The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [84]
The shrinking factor g looks a bit arbitrary, but it has a very important physical interpretation. It is evidently related to the probability that an electron will emit a photon, and this encodes the strength of the electromagnetic force. Somewhere in our calculation we had to introduce a connection with the real world because we are calculating real things and, just as Newton’s gravitational constant G carries all the information about the strength of gravity, so g carries all the information about the strength of the electromagnetic force.4
If we were actually doing the full calculation, we’d now turn our attention to the second diagram, which represents another way that our original pair of electrons can make their way to the same points, X and Y. The second diagram is very similar to the first in that the electrons start out from the same places, but now the photon is emitted from the upper electron at a different point in space and at a different time and it is absorbed by the lower electron at some other new place and time. Otherwise things run through in precisely the same way and we’ll get a second clock, ‘clock 2’, denoted C2.
Then, on we’d go, repeating the entire process again and again for each and every possible place where the photon can be emitted and each and every possible place where it can be absorbed. We should also account for the fact that the electrons can start out from a variety of different possible starting positions. The key idea is that each and every way of delivering electrons to X and Y needs to be considered, and each is associated with its own clock. Once we have collected together all of the clocks, we ‘simply’ add them all together, to produce one final clock whose size tells us the probability of finding one electron at X and a second at Y. Then we are finished – we will have figured out how two electrons interact with each other because we can do no better than compute probabilities.
What we have just described really is the heart of QED, and the other forces in Nature admit a satisfyingly similar description. We will come on to those shortly, but first we have a little more to discover.
Firstly, a paragraph describing two small, but important, details. Number 1: we have simplified matters by ignoring the fact that electrons have spin and therefore come in two types. Not only that, photons also have spin (they are bosons) and come in three types. This just makes the calculations a little more messy because we need to keep track of which types of photon and electron we are dealing with at every stage of the hopping and branching. Number 2: if you have been reading carefully then you may have spotted the minus signs in front of a couple of the pictures in Figure 10.1. They are there because we are talking about identical electrons hopping their way to X and Y and the two pictures with the minus sign correspond to an interchange of the electrons relative to the other pictures, which is to say that an electron which started out at one of the upper cluster of points ends up at Y whilst the other, lower, electron ends up at X. And as we argued in Chapter 7, these swapped configurations get combined only after an extra 6-hour wind of their clocks – hence the minus sign.
You may also have spotted a possible flaw in our plan – there are an infinite number of diagrams describing how two electrons can make their way to X and Y, and summing an infinite number of clocks might seem onerous to say the least. Fortunately, every appearance of a photon–electron branching introduces another factor of g into the calculation, and this shrinks the size of the resultant clock. This means that the more complicated the diagram, the smaller the clock it will contribute and the less important it will be when we come to add all the clocks up. For QED, g is quite a small number (it’s around 0.3), and so the shrinking is pretty severe as the number of branchings