We must now see if we can dispense not only with time but also with absolute space in quantum mechanics. In a timeless system the energy E is zero, and the condition in Box 13 says simply that at every point of Q the sum of the curvature number and the potential number is zero. The potential number is already in the form we need. For any possible relative configuration, the potential has a unique value: it depends on nothing else. To find the potential number, we simply calculate the potential V for each configuration and then multiply by 
, getting This part of the calculations is pleasingly self-contained because V depends only on the relative configuration. Each structure has its own potential irrespective of how we imagine the structure to be embedded in space.
However, a lack of ‘self-containment’ shows up in the curvature number. To find it, we must know how varies from position to position in the configuration space Q. This is not a self-contained process in Schrödinger’s equation because the points of his Q are defined by the particles’ positions in absolute space, which is used crucially in Q, making it hybrid. The all-important curvatures of are ultimately determined by position differences in absolute space. As a result, in standard quantum mechanics the orientations are in general entangled with the relative data that specify the particle separations. Now, besides positions, momenta and energy there is another very important quantity in quantum mechanics – angular momentum, which, being an action, always has discrete eigenvalues. It owes its existence in quantum mechanics to absolute space. We have not yet escaped from Newton’s framework.
We are now coming to another critical point. We have seen that in classical physics the action is a kind of ‘distance’ between two configurations that are nearly but not exactly the same. Absolute space is an auxiliary device that makes it possible to define such ‘distances’. This is why angular momentum exists in classical and quantum physics. However, in Chapter 7 we found an alternative definition of ‘distances’ that works in the purely relative configuration space – in Platonia – and owes nothing to absolute space. They are defined by the best-matching procedure, which uses relative configurations and nothing else. In classical physics, this makes it possible to create a purely relative and hence self-contained dynamics. We also found that a sophisticated form of best matching lies at the heart of general relativity. Best matching would appear to be a basic rule of the world.
It is therefore very tempting to see whether it can be applied in quantum mechanics. What we would like to do is establish rules for operating on wave functions defined solely on the relative configuration space. For example, for three bodies we would want to eliminate the six dimensions associated with their position and orientation in absolute space, and work just with the sides of the triangle. We shall then have a wave function defined on a three-dimensional Platonia. For that, we shall want to calculate a curvature number and a potential number. The latter will present no difficulty, since it will be the same as in ordinary quantum mechanics. The difficulty is in the curvature number. What, after all, is curvature? For any given curve, it is the rate at which its slope changes. But the key thing about a rate of change is that it is with respect to something. That something is all-important. It is a kind of ‘distance’. The ordinary quantum-mechanical ‘distance’ is simply distance in absolute space (times the mass of the particle considered). To eliminate absolute space in classical physics, we replaced it by the Machian best-matching distance. There is no reason why we should not do the same in quantum physics.
This is where the unfolding of quantum mechanics on configuration space is so important. To retain that essential property of it – the huge step that Schrödinger took – we must pass from his hybrid Q to Platonia. If we are to succeed in formulating quantum mechanics in the new arena, there must be ‘distances’ in it. But that is precisely what the best-matching idea was developed to provide. Exactly the same ‘distances’ needed to realize Mach’s principle in classical physics can be used in a version of wave mechanics for a universe without absolute space. All we have to do is measure curvatures with respect to the Machian distances created on Platonia by best matching. We then add curvatures measured in as many mutually perpendicular directions as there are dimensions in that timeless arena, and set the sum equal to minus the potential number.
In fact, it is quite easy to see that the wave functions that satisfy the Schrödinger conditions in this Machian case are precisely the eigenfunctions of ordinary quantum mechanics for which the angular-momentum eigenvalues are zero. This exactly matches our result in classical mechanics – that the best-matching condition leads to solutions identical to the Newtonian solutions with angular momentum zero. We have already seen why they must be static solutions.
The picture that emerges is very simple. The quantum counterpart of Machian classical dynamics is a static wave function ψ on Platonia. The rules that govern its variation from point to point in Platonia involve only the potential and the best-matching ‘distance’. Both are ‘topographic features’ of the timeless arena. Surveyors sent to map it would find them. They would see that the mists of Platonia respect its topography. It determines where the mist collects.
CHAPTER 16
‘That Damned Equation’
The year 1980 was another turning point in my life. It was when Bruno Bertotti and I thought we might have found a new theory of gravitation, only to learn that the two ideas on which we had based it were already an integral part of Einstein’s theory. Karel Kuchař’s intervention rounded off our work but also brought it to an end. It was something of an anticlimax. Bruno became increasingly involved in experiments using spacecraft, aimed at detecting the gravitational waves predicted by Einstein’s theory. For a year or two I actually stopped doing physics and became politically active in the newly founded Social Democratic Party (the SDP). However, the old interests soon revived. Margaret Thatcher’s decisive general election victory in 1983 hastened the process.
Two things occupied me through the 1980s. First, I wrote the book from which I quoted the comments about Kepler. It had always been my ambition to write about absolute and relative motion, and in 1984 I signed a contract with Cambridge University Press for a book of four hundred pages covering the period from Newton to Einstein and including an account of my work with Bruno. When I embarked upon it, it occurred to me that I ought to find out why Newton had said what he had. What had given him the idea of absolute space? Might it not be an idea to look at what Galileo had said? I made a wonderful mistake by asking those questions. Before I knew what was happening, my research into Galileo dragged me ever further into past history, through the Copernican revolution to the work of Ptolemy and all the way back to the pre-Socratic philosophers. By reading the actual works of scientists such as Ptolemy, Kepler and Galileo, I found that the early history of mechanics and astronomy was far more interesting than any account of it I could find by the professional historians of science. They had missed all sorts of fascinating things, and their histories were quite inadequate. Inspired by Kepler’s comment that the ways by which men discover things in the heavens are almost as interesting as the things themselves, I started to write about all the early work. I spent from 1985 to 1988 writing a completely unplanned book: The Discovery of Dynamics. My sympathetic and understanding editor at Cambridge, Simon Capelin, agreed to publish it as the first of a two-volume work. The second volume was to be the book originally proposed and should have been completed a year or two later. However, that got badly delayed by a parallel development that turned my interest to physics that does not yet exist at the same time as I was working backward to the early history.