Julian Barbour - The End of Time - The Next Revolution in Physics

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.

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.