The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [82]
We are free to design our clock multiplication rule to make this all work: multiply the sizes of the two clocks (1 × 0.2 = 0.2) and combine the times on the two clocks such that we wind the first clock backwards by 12 o’clock minus 10 o’clock = 2 hours. This does sound a little bit like we are over-elaborating, and it is clearly not necessary when we only have one particle to think about. But physicists are lazy, and they wouldn’t go to all this trouble unless it saved time in the long run. This little bit of notation proves to be a very useful way of keeping track of all the winding and shrinking when we come to the more interesting case where there are multiple particles in the problem – the hydrogen atom, for example.
Regardless of the details, there are just two key elements in our method of figuring out the chances to find a lone particle somewhere in the Universe. First, we need to specify the array of initial clocks which codify the information about where the particle is likely to be found at time zero. Second, we need to know the propagator P(A,B,T), which is itself a clock encoding the rule for shrinking and turning as a particle leaps from A to B. Once we know what the propagator looks like for any pair of start and end points then we know everything there is to know, and we can confidently figure out the magnificently dull dynamics of a Universe containing a single particle. But we should not be so disparaging, because this simple state of affairs doesn’t get much more complicated when we add particle interactions into the game. So let’s do that now.
Figure 10.1 illustrates pictorially all of the key ideas we want to discuss. It is our first encounter with Feynman diagrams, the calculational tool of the professional particle physicist. The task we are charged with is to work out the probability of finding a pair of electrons at the points X and Y at some time T. As our starting point we are told where the electrons are at time zero, i.e. we are told what their initial clock clusters look like. This is important because being able to answer this type of question is tantamount to being able to know ‘what happens in a Universe containing two electrons’. That may not sound like much progress, but once we have figured this out the world is our oyster, because we will know how the basic building blocks of Nature interact with each other.
To simplify the picture, we’ve drawn only one dimension in space, and time advances from left to right. This won’t affect our conclusions at all. Let’s start out by describing the first of the series of pictures in Figure 10.1. The little dots at T = 0 correspond to the possible locations of the two electrons at time zero. For the purposes of illustration, we’ve assumed that the upper electron can be in one of three locations, whilst the lower is in one of two locations (in the real world we must deal with electrons that can be located in an infinity of possible locations, but we’d run out of ink if we had to draw that). The upper electron hops to A at some later time whereupon it does something interesting: it emits a photon (represented by the wavy line). The photon then hops to B where it gets absorbed by the other electron. The upper electron then hops from A to X whilst the lower electron hops from B to Y. That is just one of an infinite number of ways that our original pair of electrons can make their way to points X and Y. We can associate a clock with this entire process – let’s call it ‘clock 1’ or C1 for short. The job of QED is to provide us with the rules of the game that will allow us to deduce this clock.
Figure 10.1. Some of the ways that a pair of electrons can scatter off each other. The electrons start out on the left and always end up at the same pair of points,