If the world were purely classical, I think we would have to say no, and that the unity Minkowski proclaimed so confidently is the deepest truth of space-time. The 3-spaces out of which it can be built up in so many different ways are knitted together by extraordinarily taut interwoven bonds. This is where the deep dilemma lies. Four decades of research by some of the best minds in the world have failed to resolve it. On the one hand, dynamics presupposes – at the foundation of things – three-dimensional entities. Knowing nothing about general relativity, someone like Poincaré could easily have outlined a form of dynamics that was maximally predictive, flexible, refined and made no use of eternal space or time. Such dynamics, constrained only by the idea that there are distinct things, must have a certain general form. A whole family of theories can be created in the same Machian mould.
On the other hand, a truly inspired genius might just have hit on one further condition. Let dynamics do all those things with whatever three-dimensional entities it may care to start from. But let there be one supreme overarching principle, an even deeper unity. All the three-dimensional things are to be, simultaneously with all their dynamical properties, mere aspects of a higher four-dimensional unity and symmetry.
If certain simplicity conditions are imposed, only one theory out of the general family meets this condition. It is general relativity. It is this deeper unity that creates the criss-cross fabric of space-time and the great dilemma in the creation of quantum gravity. As we shall see, quantum mechanics needs to deal with three-dimensional things. The dynamical structure of general relativity suggests – and sufficiently strongly for Dirac to have made his ‘counter-revolutionary’ remark – that this may be possible. Yet general relativity sends ambivalent signals. Its dynamical structure says ‘Pull me apart’, but the four-dimensional symmetry revealed by Minkowski says ‘Leave me intact.’ Only a mighty supervening force can shatter space-time.
Note added for this printing. New work summarized on p. 358 could significantly change the situation discussed in this final section of the chapter. It suggests that the timeless Machian approach is capable of leading to a complete derivation of general relativity and that it is not necessary to presuppose ‘a higher tour-dimensional unity and symmetry.’ Since this new work has only just been published and has not yet been exposed to critical examination, I decided to leave the original text intact. However, as already indicated in the note at the end of the Preface, this new work does have the potential to strengthen considerably the arguments for the nonexistance of time.
PART 4
Quantum Mechanics and Quantum Cosmology
If the difference between Newtonian and Einsteinian physics is great, quantum mechanics seems separated from both by a chasm. Most accounts of it, however, do not question the framework, essentially absolute space and time, in which it was formulated. They describe how very small systems – mostly atoms and molecules – behave in an external framework. This may make quantum mechanics appear more baffling than need be.
If quantum mechanics is universally true and applies not only to atoms and molecules but also to apples, the Moon, the stars and ultimately the universe, then we ought to consider quantum cosmology. What does the quantum mechanics of the universe look like? It cannot be formulated in an external framework. Like classical physics, quantum cosmology needs a description without a framework. We shall see that many apparent differences between classical and quantum mechanics then appear in a different light. What remains is one huge difference. We shall soon begin to get to grips with it.
CHAPTER 12
The Discovery of Quantum Mechanics
About a hundred years ago, a dualistic picture of the world took shape. The electron had just been discovered, and it was believed that two quite different kinds of thing existed: charged particles and the electromagnetic field. Particles were pictured as little billiard balls, possessing always definite positions and velocities, whereas electromagnetic fields permeated space and behaved like waves. Waves interfere, and recognition of this had led Thomas Young to the wave theory of light (Figure 22).
By the end of the nineteenth century, the evidence for the wave theory of light was very strong. However, it was precisely the failure of light, as electromagnetic radiation, to behave in all respects in a continuous wavelike manner that led first Max Planck in 1900 and then Einstein in 1905 to the revolutionary proposals that eventually spawned quantum mechanics. A problem had arisen in the theory of ovens, in which radiation is in thermal equilibrium with the oven walls at some temperature. Boltzmann’s statistical methods, which had worked so well for gases, suggested that this could not happen, and that to heat an oven an infinite amount of energy would be needed. The point is that radiation can have any wavelength, so radiation with infinitely many different wavelengths should be present in the oven. At the same time, the statistical arguments suggested that, on average, the same finite amount of energy should be associated with the radiation when in equilibrium. Therefore there would be an infinite amount of energy in the oven – clearly an impossibility. Baking ovens broke the laws of physics! Planck was driven to assume that energy is transferred between the oven walls and the radiation not continuously but in ‘lumps’, or ‘quanta’.
Accordingly, he introduced a new constant of nature, the quantum of action, now called Planck’s constant, because the same kind of quantity appears in the principle of least action. Until Planck’s work, it had been universally assumed that all physical quantities vary continuously. But in the quantum world, action is always ‘quantized’: any action ever measured has one of the values 0, ½h, h, ¾h, 2h, .... Here h is Planck’s constant. (The fact that half-integer values of h, i.e. ½h, ¾h, ..., can occur in nature was established long after Planck’s original discovery. By then it was too late to take half the original quantity as the basic unit.) The value of h is tiny.
Most people are familiar with the speed of light, which goes seven times round the world in a second or to the Moon and back in two and a half seconds. The smallness of Planck’s constant is less well known. Comparison with the number of atoms in a pea brings it home. Angular momentum is an action and can be increased only in ‘jerks’ that are multiples of h. Suppose we thread a pea on a string 30cm long and swing it in a circle once a second. Then the pea’s action is about 1032 times h. As we saw, the atoms in a pea, represented as dots a millimetre apart, would comfortably cover the British Isles to a depth of a kilometre. The number 1032, represented in the same way, would fill the Earth – not once but a hundred times. Double the speed of rotation, and you will have put the same number of action quanta into the pea’s angular momentum. It is hardly surprising that you do not notice the individual ‘jerks’ of the hs as they are added.
When people explain how our normal experiences give no inkling of relativity and quantum mechanics, the great speed of light and the tiny action quantum are often invoked. Relativity was discovered so late because all normal speeds are so small compared with light’s. Similarly, quantum mechanics was not discovered earlier because all normal actions are huge compared with h. This is true, but in a sense it is also misleading. For physicists at least, relativity is completely comprehensible. The mismatch between the relativistic world and its non-relativistic appearance to us is entirely explained by the speed of light. In contrast, the mere smallness of Planck’s constant does not fully explain the classical appearance of the quantum world. There is a mystery. It is, I believe, intimately tied up with the nature of time. But we must first learn more about the quantum.