The End of Time: The Next Revolution in Physics

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.