The End of Time: The Next Revolution in Physics

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.