The End of Time: The Next Revolution in Physics

Indeed, as Dirac and ADM got to grips with the dynamics of general relativity, the problem began to take on a more concrete shape. The first fact to emerge clearly was the nature of the ‘things that change’. This was very important, since it is the ‘things that change’ that must be quantized. They turned out to be 3-spaces – everything in the universe on one simultaneity hypersurface, including the geometrical relationships that hold within it. These are the analogue of particle positions in elementary quantum mechanics. As I have mentioned, Dirac was quite startled by this discovery – it clearly surprised him that dynamics should distinguish three-dimensional structures in a theory of four-dimensional space-time. I am surprised how few theoreticians have taken on board Dirac’s comments. Many carry on talking about the quantization of space-time rather than space (and the things within it). It is as if Dirac and ADM had never done their work. Theoreticians are loath to dismantle the space-time concept that Minkowski introduced. I am not suggesting anything that he did is wrong, but it may be necessary to accommodate his insight to the quantum world in unexpected ways. One way or another, something drastic must be done.

As explained in Chapter 11, in general relativity four-dimensional space-time is constructed out of three-dimensional spaces. It turns out that their geometry – the way in which they are curved – is described by three numbers at each point of space. This fact of there being three numbers acquired a significance for quantum gravity a bit like the Trinity has for devout Christians. Intriguingly, the issue at stake is somewhat similar – is this trinity one and indivisible? Is one member of the trinity different in nature from the other two? The reason why the three numbers at each space point turned into such an issue is because it seems to be in conflict with a fact of quantum theory that I need to explain briefly.

I mentioned in Chapter 12 the ‘zoo’ of quantum particles, which are excitations of associated fields. The typical example is the photon – the particle conjectured by Einstein and associated with Maxwell’s electromagnetic field. An important property of particles is rest mass. Some have it, others do not. The massless particles must travel at the speed of light – as the massless photon does. In contrast, electrons have mass and can travel at any speeds less than the speed of light.

Now, massless particles are described by fewer variables (numbers) than you might suppose. Quantum mechanically, a photon with mass would be associated with vibrations, or oscillations, in three directions: along the direction of its motion (longitudinal vibrations) and along two mutually perpendicular directions at right angles to it (transverse vibrations). However, for the massless photon the longitudinal vibrations are ‘frozen out’ by the effects of relativity, and the only physical vibrations are the two transverse ones. These are called the two true degrees of freedom. They correspond to the two independent polarizations of light. This remark may make these rather abstract things a bit more real for the non-physicist. Humans cannot register the polarization of light, but bees can and use it for orientation.

There are many similarities between Maxwell’s theory of the electromagnetic field and Einstein’s theory of space-time. During the 1950s this led several people – the American physicist Richard Feynman was the most famous, and he was followed by Steven Weinberg (another Nobel Laureate and author of The First Three Minutes) – to conjecture that, just as the electromagnetic field has its massless photon, the gravitational field must have an analogous massless particle, the graviton. It was automatically assumed that the graviton – and with it the gravitational field – would also have just two true degrees of freedom.

From 1955 to about 1970, much work was done along these lines in studies of a space-time which is almost flat and therefore very like Minkowski space (I did my own Ph.D. in this field). In this case, the parallel between Einstein’s gravitational field and Maxwell’s electromagnetic field becomes very close, and a moderately successful theory (experimental verification is at present out of the question, gravity being so weak) was constructed for it. Within this theory it is certainly possible to talk about gravitons; like photons, they have only two degrees of freedom. However, Dirac and ADM had set their sights on a significantly more ambitious goal – a quantum theory of gravity valid in all cases. Here things did not match up. The expected two true degrees of freedom did not tally with the three found from the analysis of general relativity as a dynamical theory – as geometrodynamics.