The End of Time: The Next Revolution in Physics

Imagine yourself on a wide sandy beach on which the receding tide has left a static pattern of waves. As you are a free agent, nothing can stop you from laying out a rectangular grid on the beach and calling the direction along one axis ‘space’ and that along the perpendicular axis ‘time’. For each value of the ‘time coordinate’, you can examine the wave pattern along the one-dimensional line of ‘space’ at that ‘time’. When you move to the neighbouring line on the beach corresponding to ‘space’ at a slightly later ‘time’, you will find that the wave pattern has changed. Simply by laying out your grid and calling one direction ‘space’ and the other ‘time’, you have transformed – in your mind’s eye – a two-dimensional static picture into wave dynamics in one dimension. This can be done with wave patterns in spaces of any dimension N. One direction can always be called ‘time’, and this automatically creates ‘evolution’ in the remaining N – 1 dimensions.

Of course, if the original wave pattern is ‘choppy’ and has not been created by some rule, the choice of the ‘time’ direction will be arbitrary. Any choice will create the impression of evolution in the remaining N – 1 dimensions, but it will not obey any definite and simple law. In the semiclassical approach, there are two decisive differences from the arbitrary situation. First, the static wave pattern is the solution of a definite equation. Second, it is a somewhat special solution – called a semiclassical solution – in that it exhibits a more or less regular wave pattern. This assumption will be considered later. However, if the wave pattern satisfies the assumption, it automatically selects a direction that it is natural to call time. With respect to this direction, a genuine appearance of dynamics arises in a static situation (Box 14). The result is this. Two static wave patterns (in a space of arbitrarily many dimensions) can, under the appropriate conditions, be interpreted as an evolution in time of the kind expected in accordance with the time-dependent Schrödinger equation. The appearance of time and evolution can arise from timelessness.

BOX 14 The Semiclassical Approach

This box provides some necessary details about the semiclassical approach. It is important here that the quantum wave function is not one wave pattern but two (the red and green ‘mists’). I mentioned the ‘tennis’ played between them – the rate of change in time of the red mist is determined by the curvatures of the green mist, and vice versa. This leads to the characteristic form of a momentum eigenstate, in which both mists have perfectly regular wave behaviour but with wave crests displaced relative to each other by a quarter of a wavelength. If the red crests are a quarter of a wavelength ahead of the green crests, the waves propagate in one direction and the momentum is in that direction. If the red crests are a quarter of a wavelength behind, the waves travel in the opposite direction and the momentum is reversed. We can call this phase locking. In a momentum eigenstate, there is perfect phase locking.

The semiclassical approach shows how two approximately phase-locked static waves can mimic evolution described by the time-dependent Schrödinger equation. In Figure 44 each of the two-dimensional wave patterns is nearly sinusoidal, and they are approximately phase-locked. These waves, being solutions of the stationary Schrödinger equation, are static – they do not move. But there is nothing to stop us (as in the example of the waves on the beach) from calling the direction along the axis perpendicular to the wave crests ‘time’ and the direction along the crests ‘space’.

Figure 44 Two nearly sinusoidal wave patterns.

The key step now is to divide the total pattern of each wave into a regular part, corresponding to an imagined perfectly sinusoidal behaviour, and a remainder that is the difference between it and the actual (nearly sinusoidal) behaviour. Call this the difference pattern (there is one for each mist). If the condition of approximate phase locking holds, it turns out that the difference patterns satisfy with respect to our ‘space’ and ‘time’ an equation of the same form as the time-dependent Schrödinger equation, except for the appearance of one additional term. This term will have less and less importance, though, the more closely the assumptions of the semiclassical approach are satisfied.