But quantum cosmology gives us only the Wheeler-DeWitt equation. It is the primary equation, but as it stands it will give only a blue mist. We cannot assume some deeper equation hiding behind it that will give phase-locked red and green mists. The truth is that this part of the semiclassical approach assumes something that should be derived. Luckily, this difficulty threatens to undermine only that part of the semiclassical approach in which the specific structure of the time-dependent Schrödinger equation is recovered. The broad picture in which ‘time’ emerges from timelessness is not threatened. In fact, complex numbers, which appear in my account in the guise of the red and green mists, are so deeply ingrained in quantum mechanics that I feel fairly confident that this problem will be sorted out. What is needed is some independent argument which enforces the appearance of a complex wave function and a coupling between its components. That would then ensure the necessary phase locking.
Nevertheless, we must take care not to introduce inadvertently into quantum cosmology assumptions that may be valid only in ordinary quantum mechanics. This brings me to the second difficulty with the semiclassical approach. It concerns motion and our conviction that we experience it, and simultaneously the issue of where our sense of the passage of time comes from. To understand the answer to this question is to understand time. It is all very well for me to speak about static wave patterns in a mist that hangs over Platonia. Such patterns will indeed, where they are sufficiently regular, define unambiguously a direction that may be called ‘time’. But even if the wave pattern is rather regular, we could not look at it and say that it distinguishes a direction of time. The one direction at right angles to the wave crests will look the same whichever way we face. There will be no signs set up on the distant horizons saying PAST and FUTURE. This is the issue we must now address.
It will be helpful to think about what it is that determines which way quantum wave packets move. Quantum mechanics is very different from classical mechanics in this respect. A classical initial condition consists of an initial position and an initial velocity. You know which way a particle will move because it is specified in the velocity. However, in quantum mechanics the initial condition is simply the values of the two components of the wave function, the red and green mists, everywhere at the initial time. Data like this seem to correspond to giving only the position in classical mechanics. Yet wave packets move under the rules Schrödinger prescribed.
In fact, the way in which a wave packet will move is coded in the relative positioning of the crests and troughs of the red and green mists. We see this most clearly in momentum eigenstates. If the red crests are ahead of the green crests, they go one way, but if the crest positioning is reversed, they go the other way.
As we have seen, states very like momentum eigenstates play a crucial role in the semiclassical approach. In all the original papers, these states also played another role – they were used to model situations corresponding to either expanding or contracting universes. All physicists and astronomers are convinced that we live in an expanding universe. There is certainly very good evidence in many different forms to support this view. The formalism of quantum cosmology must be capable of reflecting this aspect of the observed universe. There must be something that codes expansion or contraction of the universe or, rather (in the timeless interpretation), codes the observed evidence that leads us to say the universe is expanding.
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.