All the models of Platonia that we have considered include a dimension that we may call the ‘size’ of the universe. In fact, instead of representing Triangle Land by means of the sides of triangles, we could equally well – and more appropriately here – use two angles (the third is found by subtracting their sum from 180°) and the area of the triangles. The area is one direction, or dimension, in Triangle Land. Expansion or contraction of the universe then corresponds to motion along the line of increasing or decreasing size. The size dimension begins at the point of zero size – what I have called Alpha, or the centre of Platonia – and then proceeds all the way to infinity.
In the semiclassical approach, it was rather reasonably assumed that the regular wave pattern needed for ‘time’ to emerge from timelessness would develop along the direction of increasing or decreasing size. This is a fair working hypothesis. What worried me was the way in which expanding and contracting universes were modelled – by analogy with momentum eigenstates in ordinary quantum mechanics. Expansion or contraction were supposed to be coded in the relative positions of wave crests.
It is certainly possible to imagine two static wave patterns – our red and green mists – whose crests are perpendicular to lines that seem to emanate from Alpha. This was done by nearly all the researchers who used the semiclassical approach, and they assumed that one relative positioning of the ‘red’ and ‘green’ crests would model a universe expanding out of the Big Bang, while the opposite positioning would model a universe headed for the Big Crunch (the name given to one possible fate of the universe, in which it recollapses to a state of infinite density and zero size). Thus, momentum-like semiclassical states were used to achieve three different things at once: the emergence of ‘time’, the recovery of the time-dependent Schrödinger equation, and modelling expanding and contracting universes. I believe that only the first is soundly based. I have some concern about the second. I think the third is definitely wrong.
The point is that the position of the ‘green crests’ ahead of or behind the ‘red crests’ by itself has no significance. In ordinary quantum mechanics the wave function depends not only on the spatial position but also on the time. What really moves wave packets is the relation of the time dependence to the space dependence. It is not the case that if in some wave packet the green crests are ahead of the red crests then the wave packet is bound to move one way. This happens only because the time-dependent Schrödinger equation is written in a particular form. But this is a pure convention. All observed phenomena are described just as well by an alternative choice, analogous to changing ends in tennis. The two choices are identical in their consequences. They only differ in the relative positions of the red and green crests, but this is offset by reversing the time dependence. The real physics is unchanged. Without the time dependence, the positions of the crests cannot determine the direction of motion.
But this presents us with a real dilemma in static quantum cosmology, in which there is no external time and no time dependence to determine which way wave packets move. There is simply no motion or change at all. We have to find a different explanation for why we think there is motion in the world and that the universe expands.
One thing is clear: the origin of our belief that the universe is expanding cannot be coded in the relative positioning of the crests of the two waves, for the designations ‘red’ and ‘green’ are purely conventional. The ‘colours’ could be swapped, and nothing observable would change. The argument that mere static positioning of crests can correspond to what we call expansion of the universe is a chimera. This was clearly recognized in 1986 by my German physicist friend Dieter Zeh, who commented that it has meaning only if an absolute time exists. It really is necessary to think very differently about these things if time is abolished once and for all as an independent element of reality.
From 1988 to 1991 I was absorbed by this issue. I became more and more convinced that a decisive new idea was needed, but for a long time could find no answer that satisfied me. I formulated the problem this way. I imagined myself watching some phenomena involving motion in a very essential and vital way – a display of acrobatics, say, or the flight of a kingfisher. I then imagined being struck dead instantaneously and my ‘soul’ being carried down to a kind of Plato’s cave. Here I would find omniscient mathematicians examining a model of Platonia all covered with these red, green and blue quantum mists that I have asked you to conjure up in your mind’s eye. They are examining the solution of the Wheeler-DeWitt equation corresponding to the universe in which I had just been taken from life. I then asked myself this: what precise thing in that mysterious pattern of mists blanketing Platonia corresponds to my being aware of seeing the kingfisher in flight? Where – in a timeless static world – is the appearance of motion coded? Where can I see the kingfisher’s colours flashing in the sunlight?
As we have noted, in standard quantum mechanics the information about wave-packet motion is coded in the relative positioning of the red and green mists. This was the questionable assumption taken over in the semiclassical approach. However, there is much more to quantum mechanics than just the wave function at one instant (the pattern of red and green mists). We have already seen how time is needed if such relationships are to be translated into wave-packet motion. But even that is not enough, for the wave function acquires definite meaning only through prescriptions about the measurements that will be made on the system. These take the form of statements about the positions and construction of measuring instruments that behave classically and are external to the quantum system.
It is obvious that in quantum cosmology the whole superstructure of an external time, and of measuring instruments outside the considered system, must go. The instruments must be subsumed into the quantum system (which becomes the complete universe), and we must get to grips with a static wave function. Does this leave any scope for making a connection between actual experiences and the bare bones of embryonic quantum gravity as found by DeWitt?
I believe it does. Is not our most primitive experience always that we seem to find ourselves, in any instant, surrounded by objects in definite positions? Each experienced instant is thus of the nature of an observation, a discovery, even – we establish where we are. Moreover, what we observe is always a collection, or totality, of things. We see many things at once. In fact, most humans, indeed nearly all animals, have a wonderfully developed spatial awareness. In writing this book I have relied heavily on you possessing this gift – time and again I have asked you to imagine configurations of the universe as entities. They are all the places in Platonia.
When, therefore, I find myself in Plato’s cave and see his demesne of Platonia laid out before me, I can, using my vivid memory of the kingfisher flashing between the banks of the stream where I stood, identify the instant in which death took me. By ‘identify the instant’, I mean recognize the configuration of riverbank, sunlight and shadow, rippled water and kingfisher’s wings – all frozen in the position I last witnessed. As always, I insist that instant of time simply means configuration of the universe. This part of the problem of finding a connection between the psychical experience and the model of physical reality is relatively straightforward. There is little or no problem in the representation of position.
The real problem, then, is in the representation of motion. We seem to have exhausted all the resources of static quantum cosmology simply to put everything into place on the riverbank. Quantum mechanics does permit us to gain total information about position, but only at the expense of total loss of information about motion. We seem to have nothing left over to enable the kingfisher to fly. This is the crux of the matter. Classical physics presupposes both positions and motions, matching our experience that we see both at once. But quantum mechanics – in its present standard form – has this curious halving of the accessible data.