In the Newtonian picture, in which history is a curve in configuration space, it is extremely hard to understand how records arise. Even if a single curve is realized, any point on it could have any number of histories passing through it. How can one instantaneous configuration of particles suggest the motions that they have? However, if we keep an open mind about the laws that determine things, a fraction of a pea’s atoms may well seem to record a history of its large-scale features. This does not mean that all its atoms had a unique history. Without change in the pea’s large-scale structure, the same large-scale history could be coded in innumerable different ways by only a tiny fraction of its atoms. In the imagery of Figure 52, there will be a whole cloud of points X in the configuration space that correspond to the same large-scale configuration and to the same history up to it. The different points in the cloud simply code the same history in different ways. What is more, for each point along the large-scale history AB there will be a corresponding cloud of points that record the same history up to that point in different ways. There will be a ‘tube’ of such points in the configuration space. No continuous ‘thread’ joins up these points in the tube into Newtonian histories. The points are more like sand grains that fill a glass tube. Each grain tells its story independently of its neighbours. In any section of the tube, the grains all tell essentially the same story but in different ways, though some may tell it with small variations.
Figure 52 The division of Platonia. The horizontal dimensions represent the large-scale configuration space, and the single vertical dimension represents the small-scale space. The ‘horizontal curves’ (AEB and CFD) represent histories of the large-scale features. Each point like A represents, say, the overall position and size of a swarm of bees. In contrast, each of the points Q, P, X, V on the vertical lines represents the huge number of small-scale details.
I think this is the way to think about history in quantum stasis. Could we but see the picture – all Platonia with its misty crannies – we should see it as it is: the lawful definite world for which Einstein, like so many physicists, longed. But it is a timeless book full of different stories that tell of time. Quantum mechanics can create the appearance of multiple histories. However, will it in quantum cosmology? Its conditions are not quite Mott and Heisenberg’s conditions.
CHAPTER 22
The Emergence of Time and its Arrow
CAUSALITY IN QUANTUM COSMOLOGY
John Bell’s account of time-capsule selection contains a very large configuration space, time, the wave function and its equation (the time-dependent Schrödinger equation) and a special initial state. This last is most important. If quantum cosmology is static, something else must replace it. We cannot impose an initial condition in the past because there is no past. But we can try something similar. Suppose that the universal configuration space had only three dimensions and not the monstrous number I have so often asked you to consider. We could then specify the wave function on a two-dimensional plane in that three-dimensional space, and use the equation satisfied by the wave function to find it at other points. This is like evolving a state in time except that the evolution is in the third, spatial direction.
If we attempt this in ordinary quantum mechanics with the stationary Schrödinger equation, which in some respects at least is like the Wheeler-DeWitt equation, the wave function starts to misbehave sooner or later. Either it becomes infinite, or it cannot be evolved continuously, or some other disaster happens: it ceases to be ‘well behaved’. The remarkable and exciting discovery that Schrödinger made was that the hydrogen atom does have a very special set of solutions that are well behaved everywhere and for which therefore no disaster happens. These very special states correspond exactly to the negative-energy states of the hydrogen atom. He had explained what had hitherto been one of the deepest mysteries of physics – the spectral lines of atoms and molecules.
My main interest here is the transformation of our notions about causality that a solution of this kind could represent. The traditional view is that what happens now was ‘caused’ by some state in the past. There is always arbitrariness in this picture because the past state is arbitrary. But suppose the world is described instead by a solution of some Wheeler-De Witt equation that is everywhere well behaved in Schrödinger’s sense. I have already pointed out that such solutions are ultra-sensitive to the domain on which they are defined – otherwise they could not remain well behaved everywhere. Such solutions present a kind of pre-established harmony.
The Wheeler-DeWitt equation then constitutes the rules of a game played in eternity. The wave function is the ball, Platonia is the pitch. If a well-behaved solution exists, then only two things can have conspired to create it: the rules of the game and the shape (the topography) of the pitch. In contrast, Bell’s time capsules are created by the rules, time, the topography and a special initial condition. What a prize if we could create time capsules by the rules and the shape of the pitch alone! Arbitrary, vertical causality (through time) would then be replaced by timeless horizontal and rational causation – across Platonia.
SOCCER IN THE MATTERHORN
It is possible. There are plenty of time capsules in Platonia. It is not just time and the special initial conditions that enable the wave function to find time capsules. The rules of the game and, above all, the pitch size and topography are most conducive to it. Indeed, the configuration space is a prerequisite. As Nevill Mott remarked, ‘The difficulty that we have in picturing how it is that a spherical wave can produce a straight track arises from our tendency to picture the wave as existing in ordinary three-dimensional space, whereas we are really dealing with wave functions in the multispace formed by the co-ordinates both of the alpha-particle and of every atom in the Wilson chamber.’ What interests me now is not so much the dimensions as the pitch’s shape. What follows is speculation. Mine. I am not aware that anyone else has made it (though Dieter Zeh considered something rather similar). I have lectured several times on the idea, and in 1994 published quite a long paper on it in the journal Classical and Quantum Gravity. A problem with the idea is that as yet it is purely qualitative. Physicists rightly want to see real calculations (which, alas, are bound to be difficult), not mere speculation, before they endorse an idea. But the more I think about the idea, the more plausible, indeed almost inescapable, it appears to be. It is about the origin of the arrow of time – and time itself.
The arrow of time, manifested in the ubiquity of time capsules, is a colossal asymmetry. It is well-nigh inexplicable in time-symmetric physics, the present rules. Since Boltzmann’s age, it has towered there, an unsealed Everest. It can be described but not yet explained.
This book has been one long, sustained effort to shed redundant concepts. We now are down to two: a static but well-behaved wave function and the configuration space. The latter is Platonia, our pitch. I look at it as a child might – what a lopsided thing it is! However I turn in my mind the notion of ‘thing’, the space of all things constructed according to one rule comes out asymmetric. All the mathematical structures built by physicists to model the world have this inherent asymmetry. One rule creates triangles, but they are all different. No matter how you arrange them, their configuration space falls out oddly. Have another look at Triangle Land (Figures 3 and 4), which is just about the simplest Platonia there can be. And what does it look like? An upturned Matterhorn. Imagine trying to play football on that pitch.