The End of Time: The Next Revolution in Physics

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.

Einstein went further than Planck in embracing discreteness. His 1905 paper, written several months before the relativity paper, is extraordinarily prescient and a wonderful demonstration of his ability to draw far-reaching conclusions from general principles. He showed that in some respects radiation behaved as if it consisted of particles. In a bold move, he then suggested that ‘the energy of a beam of light emanating from a certain point is not distributed continuously in an ever increasing volume but is made up of a finite number of indivisible quanta of energy that are absorbed or emitted only as wholes’. Einstein called the putative particles light quanta (much later they were called photons). In a particularly beautiful argument, Einstein showed that their energy E must be the radiation frequency ω times Planck’s constant: E = hω. This has become one of the most fundamental equations in physics, just as significant as the famous E = mc².

The idea of light quanta was very daring, since a great many phenomena, above all the diffraction, refraction, reflection and dispersion of light, had all been perfectly explained during the nineteenth century in terms of the wave hypothesis and associated interference effects. However, Einstein pointed out that the intensity distributions measured in optical experiments were invariably averages accumulated over finite times and could therefore be the outcome of innumerable ‘hits’ of individual light quanta. Then Maxwell’s theory would correctly describe only the averaged distributions, not the behaviour of the individual quanta. Einstein showed that other phenomena not belonging to the classical successes of the wave theory could be explained better by the quantum idea. He explained and predicted effects in ovens, the generation of cathode rays by ultraviolet radiation (the photoelectric effect), and photoluminescence, all of which defied classical explanation. It was for his quantum paper, not relativity, that Einstein was awarded the 1921 Nobel Prize for Physics.