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Before getting into the details, let’s sketch how this is going to pan out. The uppermost picture represents one of the myriad ways that the initial pair of electrons can make their way to X and Y. The other pictures represent some of those ways. The crucial idea is that for each possible way that the electrons can get to X and Y we are to identify a quantum clock – C1 is the first in a long list of clocks.2 When we’ve got all of the clocks, we are to add them all together and obtain one ‘master’ clock. The size of that clock (squared) tells us the probability of finding the pair of electrons at X and Y. So once again we are to imagine that the electrons make their way to X and Y not by one particular route, but rather by scattering off each other in every possible way. If we look at the final few pictures in the figure, we can see a variety of more elaborate ways for the electrons to scatter. The electrons not only exchange photons, they can emit and reabsorb a photon themselves, and in the final two figures something very odd is happening. These pictures include the scenario where a photon appears to emit an electron which ‘goes in a circle’ before ending up where it started out – we shall have more to say about that in a little while. For now, we can simply imagine a series of increasingly complicated diagrams corresponding to cases where the electrons emit and absorb huge numbers of photons before finally ending up at X and Y. We’ll need to contemplate the multifarious ways that the electrons can end up at X and Y, but there are two very clear rules: electrons can only hop from place to place and emit or absorb a single photon. That’s really all there is to it; electrons can hop or they can branch. Closer inspection should reveal that none of the pictures we have drawn contravenes those two rules because they never involve anything more complicated than a junction involving two electrons and a photon. We must now explain how to go about computing the corresponding clocks, one for each picture in Figure 10.1.

Let’s focus on the uppermost picture and explain how to determine what the clock associated with it (clock C1) looks like. Right at the start of the process, there are two electrons sitting there, and they will each have a clock. We should start out by multiplying them together according to the clock multiplication rule to get a new, single clock, which we will denote by the symbol C. Multiplying them makes sense because we should remember that the clocks are actually encoding probabilities, and if we have two independent probabilities then the way to combine them is to multiply them together. For example, the probability that two coins will both come up heads is ½ × ½ = ¼. Likewise, the combined clock, C, tells us the probability to find the two electrons at their initial locations.

The rest is just more clock multiplication. The upper electron hops to A, so there is a clock associated with that; let’s call it P(1,A) (i.e. ‘particle 1 hops to A’). Meanwhile the lower electron hops to B and we have a clock for that too; call it P(2,B). Likewise there are two more clocks corresponding to the electrons hopping to their final destinations; we shall denote them by P(A,X) and P(B,Y). Finally, we also have a clock associated with the photon, which hops from A to B. Since the photon is not an electron, the rule for photon propagation could be different for the rule for electron propagation so we should use a different symbol for its clock. Let’s denote the clock corresponding to the photon hop L(A,B).3 Now we simply multiply all the clocks together to produce one ‘master’ clock: R = C × P(1, A) × P(2, B) × P(A, X) × P(B, Y) × L(A, B). We are very nearly done now, but there remains some additional clock shrinking to do because the QED rule for what happens when an electron emits or absorbs a photon says that we should introduce a shrinking factor, g. In our diagram, the upper electron emits the photon and the