The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [62]
So, we have an interesting conclusion to draw. Even though we started out by allowing ourselves complete freedom, we have discovered that, because there is no way of telling the particles apart, there are in fact only two ways we can combine the clocks: we can either add them or we can add them after first winding one or the other by 180 degrees. The truly delightful thing is that Nature exploits both possibilities.
For electrons, we have to incorporate the extra twist before adding the clocks. For particles like photons, or Higgs bosons, we have to add clocks without the twist. And so it is that Nature’s particles come in two types: those which need the twist are called fermions and those without the twist are called bosons. What determines whether a particular particle is a fermion or a boson? It is the spin.
The spin is, as the name suggests, a measure of the angular momentum of a particle and it is a matter of fact that fermions always have a spin equal to some half-integer value3 while bosons always have integer spin. We say that the electron has spin-half, the photon has spin-one and the Higgs boson has spin-zero. We have been avoiding dealing with the details of spin in this book, because it is a technical detail most of the time. However, we did need the result that electrons can come in two types, corresponding to the two possible values of their angular momentum (spin up and spin down), when we were discussing the periodic table. This is an example of a general rule that says particles of spin s generally come in 2s + 1 types, e.g. spin ½ particles (like electrons) come in two types, spin 1 particles come in three types and spin 0 particles come in one type. The relationship between the angular momentum of a particle and the way we are to combine clocks is known as the spin-statistics theorem, and it emerges when quantum theory is formulated so that it is consistent with Einstein’s Theory of Special Relativity. More specifically, it is a direct result of making sure that the law of cause and effect is not violated. Unfortunately, deriving the spin-statistics theorem is beyond the level of this book – actually it is beyond the level of many books. In The Feynman Lectures on Physics, Richard Feynman has this to say:
We apologise for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments at an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation.
Bearing in mind that Richard Feynman wrote this in a university-level textbook, we must hold up our hands and concur. But the rule is simple, and you must take our word for it that it can be proved: for fermions, you have to give a twist, and for bosons you don’t. It turns out that the twist is the reason for the Exclusion Principle, and therefore for the structure of atoms; and, after all our hard work, this is now something that we can explain very simply.
Imagine moving points A and B in Figure 7.3 closer and closer together. When they are very close together, clock 1 and clock 2 must be of nearly the same size and read nearly the same time. When A and B are right on top of each other then the clocks must be identical. That should be obvious, because clock 1 corresponds to particle 1 ending up at point A and clock 2 is, in this special case, representing exactly the same thing because points A and B are on top of each other. Nevertheless, we do still have two clocks, and we must still add them together. But here is the catch: for fermions, we must give