite position in a room, which can represent absolute space. There are three digital displays – I shall call them ψ meters – that show red, green and blue numbers on the wall. These numbers give the intensities of the three ‘mists’ represented by ψ for the system at the time considered. Suppose you take just one ball, representing one particle of the system, and detach it from the model. Keeping all the other balls fixed, you can move the one ball around and, courtesy of the ψ meters, see how ψ changes. As you move in each direction in space, each ψ value will change. For each point of space you can find the value of ψ. The blue ψ meter will always tell you the positions for which the probability is high or low. Suppose you do this and then return the ball to its original place.

Now move a second ball to a slightly different position, and leave it there. The ψ meters will change to new values. Once again, explore space with the first ball, watching the ψ meters. The values of ψ will be (in general) quite different. The ψ values on the displays embody information. The amount is staggering. For every single position in space to which you move any one of the other balls, you get a complete new set of values in space for the ball chosen as the ‘explorer’. And any ball can be the explorer. Each explorer will have its own distinctive three-dimensional patterns of ψ for every conceivable set of positions of the others.

Now, what is a molecule? When Richard Dawkins described the haemoglobin molecule and its six thousand million million million perfect copies in our body, he said that in its intricate thornbush structure there is ‘not a twig nor a twist out of place’. That is in a molecule containing perhaps twenty thousand atoms. But molecules are even more remarkable than that. The twig and the twist are averaged structures corresponding to the most probable configuration in which the molecule will be found. In the Schrödinger picture, the molecule is not just one structure but a huge collection of potentially present structures, each with its own probability.

In fact, the complete structure of complicated protein molecules like haemoglobin cannot be understood solely on the basis of wave mechanics. This is because of the way they are put together from amino acid units. But for simpler molecules, which may still contain many particles, you could (in imagination at least) do what I have just described for the ball-and-strut model. Start with one of the model configurations shown in chemistry textbooks, and look at the ψ meters, especially the blue one. It will give a high reading. Around that highly probable structure are other similar structures, all with a high – but not quite so high – blue intensity. Individual units of the structure – simpler forms of Dawkins’s ‘twigs’ – could be moved as a whole, say by twisting them, from the most probable configuration, and the blue intensity would drop. It would also drop if one atom of the few dozen within the twig were moved from the most-favoured position. The molecule is not just the most probable configuration. It is all possible configurations with their ψ values, held in balance by the laws of wave mechanics. The existence and most-favoured shape of molecules can be understood in no other way.

Contrary to the impression given in many books, quantum mechanics is not about particles in space: it is about systems being in configurations – at ‘points’ in a Q, or ‘hybrid Platonia’. That is something quite different from individual probabilities for individual particles being at different points of ordinary space. Each ‘point’ is a whole configuration – a ‘universe’. The arena formed by the ‘points’ is unimaginably large. And classical physics puts the system at just one point in the arena. The wave function, in contrast, is in principle everywhere.

This is what I mean by saying that Schrödinger opened the door onto a vast new arena. Compared with Schrödinger’s vistas, grander than any Wagnerian entrance into Valhalla, the Heisenberg uncertainty relation for a single particle captures little of quantum mechanics. All revolutions in physics pale into insignificance beside Schrödinger’s step into the configuration space Q. Not that he did it happily.

CORRELATIONS AND ENTANGLEMENT

It is not possible to observe the extraordinary quantum arena directly. Some people do not believe it exists at all. To a large degree it has been deduced, or surmised, from phenomena observed in systems of a few particles. Getting clear, direct evidence for the quantum behaviour of single particles was difficult. It was long after Dirac made his memorable remark about each photon interfering with itself that the development of sources which release individual particles with long time intervals between releases confirmed the build-up of interference patterns in individual ‘hits’. In the last two decades, it has become possible to create in the laboratory pure quantum states of two particles, whose Q therefore has six dimensions. The quantum predictions, all verified, are not easy to explain in many words, let alone a few, and a serious attempt to do so would take me too far from my main story. The simplest possible illustration is given by two particles moving on a single line; each has a one-dimensional Q and together they have a two-dimensional configuration space (Figure 39).

As for a single particle, the maximally informative description of a quantum system at any instant t is specified by a complex wave function ψ which, in principle, has a different value at each point of the configuration space. As t changes, ψ changes. All information that can be known about the system at t is encoded in ψ at f, and consists of predictions that can be made about it. Many different kinds of prediction can be made, but they are often mutually exclusive. In a very essential way, the predictions refer to the system, not its parts.

Let us start with position predictions. Just as we did for a single particle, we can form from ψ the sum of the squares of its intensities, finding the intensity of the ‘blue mist’ (Figure 40). This gives the relative probability that the system will be found at the corresponding point in Q if an appropriate measurement is made. The important thing is that a single point in Q corresponds to positions of both particles. Anyone who has not understood this has not understood quantum mechanics. It is this fact, coupled with complementarity, that leads to the most startling quantum phenomena.

Figure 39 The two-dimensional configuration space Q of two particles on one line. The line is shown in multiple copies on the left. Nine different configurations of the two particles on it are shown. The positions of particles 1 and 2 are indicated by the black and white triangles, respectively. The axes of Q on the right show the distances of particle 1 (horizontal axis) and particle 2 (vertical axis) from the left-hand end of the line. The points on the 45° diagonal in Q correspond to configurations for which the two particles coincide (points 1, 4 and 8). You might like to check how the nine configurations on the left are represented by the nine corresponding points on the right.

In Chapter 3 we imagined tipping triangles out of a bag. That exercise was presented because it mimics one of the ways in which we can interpret a quantum state. Imagine now that the blue mist has the distribution shown in Figure 40. To avoid problems with infinite numbers of configurations, we divide up Q by a grid of cells sufficiently fine that ψ hardly changes within any one of them (Figure 40, on the right). The intensity of the blue mist at the central point of each cell then gives the relative probability of the nearly identical configurations in that cell. On a piece of cardboard, let us depict one of these configurations (as shown on the left in Figures 39 and 40). This will serve as the representative of all the configurations of that cell. For the grid in the figure with 100 cells, there are 100 relative probabilities whose sum should be conveniently large, say a million. Then we shall not distort things seriously by replacing exact relative probabilities like 127.8543... by the rounded-up integer 128.

We now imagine putting into a bag the number of copies of each representative configuration equal to its rounded probability, 128 for example. In quantum mechanics, performing a measurement to determine the positions of both particles is like drawing at random one piece of cardboard from the bag. We get some definite configuration. In the process, we destroy the wave function and replace it by one entirely concentrated around the configuration we have found. If we recreate the original wave function, by repeating the operations that we used to set it up, and repeat the experiment millions of times, then the relative frequencies with which the various configurations are ‘drawn from the bag’ will match, statistically, the calculated relative probabilities.

Figure 40 Like Figure 39, this shows nine different configurations of two particles (black and white triangles) on a line and the points corresponding to them in the configuration space Q (on which a grid has been drawn). A possible distribution of the intensity of the blue probability mist is shown as the height of a surface over Q in the top part of the figure (you are seeing the surface in perspective from above, and rotated). In the state of the system shown here, the probabilities for configurations 4, 6 and 9 are high, while 5 has a very low probability.

Figure 41 (a) The effect of measuring, for the probability density of Figure 40, the position of the particle represented by the horizontal axis, and finding that it lies in the interval on which the vertical strip stands. All the wave function outside the strip is instantaneously collapsed.

This is only the start. We can select from a menu of different kinds of measurement. For example, we can opt to find the position of only one particle, which has remarkable implications for what we can say about the other one. Suppose first that we measure the position of just one of the particles. According to the quantum rules, this instantaneously collapses the wave function from its original two-dimensional ‘cloud’ to a one-dimensional profile (Figure 41). The point is that we now know the position of one particle to within some small error, so none of the wave function outside the narrow strip is relevant any longer. It is annihilated. If the particle whose position is measured is represented along the horizontal axis, only a vertical strip of ψ survives (Figure 41(a)); if the position of the other particle is measured, only a horizontal strip survives (Figure 41(b)).

Figure 41 (b) The same for a position measurement of the other particle.

Either profile then gives conditional information. If we know where one particle is, the possible positions of the other are restricted to a narrow strip. The relative probabilities for the position of the second particle are determined by the values of ψ within the strip. Provided we know the original wave function, acquired knowledge about one particle sharpens our knowledge – instantaneously – about the other. This is the place to explain entangled states, or quantum inseparability (Box 12).

BOX 12 Entangled States

Figure 42 again shows our two-particle Q and two different quantum states. In the unentangled state at the bottom, all the horizontal ψ profiles are identical, and so are all the vertical profiles (only their shapes count). Such a wave function is said to be unentangled because if we gain information about particle 1 (black triangles) – that it is at some definite position – we do not gain any new information about particle 2 (white triangles). This is because all the horizontal profiles and all the vertical profiles are identical: they give identical relative probabilities. This is shown by two profiles that result from an exact position measurement of particle 1. Measurement on particle 1 leads to no new information about particle 2.

Figure 42 Entangled (top) and unentangled (bottom) states for two particles (black and white triangles).

Much more interesting is the entangled state at the top, for which the horizontal and vertical profiles are not identical. Figure 42 shows two profiles that result from exact position measurement of particle 1. They give very different probability distributions for particle 2: the gain in information about particle 2 is considerable. This is typical of quantum mechanics, since virtually all wave functions are entangled to a greater or lesser extent.

A particular feature of entangled states should be noted. Particles normally interact (affect each other) when they are close to each other. For the two-particle Q in Figure 39, the particles coincide on the diagonal line, and the region in which the particles are close to each other and can interact strongly is therefore a narrow strip around that line. However, the wave function of an entangled state may be located completely outside this region – the particles may be very far apart when one of them is observed. Yet the other particle is apparently immediately affected. It can jump to one or other of two hugely different possibilities. Moreover, if the ‘ridge’ in the top part of Figure 42 is made thinner and thinner, shrinking to a line, then position determination of one particle immediately determines the position of the other to perfect accuracy. Such situations are not easy to engineer for position measurements (and in general will not persist because of wave-packet spreading), but there are analogous situations for momentum and angular-momentum measurements that are easy to set up.

The facts discussed in Box 12 are immensely puzzling if we wish to find a physical and causal mechanism to explain how measurement on one particle can have an immediate effect on a distant particle. As I have already explained, innumerable interference phenomena indicate that, in some sense, the particles are, before any measurement is made, simultaneously present wherever ψ extends. Since there is no restriction on the distance between the particles, any causal effect on the second particle after the first has been observed would have to be transmitted instantaneously. However, relativity theory is supposed to rule out all causal effects that travel faster than the speed of light. Moreover, in the mid-1980s Alain Aspect in Paris performed some very famous experiments in which such wave-function collapses were tested, and the predictions of quantum mechanics confirmed with great accuracy. The experiments were so arranged that any physical effect would have had to be transmitted faster than the speed of light to bring about the collapse.

The situation is actually delicate and intriguing. Relativity absolutely prohibits the transmission of information faster than light. But, curiously, wave-function collapse does not transmit information. When information about particle 1 has been obtained by an experimentalist, he or she will know immediately what a distant experimentalist can learn about particle 2. But there is no way such information can be transmitted faster than light. There is no conflict with the rules of relativity, though many physicists are concerned that its ‘spirit’ is violated.

So far, we have considered only position measurements on a two-particle system. But we can also consider many other measurements, of momentum, for example. Given ψ in Q, we directly obtain predictions for positions. But Dirac’s transformation theory enables us to pass to the complementary momentum space, which gives direct predictions for momentum measurements. If the wave function is tightly entangled with respect to momentum, measuring the momentum of particle 1 would immediately tell us the momentum of particle 2. And this despite the fact that before any measurements are made there is considerable uncertainty about the momenta of the particles. However, what is certain is that they are entangled, or correlated. This brings us to the EPR paradox.

THE EPR PARADOX

The nub of the Einstein-Podolsky-Rosen (EPR) paradox, formulated in 1935 by Einstein and collaborators Boris Podolsky and Nathan Rosen, is that two particles can be in a state in which they are perfectly correlated (entangled) as regards both their position and their momentum. The actual example of such a state that EPR found is rather unrealistic, but in 1952 David Bohm, an American theoretical physicist who later worked in London for many years, proposed a much more readily realized state using spin, the intrinsic angular momentum associated with quantum particles. Alain Aspect performed his experiments on such a system. What puzzled EPR about their state was that if the position of one particle was measured, the position of the other particle could be immediately established with certainty because of the perfect correlation. Since the second particle, being far away, could not be physically affected by the measurement, but it was known for certain where it would be found, EPR concluded that it must have had this definite property before the measurement on the first particle.

But, it could just as well have been decided to measure momentum. The measurement of one momentum will then instantaneously determine the other momentum with certainty. By the same argument as before, the particle must have possessed that momentum before the measurement on the first particle. Finally, the choice between momentum or position measurement is a matter of our whim, about which the second particle can know nothing. The only conclusion to draw is that the second particle must have possessed definite position and momentum before any measurements were made at all. However, according to the fundamental rules of quantum mechanics, as exemplified in the Heisenberg uncertainty principle, a quantum particle cannot possess definite momentum and position simultaneously. EPR concluded there must be something wrong – quantum mechanics must be incomplete.

Niels Bohr actually answered EPR quite easily, though not to everyone’s satisfaction. His essential point was that quantum mechanics predicts results made in a definite experimental context. We must not think that the two-particle system exists in its own right, with definite properties and independent of the rest of the world. To make position or momentum measurements, we must set up different instruments in the laboratory. Then the total system, consisting of the quantum system and the measuring system, is different in the two cases. Nature arranges for things to come out differently in the two cases. Nature is holistic: it is not for us to dictate what Nature is or does. Quantum mechanics is merely a set of rules that brings order into our observations. Einstein never found an answer to this extreme operationalism of Bohr, and remained deeply dissatisfied.

I feel sure that Bohr got closer to the truth than Einstein. However, Bohr too adopted a stance that I believe is ultimately untenable. He insisted that it was wrong to attempt to describe the instruments used in quantum experiments within the framework of quantum theory. The classical world of instruments, space and time must be presupposed if we are ever to talk about quantum experiments and communicate meaningfully with one another. Just as Schrödinger made his Kantian appeal to space and time as necessary forms of thought, Bohr made an equally Kantian appeal to macroscopic objects that behave classically. Without them, he argued, scientific discourse would be impossible. He is right in that, but in the final chapters I shall argue that it may be possible to achieve a quantum understanding of macroscopic instruments and their interaction with microscopic systems. Here it will help to consider why Einstein thought the way he did.

Referring to their demonstration that distant measurement on the first system, ‘which does not disturb the second in any way’, nevertheless seems to affect it drastically, EPR commented that ‘No reasonable definition of reality could be expected to permit this.’ These words show what is at stake – it is the atomistic picture of reality. Despite the sophistication of all his work, in both relativity and quantum mechanics, Einstein retained a naive atomistic philosophy. There are space and time, and distinct autonomous things moving in them. This is the picture of the world that underlies the EPR analysis. In 1949 Einstein said he believed in a ‘world of things existing as real objects’. This is his creed in seven words. But what are ‘real objects’?

To look at this question, we first accept that distinct identifiable particles can exist. Imagine three of them. There are two possible realities. In the Machian view, the properties of the system are exhausted by the masses of the particles and their separations, but the separations are mutual properties. Apart from the masses, the particles have no attributes that are exclusively their own. They – in the form of a triangle – are a single thing. In the Newtonian view, the particles exist in absolute space and time. These external elements lend the particles attributes – position, momentum, angular momentum – denied in the Machian view. The particles become three things. Absolute space and time are an essential part of atomism.

The lent properties are the building blocks of both classical and quantum mechanics. Classically, each particle has a unique set of them, defining the state of each particle at any instant. This is the ideal to which realists like Einstein aspire. The lent properties also occur in quantum mechanics. They are generally not the state itself, but superpositions of them are. If a quantum system is considered in isolation from the instruments used to study it, its basic elements still derive from a Newtonian ontology. This is what misled EPR into thinking they could outwit Bohr. Einstein’s defeat by Bohr is a clear hint that we shall only understand quantum mechanics when we comprehend Mach’s ‘overpowering unity of the All’.

BELL’S INEQUALITIES

Strong confirmation for quantum mechanics being holistic in a very deep sense was obtained in the 1960s, when John Bell, a British physicist from Belfast, achieved a significant sharpening of the EPR paradox. The essence of the original paradox is the existence of correlations between pairs of quantities – pairs of positions or pairs of momenta – that are always verified if one correlation or the other is tested. By itself, some degree of correlation between the two particles is not mysterious. The EPR-type correlated states are generally created from known uncorrelated states of two particles that are then allowed to interact. Even in classical physics, interaction under such circumstances is bound to lead to correlations. Bell posed a sharper question than EPR: is the extent of the quantum correlations compatible with the idea that, before any measurement is made, the system being considered already possesses all the definite properties that could be established by all the measurements that, when performed separately, always lead to a definite result?

Bell’s question perfectly reflects Einstein’s ‘robust realism’ – that the two-particle system ought to consist of two separate entities that possess definite properties before any measurements are made. Assuming this, Bell proceeded to derive certain inequalities, justly famous, that impose upper limits on the degree of the correlations that such ‘classical’ entities could exhibit (tighter correlations would simply be a logical impossibility). He also showed that quantum mechanics can violate these inequalities: the quantum world can be more tightly correlated than any conceivable ‘classical world’. Aspect’s experiments specifically tested the Bell inequalities and triumphantly confirmed the quantum predictions. The only way in which the atomized world after which Einstein hankered can be saved is by a physical interaction that has so far completely escaped detection and is, moreover, propagated faster than light. Einstein could hardly have taken comfort from this straw. Far better, it seems to me, is to seek understanding of the Here in Mach’s All. I shall give some indication of what I mean by this after we have considered the next topic.

THE MANY-WORLDS INTERPRETATION

In 1957, Hugh Everett, a student of John Wheeler at Princeton, proposed a novel interpretation of quantum mechanics. Its implications are startling, but for over a decade it attracted little interest until Bryce DeWitt drew wide attention to it, especially by his coinage many worlds to describe the main idea. Everett had used the sober title ‘Relative state formulation of quantum mechanics’. One well-known physicist was prompted to call it the ‘best-kept secret in physics’. So far as I know, Everett published no other scientific paper. He was already working for the Weapons Systems Evaluation Group at the Pentagon when his paper was published. He was apparently a chain smoker, and died in his early fifties.

Everett noted that in quantum mechanics ‘there are two fundamentally different ways in which the state function can change’: through continuous causal evolution and through the notorious collapse at a measurement. He aimed to eliminate this dichotomy, and show that the very phenomenon that collapse had been introduced to explain – our invariable observation of only one of many different possibilities that quantum mechanics seems to allow – is actually predicted by pure wave mechanics. Collapse is redundant.

The basis of Everett’s interpretation is the endemic phenomenon of entanglement. By its very nature, entanglement can arise only in composite systems – those that consist of two or more parts. In fact, an essential element of the many-worlds interpretation as it is now almost universally understood is that the universe can and must be divided into at least two parts – an observing part and an observed part. However, Everett himself looked forward to the application of his ideas in the context of unified field theories, ‘where there is no question of ever isolating observers and object systems. They are all represented in a single structure, the field.’ That is ultimately the kind of situation that we must consider, but for the moment we shall look at the familiar form of the interpretation.

The simplest two-particle system can be used to explain a quantum measurement. The core idea is all that counts. One particle, called the pointer, is used to establish the location of the other particle, called the object. Figure 43 shows things with which we are already familiar. At an initial time t_set-up, the pointer (horizontal axis) and object (vertical axis) are not entangled. For any of the small range of possible pointer positions, the object has identical ranges of possible positions, shown schematically by points 1 to 6 on the left. Determination of the pointer position in this state would tell us nothing about the object. But the interactions of the particles are so arranged that by the later time t_measurement the wave function, passing through the interaction region, has ‘swung round’ into the position shown on the right. Remembering how points in Q translate into positions in space, we see that the object still has its original range of positions 1 to 6, but that the new pointer positions are strongly correlated with them.

Figure 43 The initially unentangled state at t_set-up, shown schematically at the bottom left by the vertical column of positions 1, 2, 3, 4, 5, 6, and the first probability density above it, evolves into the entangled state at t_measurement, indicated on the right by the inclined numbers 1, 2, 3, 4, 5, 6 and the upper right probability density.

If we operate detectors at t_measurement to find the pointer’s position (by letting it strike an emulsion or even, in principle, by acting ourselves as observers and placing our eyes at the appropriate places), the standard quantum rules immediately tell us the object’s position, for different pointer positions are now correlated with different object positions. Quantum measurement consists of two stages: the creation from an unentangled state of a strongly entangled state (creating the conditions of a so-called good measurement) followed by the exploitation of the correlations in that state (using the determination of the pointer position to deduce – measure – the object’s position). The existence of such correlations has now been wonderfully well confirmed by experiments. If detectors have been used to find the pointer and are then used to locate the object directly, the quantum correlations predicted are invariably confirmed. Measurement theory is really verification of the correlations associated with entanglement. Personally, I think that the term ‘measurement’ has generated misunderstanding, and that it would be better simply to speak of verification of correlations.

The measurement problem of quantum mechanics is this: how does the entangled state of many possibilities collapse down to just one, and when does it happen? Is it when the pointer strikes the emulsion, or when the human observer sees a mark on the emulsion? I won’t go into all the complications, which depend on how much of the world we wish to describe quantum mechanically. It leads to a vicious infinite regress. You can go on asking quantum mechanics again and again to say when collapse occurs, but it never gives an answer. The different possibilities already represented at t_set-up in the different positions of the object system can never be eradicated, and simply ‘infect’ the rest of the world – first the pointer, then the emulsion, then the retina of the experimentalist’s eye, finally his or her conscious state. All that the Copenhagen interpretation can say is that collapse occurs at the latest in the perceptions of the experimentalist. When it happens no one can say – it can only be said that if collapse does not happen we cannot explain the observed phenomena.

But must it happen? Everett came up with a simple – with hindsight obvious – alternative. Collapse does not happen at all: the multiple possibilities represented in the entangled state continue to coexist. In each possibility the observer, in different incarnations, sees something different, but what is seen is definite in each case. Each incarnation of the observer sees one of the possible outcomes that the Copenhagen interpretation assumes is created by collapse. The implications of this are startling. A single atomic particle – the object particle in Figure 43 – can, by becoming entangled with first the pointer and then the emulsion, and finally the conscious observer, split that observer (indeed the universe) into many different incarnations. In his paper in 1970 that at last brought Everett’s idea to wide notice, Bryce DeWitt wrote:

I still recall vividly the shock I experienced on first encountering this multiworld concept. The idea of 10¹⁰⁰⁺ slightly imperfect copies of oneself all constantly splitting into further copies, which ultimately become unrecognizable, is not easy to reconcile with common sense. Here is schizophrenia with a vengeance.

Everett’s proposal raises two questions. If many worlds do exist, why do we see only one and not all? Why do we not feel the world splitting? Everett answered both by an important property of quantum mechanics called linearity, or the superposition principle. It means that two processes can take place simultaneously without affecting each other. Consider, for example, Young’s explanation of interference between two wave sources. Each source, when active alone, gives rise to a certain wave pattern. If both sources are active, the processes they generate could disturb each other drastically. But this does not happen. The wave pattern when both sources are active is found simply by adding the two wave patterns together. The total effect is very different from either of the individual processes, but in a real sense each continues unaffected by the presence of the other. This is by no means always the case; in so-called non-linear wave processes, the wave pattern from two or more sources cannot be found by simple addition of the patterns from the separate sources acting alone. However, quantum mechanics is linear, so the much simpler situation occurs.

As a result, quantum processes can be regarded as being made up of many individual subprocesses taking place independently of one another. In Figure 43 the total process of the wave function ‘swinging round’ and becoming entangled is represented symbolically by the arrows as six individual subprocesses (or branches, to use Everett’s terminology). In all of them, the pointer starts in the same position but ends in a different position. Everett makes the key assumption that conscious awareness is always associated with the branches, not the process as a whole. Each subprocess is, so to speak, aware only of itself. There is a beautiful logic to this, since each subprocess is fully described by the quantum laws. There is nothing within the branch as such to indicate that it alone does not constitute the entire history of the universe. It carries on in blithe ignorance of the other branches, which are ‘parallel worlds’ of which it sees nothing. The branches can nevertheless be very complicated. An impressive part of Everett’s paper demonstrates how an observer (modelled by an inanimate computer) within one such branch could well have the experience of being all alone in such a multiworld, doing quantum experiments and finding that the quantum statistical predictions are verified.

Any scientific theory must establish a postulate of psychophysical parallelism: it is necessary to say what elements of the physical theory correspond to actual conscious experience. Given our current meagre understanding of consciousness, we have considerable freedom in the choice we make. Everett exploited the linearity of quantum mechanics to make his particular choice. It leads, however, to what is now widely seen as a serious technical problem.

Right at the start, Everett stated that ‘The wave function is taken as the basic physical entity with no a priori interpretation.’ He aimed to show that the interpretation of the theory emerges from ‘an investigation of the logical structure of the theory’. This aim, coupled with his insistence that the wave function is the only thing that exists, creates the difficulty, since the logical structure of the theory is generally reckoned to be represented by Dirac’s transformation theory. According to it, any quantum state can indeed be regarded as made up of other states – branches in an Everett-type ‘many-worlds’ picture. The difficulty is that this representation is not unique. There are many different ways in which one and the same state, formed from the same two ‘observer’ and ‘object’ systems, can be represented as being made up of other states. We can, for example, use position states, but we can equally well use momentum states.

The fact is that quantum mechanics is doubly indefinite. First, if states of a definite kind are chosen, any state of a composite system is a unique sum of states of its subsystems. For position states, this is shown in Figures 40 to 43. The probability distribution is spread out over a huge range of possibilities in which one particle has one definite position and the other particle has another definite position. Positions are always paired together in this way. Everett resolved the apparent conflict between our experience of a unique world and this multiplicity of possibilities by associating a separate and autonomous experience with each. However, he did not address the second indefiniteness: the states shown as positions in Figures 40 to 43 could equally well be represented by, for example, momentum states. Then pairs of momentum states result. Depending on the representation, different sets of parallel worlds are obtained: ‘position histories’ in the one case, ‘momentum histories’ in the other. One quantum evolution yields not only many histories but also many families of different kinds of history.

It was surprisingly long before this difficulty was clearly recognized as the preferred-basis problem: a definite kind of history will be obtained only if there exists some distinguished, or preferred, choice of the basis, by which is meant the kind of states used in the representation. The preferred basis problem is the EPR paradox in a different guise. Everett may have instinctively assumed that the position basis is somehow naturally singled out, but there is little evidence in his paper to confirm this.

The first question that must be addressed is surely this: what is real? Everett took the wave function to be the only physical entity. The price for this wave-function monism is the preferred-basis problem. Because the wave functions of composite systems can be represented in so many ways, the application of Everett’s ideas to different kinds of representation suggests that one and the same wave function contains not only many histories, but also many different kinds of history. It leads to a ‘many-many-worlds’ interpretation. Some accept this, but I feel there is a more attractive alternative.

A DUALISTIC PICTURE

The purists among the quantum ‘founding fathers’, above all Dirac and Heisenberg, saw a close parallel between the representation of one and the same quantum state in many ways and the possibility of putting many different coordinate systems on one and the same space-time. In relativity, this corresponds to splitting space-time into space and time in different ways. After Einstein’s great triumph, no physicist would dream of saying that this could be done in one way only. Similarly, Dirac and Heisenberg argued, there is nothing in quantum theory to suggest that there is a preferred way to represent quantum states. However, the parallel may not be accurate.

First, in classical relativity, space-time represents all reality – the complete universe. In contrast, a quantum state by itself has no definite meaning until the strategic decision – say, to measure position or momentum – has been taken. The state acquires its full meaning only in conjunction with actual measuring apparatus outside the system. The system must interact with the apparatus to reveal its latent potentialities. At present, its interaction with an apparatus – essentially the rest of the universe – is not fully understood. The quantum state by itself is only part of the story. It may be premature to draw conclusions about the quantum universe from incomplete quantum descriptions of subsystems of it.

Second, quantum mechanics as presently formulated needs an external framework. Indeed, the most basic observables, those for position, momentum and angular momentum, all correspond to the ‘lent’ properties mentioned in the discussion of the EPR paradox. They could not exist without the framework of absolute space, and Mach’s principle suggests strongly it is determined by the instantaneous configurations of the universe. Time, moreover, plays an essential role in quantum mechanics yet stands quite outside the description of the quantum state. But we saw in Chapter 6 that time is really just a shorthand for the position of everything in the universe, so the configurations of the universe can be expected to play an essential and direct role in a quantum description of the universe. I cannot see how we can hope to understand the external framework of current quantum theory unless we put them into the foundations of quantum cosmology. This is what leads me to the dualistic picture of Platonia, the collection of all possible configurations of the universe, and the completely different wave function, conceived of as ‘mist’ over Platonia. In the language of Everett’s theory, this introduces a preferred basis. In answer to the question ‘what is real?’, I answer ‘configurations’. My book is the attempt to show that they explain both time and the quantum – as different sides of the same coin.