The End of Time: The Next Revolution in Physics

The true heart of quantum mechanics and the way to quantum cosmology is the way in which it describes composite systems – that is, systems consisting of several particles. It is an exciting, indeed extraordinary story, though it is seldom well told. When Schrödinger discovered wave mechanics, he said it could be generalized and ‘touches very deeply the true essence [wahre Wesen] of the quantum prescriptions’. But it was not just the Bohr quantization prescriptions that came into focus: at stake here are the rules of creation. A bold claim, but one I hope to justify as the book goes on. First, we have to see how Schrödinger opened the door onto a vast new arena.

The central concept of this book is Platonia. It is a relative configuration space. The new arena that Schrödinger introduced is something similar, a configuration space (without the ‘relative’). The notion is easily explained. Each possible relative arrangement of three particles is a triangle and corresponds to a single point in the three-dimensional Triangle Land. But now imagine the three particles located in absolute space. Besides the triangle they form, which is specified by three numbers (the lengths of its sides), we now have to consider the location of its centre of mass in absolute space, which requires three more numbers, and also its orientation in absolute space, which also requires three more numbers. Location in Triangle Land needs three numbers, in absolute space it needs nine. Just as each triangle corresponds to one point in three-dimensional Triangle Land, the triangle and its location in absolute space correspond to one point in a nine-dimensional configuration space. The tetrahedron formed by four particles corresponds to one point in six-dimensional Tetrahedron Land and one point in the corresponding twelve-dimensional configuration space. For any Platonia corresponding to the relative arrangements of a certain number of particles, the matching configuration space has six extra dimensions. Schrödinger called such a space a Q, and I shall follow his example. Such a Q is a ‘hybrid Platonia’, since it contains both absolute and relative elements. This hybrid nature is very significant, as will become apparent.

The most important thing about Schrödinger’s wave mechanics is that it is formulated not in space and time, but in a suitably chosen Q and time. This is not apparent for a single particle, for which the configuration space is ordinary space. Since most accounts of quantum mechanics consider only the behaviour of a single particle, many people are unaware that the wave function is defined on configuration space. That is where ψ lives. It makes a huge difference.

An illustration using a plastic ball-and-strut model of molecules may help to bring this home. Imagine that you are holding such a model in some definite position in a room, which can represent absolute space. There are three digital displays – I shall call them ψ meters – that show red, green and blue numbers on the wall. These numbers give the intensities of the three ‘mists’ represented by ψ for the system at the time considered. Suppose you take just one ball, representing one particle of the system, and detach it from the model. Keeping all the other balls fixed, you can move the one ball around and, courtesy of the ψ meters, see how ψ changes. As you move in each direction in space, each ψ value will change. For each point of space you can find the value of ψ. The blue ψ meter will always tell you the positions for which the probability is high or low. Suppose you do this and then return the ball to its original place.

Now move a second ball to a slightly different position, and leave it there. The ψ meters will change to new values. Once again, explore space with the first ball, watching the ψ meters. The values of ψ will be (in general) quite different. The ψ values on the displays embody information. The amount is staggering. For every single position in space to which you move any one of the other balls, you get a complete new set of values in space for the ball chosen as the ‘explorer’. And any ball can be the explorer. Each explorer will have its own distinctive three-dimensional patterns of ψ for every conceivable set of positions of the others.

Now, what is a molecule? When Richard Dawkins described the haemoglobin molecule and its six thousand million million million perfect copies in our body, he said that in its intricate thornbush structure there is ‘not a twig nor a twist out of place’. That is in a molecule containing perhaps twenty thousand atoms. But molecules are even more remarkable than that. The twig and the twist are averaged structures corresponding to the most probable configuration in which the molecule will be found. In the Schrödinger picture, the molecule is not just one structure but a huge collection of potentially present structures, each with its own probability.