As I mentioned earlier, Bruno and I had been completely concerned with classical physics. We had wanted to show that Mach had been right and that his ideas could lead to new classical physics; we had given not a moment’s thought to any quantum implications they might have. Quantum cosmology was a world beyond our ken. It is strange what sparks a desire to work on something. My lack of interest in quantum gravity was particularly odd, since it was the early work done in that field which, through the remark by Dirac, quoted in the Preface, had set me on my long trek. It was the same work that had led to the work of Baierlein, Sharp and Wheeler that Bruno and I had come to see as the implementation of Mach’s ideas within general relativity. Not even working with Karel Kuchař, one of the world’s leading experts in quantum gravity, provided the stimulus I needed. Perhaps it all seemed too daunting. I needed the example and encouragement that came from a new friend, Lee Smolin.
I first met Lee a few weeks before I travelled to Salt Lake City in the autumn of 1980. It was quite a dramatic time for me since I had just narrowly escaped death through an insidious appendix that had burst without giving me any pain. My only symptoms were tiredness, slight sickness and the merest hint of stomach pain. Luckily my vigilant doctor sent me to hospital as a precaution. An X-ray proved difficult to interpret, and after quite lengthy deliberation the doctors decided to open me up. They found that any further delay could have been fatal. Seeing my state, the surgeon apparently commented that ‘this must be a very brave man’, believing I must have been in agony. In fact, I had been cheerfully reading The Times without any discomfort only half an hour before the operation. The day after I came back from hospital still convalescing, two American physicists visiting Oxford phoned to say that they had heard from Roger Penrose about my interest in Mach’s principle. Could they come and see me? They came the next day, and I greeted them in my dressing gown.
One was Lee, then a young postdoc. The meeting changed both of our lives significantly. He proved very receptive to the ideas of Leibniz and Mach to which I introduced him, while he encouraged me to see what application they might have to the problem to which he had decided to devote himself – quantum gravity. We met several times in the next few years, and collaborated on an attempt to formulate Leibniz’s philosophical system, his ‘monadology’, in mathematical form. I think we made some real progress. Lee has written about his view of things in his The Life of the Cosmos. Certain aspects of our work together were decisive in my own elaboration of the notion of time capsules and my conviction that the ultimate and only truly real things are the instants of time. As far as I am aware, Leibnizian ideas offer the only genuine alternative to Cartesian-Newtonian materialism which is capable of expression in mathematical form. What especially attracts me to them is the importance, indeed primary status, given to structure and distinguishing attributes, and the insistence that the world does not consist of infinitely many essentially identical things – atoms moving in space – but is in reality a collection of infinitely many things, each constructed according to a common principle yet all different from one another. Space and time emerge from the way in which these ultimate entities mirror each other. I feel sure that this idea has the potential to turn physics inside out – to make the interestingly structured appear probable rather than improbable. Before he became a poet, T. S. Eliot studied philosophy. He remarked, ‘In Leibniz there are possibilities.’
In 1988, when I had finished my book on the discovery of dynamics, I spent three weeks with Lee at Yale, and began to think seriously how one might make sense of the embryonic form of quantum gravity that had been developed from about the time of Einstein’s death in 1955, leading to the publication of the Wheeler-DeWitt equation in 1967. During the next four years, Lee and I had many discussions. Although we eventually followed different paths – Lee is reluctant to give up time as a primary element in physics – the ideas I want to describe in the final part of the book crystallized during those discussions. For me, their attraction stems from the inherent plausibility of Platonia as the arena of the universe and the implication of Schrödinger’s breathtaking step into a rather similar configuration space. As I see it now, the issue is simple.
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).
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.