As I explained in Chapter 2, a time capsule, as I define it, is in itself perfectly static – it is, after all, one of Plato’s forms. However, it is so highly structured that it creates the impression of motion. In the chapters that follow, we shall see if there is any hope that static quantum cosmology will concentrate the wave function of the universe on time capsules. As logical possibilities, they are certainly out there in Platonia. But will ψ find them?
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.