The End of Time: The Next Revolution in Physics

CHAPTER 19

Latent Histories and Wave Packets

SMOOTH WAVES AND CHOPPY SEAS

All interpretations of quantum mechanics face two main issues. First, the theory implies the existence of far more ‘furniture’ in the world than we see. I have suggested that the ‘missing furniture’ is simply other instants of time that we cannot see because we experience only one at a time. The other issue is why our experiences suggest so strongly a macroscopic universe with a unique, almost classical history. In the very process of creating wave mechanics, Schrödinger found a most interesting connection between quantum and classical physics that cast a great deal of light on this problem. The interpretation he based on it was soon seen to be untenable, but it is full of possibilities and continues to play an important role. It is the starting point of other interpretations, including the one I advocate, so I should like to say something about it.

In the 1820s and 1830s, William Rowan Hamilton, whom we have already met, established a fascinating and beautiful connection between the two great paradigms of physical thought of his time – the wave theory of light and the Newtonian dynamics of particles. Cornelius Lanczos, a friend of Einstein and author of the fine book The Variational Principles of Mechanics, opens his chapter on these things with a quotation from Exodus: ‘Put off thy shoes from off thy feet, for the place whereon thou standest is holy ground.’ Let me quote Lanczos – he is not exaggerating:

We have done considerable mountain climbing. Now we are in the rarefied atmosphere of theories of excessive beauty and we are nearing a high plateau on which geometry, optics, mechanics, and wave mechanics meet on common ground. Only concentrated thinking, and a considerable amount of re-creation, will reveal the full beauty of our subject in which the last word has not yet been spoken. We start with … Hamilton’s own investigations in the realm of geometrical optics and mechanics. The combination of these two approaches leads to de Broglie’s and Schrödinger’s great discoveries, and we come to the end of our journey.

The italics are mine. Lanczos’s account does end with Schrödinger’s discoveries, but I think it can be taken one step further. By the way, do not worry about the call for ‘concentrated thinking’. If you have got this far, you will not fail now.

Hamilton made several separate discoveries, but the most fundamental result is simple and easy to visualize. Two characteristic situations are encountered in wave theory – ‘choppy’ waves, as on a squally sea, and regular wave patterns. Hamilton was studying the connection between Kepler’s early theory of light rays and the more modern wave theory introduced by Young and Fresnel. Hamilton assumed that light passing through lenses took the form of very regular, almost plane waves of one frequency (Figure 45).

In optics, many phenomena can be explained by such waves. To do this, we need to know how the wave crests are bent and how the wave intensity, which is measured by the square of the wave amplitude (Figure 45), varies. In general, when the wave is not very regular, the ways in which the wave crests bend and the amplitude varies are interconnected, and it is not possible to separate their behaviour. However, as the behaviour gets more regular, the amplitude changes less and simultaneously ceases to affect the bending of the wave crests. Hamilton found the equation that governs the disposition of the wave crests in this case. Now known as the eikonal equation, it is the foundation of all optical instruments – microscopes, telescopes – and also electron microscopes. Indeed, numerous effects in optics are fully explained by the bending of the wave crests. However, other phenomena, above all the diffraction and spreading of light when it passes through a small orifice, can be explained only by the full wave theory. In these phenomena the regular pattern of wave crests is broken up.

Figure 45 An example of a regular wave pattern, showing wave crests and the lines that run at right angles to them. Such patterns are characterized by two independent quantities – the wavelength and the amplitude (the maximum height of the wave).

We shall stick to phenomena in which the wave crests remain regular. Lines that run at right angles to such wave crests can be defined; they are easy to visualize (Figure 45). Hamilton’s work in the 1820s showed that these lines correspond to the older idea of light rays, and that there are two seemingly quite different ways of explaining the behaviour of light and the functioning of optical instruments. In the older, more primitive way, light is composed of tiny particles (corpuscles) that travel along straight lines in empty space, but are bent in air, water and optical instruments (made of glass). The theory of light corpuscles works because the paths they take, along Kepler’s light rays, coincide with the lines that run at right angles to the wave crests. This is the second of Hamilton’s great discoveries: if light is a wave phenomenon, there are nevertheless many occasions in which it can be conceived as tiny particles that travel along these rays.

This insight led to the distinction between wave optics and geometrical optics, which uses light rays. Innumerable experiments show that only wave optics, in which light is described by waves, can explain certain phenomena. The earlier theory of light rays simply fails under these circumstances. Equally, there are many cases in which geometrical optics, with its Keplerian light rays, functions perfectly well. We see here the typical situation that arises when a new theory supplants an old one. The new theory invariably uses very different concepts – it ‘inhabits a different world’ – yet it can explain why the old theory worked as well as it did and why it is that it fails where it does. Where the wave pattern becomes irregular, geometrical optics ceases to be valid.

Geometrical optics shows how theories that explain many phenomena impressively and simply can still give a misleading picture. As my daughter learned on those frosty nights, this had happened in ancient astronomy. Ptolemy’s epicycles gave a beautifully simple and successful theory of planetary motion, but were made redundant when Copernicus made the Earth mobile. Geometrical optics is another classic example of a ‘right yet wrong’ theory. In fact, with its confrontation and reconciliation of seemingly different worlds (particles and waves), it is one long, ongoing saga. It started with Kepler’s optics, continued with the rival optical theories of Newton (particles) and Huygens, Euler, Young and Fresnel (wave theory), and reached a first peak with Hamilton. It burst into life again in 1905 with Einstein’s notion of the light quantum, then went through another remarkable transformation in Schrödinger’s 1926 discovery of wave mechanics. I believe this saga has not yet run its course, as I will explain in the next chapter.

Now we come to Hamilton’s next discovery – the explanation of Fermat’s principle of least time, the idea that did more than anything else to foster the development of the principle of least action.

HISTORY WITHOUT HISTORY

Figure 46 shows the wave crests of a light wave in a medium in which the speed of light is the same in all directions but varies from point to point, causing the wave crests to bend. The speed of light is less where the crests are closer together. Obviously, if some particle wanted to get from A to F in the least time, travelling always at the local speed of light, it would follow the curve ABCDEF. The individual segments of this curve are always perpendicular to the wave crests, and any deviation would result in a longer travel time. But this is also exactly the route a light ray would follow, cutting the wave crests at right angles. This was another great discovery that Hamilton made – that when geometrical optics holds, the wave theory of light can explain both Fermat’s principle of least time and Kepler’s light rays. The rays follow the lines of least travel time, and these are simultaneously the lines that always run perpendicular to the wave crests.

One of the most interesting things about geometrical optics is a connection it establishes with particles in Newtonian mechanics. The characteristic property of a moving particle is that it traces out a path through space. When regular wave patterns are present, wave theory creates similar one-dimensional tracks without any particles being present at all – the tracks of the light rays. Of course, in a strict wave theory the rays are not really ‘there’, but they are present as theoretical constructs. And many phenomena can be explained rather well by assuming that particles really are there. As John Wheeler would say, one has ‘particles without particles’, or even ‘histories without histories’.

Figure 46 The explanation of Fermat’s principle of least time under the conditions of a regular wave pattern, so that geometrical optics holds.

In fact, work that Hamilton did about ten years after his optical discoveries shows how apt such a ‘Wheelerism’ is. As we saw in Part 2, classical physics is the story of paths in configuration spaces. They are Newtonian histories. Hamilton thought about what would happen if for them only one value of the energy is allowed, and made a remarkable discovery. He found that just as light rays, which are paths, arise from the wave theory of light when there is a regular wave pattern, the paths of Newtonian dynamical systems can arise in a similar fashion. I need to spell this out.

Working entirely within the framework of Newtonian dynamics, Hamilton introduced something he called the principal function. All you need to know about this function is that it is like the mists on configuration space: at each point of the configuration space, it has a value (intensity), the variation of which is governed by a definite equation. Hamilton showed that when, as can happen, the intensity forms a regular wave pattern, the family of paths that run at right angles to its crests are Newtonian histories which all have the same energy. They are not all the histories that have that energy, but they are a large family of them. Each regular wave pattern gives rise to a different family. Hamilton also found that the equation that governs the disposition of the wave crests, which in turn determine the Newtonian histories, has the same basic form as the analogous eikonal equation in optics. But whereas that equation operates in ordinary three-dimensional space, this new equation operates in a multidimensional configuration space.

Many physicists have wondered how the beautiful variational principles of classical physics arise. Hamilton’s work suggests an explanation. If the principle that underlies the world is some kind of wave phenomenon, then, wherever the wave falls into a regular pattern, paths that look like classical dynamical histories will emerge naturally. For this reason, waves that exhibit regular behaviour are called semiclassical. This is because of the close connection between such wave patterns and classical Newtonian physics. It also explains the name of the programme discussed in the previous chapter.

All the things that this book has been about are now beginning to come together. A review of the essential points may help. We started with Newton’s three-dimensional absolute space and the flow of absolute time. History is created by particles moving in that arena. Then we considered Platonia, a space with a huge number of dimensions, each point of which corresponds to one relative configuration of all the particles in the Newtonian arena. The great advantage of the concept of a configuration space, of which Platonia is an example, is that all possible histories can be imagined as paths. There are two ways of looking at the single Newtonian history that was believed to describe our universe. The first is as a spot of light that wanders along one path through Platonia as time flows. The spot is the image of a moving present. In the alternative view, there is neither time nor moving spot. There is simply the timeless path, which we can imagine highlighted by paint. Newtonian physics allows many paths. Why just one should be highlighted is a mystery. We have also seen that only those Newtonian paths with zero energy and angular momentum arise naturally in Platonia.

Hamilton’s studies opened up a new way to think about such paths. It works if the energy has one fixed value, which may be zero, and introduces a kind of mist that covers the configuration space with, in general, variable intensity. In those regions in which the mist happens to fall into a pattern with regular wave crests, there automatically arise a whole family of paths which all look like Newtonian histories. They are the paths that run at right angles to the wave crests. If you were some god come on a visit to the configuration space and could see these wave crests laid out over its landscape, you could start at some point and follow the unique path through the point that the wave crests determine. You would find yourself walking along a Newtonian history. However, your starting point, and the path that goes with it, would have to be chosen arbitrarily, because precisely when the pattern of wave crests becomes regular, the wave intensity (determined by the square of the wave amplitude) becomes uniform. There would be nothing in the wave intensity to suggest that you should go to one point or another.

Hamilton’s work opens up a way to reconcile contradictory pictures of the world. Quantum mechanics and the Wheeler-DeWitt equation suggest that reality is a static mist that covers Platonia. But all our personal experience and evidence we find throughout the universe speak to us with great insistence of the existence of a past – history – and a fleeting present. The paths that can be followed anywhere in Platonia where the mist does form a regular wave pattern can be seen as histories, present at least as latent possibilities.

I feel sure that the mystery of our deep sense and awareness of history can be unravelled from the timeless mists of Platonia through the latent histories that Hamilton showed can be there. But just how is the connection to be made? In the remainder of this chapter I shall explain Schrödinger’s valiant, illuminating, but unsuccessful attempt to manufacture a unique history out of Hamilton’s many latent histories. Then, in the next chapter, I shall consider the alternative – that all histories are present.

AIRY NOTHING AND A LOCAL HABITATION

When Schrödinger discovered wave mechanics he was well aware of Hamilton’s work, since de Broglie had used the deep and curious connection between wave theory and particle mechanics in his own proposal. De Broglie’s genius was to suggest that Hamilton’s principal function was not just an auxiliary mathematical construct but a real physical wave field that actually guided a particle by forcing it to run perpendicular to the wave crests. Schrödinger sought to exploit Hamilton’s work somewhat differently. His instinct was to interpret the wave function as some real physical thing – say, charge density. Of course, this could not be concentrated at a point, since its behaviour was governed by a wave equation, and waves are by nature spread out. Nevertheless, Schrödinger initially believed that his wave theory would permit relatively concentrated distributions to hold together indefinitely and move like a particle. His work led to the very fruitful notion of wave packets. These can be constructed using the most regular wave patterns of all – plane waves like the example in Figure 45. A plane wave has a direction of propagation and a definite wavelength. All the lines that run perpendicular to the wave crests are then latent, or potential, particle ‘trajectories’.

Because the Schrödinger equation has the vital property of linearity mentioned earlier, we can always add two or more solutions and get another. In particular, we can add plane waves. Although each separate solution is a regular wave throughout space, when the solutions are added the interference between them can create surprising patterns. This makes possible the beautiful construction of Schrödinger’s wave packets (Box 15).

BOX 15 Static Wave Packets

A wave with its latent classical histories perpendicular to the wave crests is shown at the top of Figure 47. Using the linearity, we add an identical wave with crests inclined by 5° to the original wave. The lower part of the computer-generated diagram shows the resulting probability density (blue mist). The superposition of the inclined waves has a dramatic effect. Ridges parallel to the bisector of the angle between them (i.e. nearly perpendicular to the original wave fields) appear, and start to ‘highlight’ the latent histories. In fact, these emergent ridges are the interference fringes that show up in the two-slit experiment (Box 11), in which two nearly plane waves are superimposed at a small angle, and also in Young’s illustration of interference (Figure 22).

Much more dramatic things happen if we add many waves, especially if they all have a crest (are in phase) at the same point. At that point all the waves add constructively, and a ‘spike’ of probability density begins to form. At other points the waves sometimes add constructively, though to a lesser extent, and sometimes destructively. Wave patterns like those shown in Figure 48 are obtained.

Figure 47 If two inclined but otherwise identical plane waves like the one at the top are added, the figure at the bottom is obtained. The ridges run along the direction of the light rays’ in the original plane waves. (The top figure shows the amplitude, the bottom the square of the added waves, since in quantum mechanics that measures the probability density.)

Figure 48 brings to mind a passage in A Midsummer Night’s Dream that has haunted poets for centures:

And, as imagination bodies forth

The forms of things unknown, the poet’s pen

Turns them to shapes, and gives to airy nothing

A local habitation, and a name.

The intersection of two wave fields does not result in any distinguished point, just a field of parallel ridges. There is no ‘local habitation’. But if the crests of three or more waves intersect at a common point – so that the waves are in phase there – and their amplitudes are varied appropriately, then a point becomes distinguished. A localized ‘blob’ is formed. As Schrödinger realized with growing excitement in the winter of 1925/6, this begins to look like a particle.

Figure 48 These wave patterns are obtained (from the bottom upward) by adding increasing numbers of plane waves oriented within a small range of directions. All waves have a crest where the ‘spike’ rises from the ‘choppy’ pattern. Their amplitudes also vary in a range, since otherwise ‘ridges’ like those in Figure 47 are obtained.

The pièce de résistance is finally achieved if the waves of different wavelengths move and do so with different speeds. This often happens in nature. In most media – above all in vacuum – light waves all propagate with the same speed. However, in some media the waves of different wavelengths travel at different speeds. Since waves of different wavelengths have different colours, this can give rise to beautiful chromatic effects. In quantum mechanics, the waves associated with ordinary matter particles like electrons, protons and neutrons always propagate at different speeds, depending on their wavelengths. The relationship between the wavelength and the speed of propagation is called their dispersion relation.

Figure 49 has been constructed using such a dispersion relation. The initial ‘spike’ (wave packet) at the bottom is the superposition of waves of different angles in a small range of wavelengths. The dispersion relation makes each wave in the superposition move at a different speed. At the initial time, the waves are all in phase at the position of the ‘spike’, but the position at which all the waves are in phase moves as the waves move. The ‘spike’ moves! Its positions are shown at three times (earliest at the bottom, last at the top). This wave packet disperses quite rapidly because relatively few waves have been used in its construction. In theoretical quantum mechanics, one often constructs so-called Gaussian wave packets, which contain infinitely many waves all perfectly matched to produce a concentrated wave packet. These persist for longer.

It is a remarkable fact about waves in general and quantum mechanics in particular that the wave packet moves with a definite speed, which is known as the group velocity and is determined by the dispersion law. It is quite different from the velocities of any of the individual waves that form the packet. Only when there is no dispersion and all the waves travel at the same speed is the velocity of the packet the same as the speed of propagation of the waves. These remarkable purely mathematical facts about superposition of waves were well known to Schrödinger at the time he made his great discoveries – one of which was that this beautiful mathematics seemed to be manifested in nature.

SCHRÖDINGER’S HEROIC FAILURE

This led him to propose the wave-packet interpretation of quantum mechanics. His main concern was to show how a theory based on waves could nevertheless create particle-like formations. A potential strength of his proposal was that particle-like behaviour could be expected only above a certain scale. Over short enough distances, within atoms or in colliding wave packets, the full wave theory would have to be used, but in many circumstances it seemed that particles should be present. With total clarity, which shines through his marvellous second paper on wave mechanics, he saw that if particles are associated with waves, then in atomic physics we must expect an exact parallel with geometrical optics. There will be many circumstances in which ordinary Newtonian particles seem to be present, but in the interior of atoms, for example, where the potential changes rapidly, we shall have to use the full wave theory. Schrödinger’s second paper contains wonderful insights.

Figure 49 A moving wave packet obtained by adding plane waves having slightly different orientations, wavelengths and propagation velocities. The initially sharply peaked packet disperses quite quickly, as shown in the two upper figures.

Unfortunately, his idea soon ran into difficulties. He had been aware of one from the start. For a single particle, the configuration space is ordinary space, and the idea that the wave function represents charge density makes sense. But he was well aware that his wave function was really defined for a system of particles and therefore had a different value for each configuration of them. I highlighted this earlier by imagining ‘wave-function meters’ in a room which showed the effect of moving individual atoms in models of molecules. It is difficult to see how the wave function can be associated with the charge density of a single particle in space.

Another problem actually killed the proposal. Although wave packets do travel as if they were a particle, they do spread. This was the one effect that Schrödinger failed to grasp initially. He actually did detailed and beautiful calculations for one special case – the two-dimensional harmonic oscillator, or conical pendulum. If a lead bob suspended on a weightless thread is pulled to one side and released, it will swing backwards and forwards like an ordinary pendulum. However, if it is given a sidewise jolt as well it will trace out an ellipse.

Schrödinger was able to show that for the quantum states corresponding to large ellipses it is possible to form wave packets that do not spread at all – the wave packets track round the ellipse for ever. This was a truly lovely piece of work, but misleading. Murphy’s law tripped up Schrödinger. The harmonic oscillator is exceptional and is essentially the only system for which wave packets hold together indefinitely. In all other cases they spread, doing so rapidly for atomic particles. This doomed the idea of explaining particle-like behaviour by the persistence of wave packets, as Heisenberg noted with some satisfaction. (Most of the founding fathers of quantum mechanics defended their own particular directions with great fervour. Schrödinger hated quantum jumps, and found the extreme abstraction of Heisenberg’s matrix mechanics ‘positively repulsive’.)

However, the notion of a wave packet is beautiful and transparent, and has been widely and effectively used. This has tended to make people think that Schrödinger’s original idea was still to a large degree right, and does explain why classical particle-like behaviour (restricted in its accuracy only by the Heisenberg uncertainty relation) is so often observed, especially in macroscopic bodies. There is one great difficulty, though. We can construct wave packets with strongly expressed particle-like properties, but we have to superimpose many different semiclassical solutions in just the right way. There must be a relatively small range of directions and wavelengths, adjustment of the wave amplitudes and, above all, coincidence of the phases of all waves at one point. Nothing in the formalism of quantum mechanics explains how this miraculous pre-established harmony should occur in nature. A single semiclassical solution might well arise spontaneously and naturally. But that will be associated with a whole family of classical trajectories, which exist only as formal constructions – they are at best latent histories. Quantum mechanics generally gives a wave function spread out in a uniform regular manner. Even if by some miracle we could ‘manufacture’ some wave packet, it would inevitably spread. Some further decisive idea is needed to explain how a universe described by quantum mechanics appears so classical and unique.