The End of Time: The Next Revolution in Physics

You can play different games in one and the same arena. You can also adjust the rules of a game as played in one arena so that it can be played in a different arena. Both general relativity and quantum mechanics are complex and highly developed theories. In the forms in which they were originally put forward, they seem to be incompatible. What I found to my surprise was that it does seem to be possible to marry the two in Platonia. The structures of both theories, stripped of their inessentials, mesh. What if Schrödinger, immediately after he had created wave mechanics, had returned to his Machian paper of only a year earlier and asked himself how Machian wave mechanics should be formulated? His Machian paper implicitly required Platonia to be the arena of the universe, while any wave mechanics simply had to be formulated on a configuration space. Such is Platonia, though it is not quite the hybrid Newtonian Q he had used. But the structure of Machian wave mechanics would surely have been immediately obvious to him, especially if he had taken to heart Mach’s comments on time. As a summary of the previous chapter, here are the steps to Machian wave mechanics in their inevitable simplicity.

For a system of N particles, the Schrödinger wave function in the Newtonian case will in general change if the relative configuration is changed, if the position of its centre of mass is changed, if its orientation is changed, and if the time is changed. Mathematicians call these things the arguments of the wave function. They constitute its arena. To see what really counts, we can write the wave function in the symbolic way that mathematicians do:

ψ (relative configuration, centre of mass, orientation, time)

(1)

But if the N particles are the complete universe, there cannot be any variation with change of centre of mass, orientation or time for the simple reason that these things do not exist. The Machian wave function of the universe has to be simply

ψ (relative configuration)

(2)

Note the grander ψ. This is the wave function of the universe. It has found its home in Platonia.

I have met distinguished theoretical physicists who complain of having tried to understand canonical quantum gravity, the formalism through which the Wheeler-DeWitt equation was found, and have given up, daunted by the formalism and its seemingly arcane complexity. But, as far as I can see, the most important part boils down simply to the passage from the hybrid (1) to the holistic (2).

‘THAT DAMNED EQUATION’

This is a bold claim, but the fact is that it still remains the most straightforward way to understand the Wheeler-DeWitt equation. To conclude Part 4, I shall say something about this remarkable equation and the manner of its conception, which unlike the hapless Tristram Shandy’s was inevitable, being rooted in the structure of general relativity. You may find this section a little difficult, which is why I have just given the simple argument by which I arrive at its conclusion. Just read over any parts you find tough.

That there was a deep problem of time in a quantum description of gravity became apparent at the end of the 1950s in the work of Dirac and Arnowitt, Deser and Misner (ADM) described in Chapter 11. The existence of the problem was – and still is – mainly attributed to general covariance. The argument goes as follows. The coordinates laid down on space-time are arbitrary. Since the coordinates include one used to label space-time in the time direction and all coordinates can be changed at whim, there is clearly no distinguished label of time. This is what leads to the plethora of paths when a single space-time is represented as histories in Platonia. However, the real root of the problem lies in the deep structure of general relativity that we considered in the same chapter.