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illustrated in Figure 10.4. Providing that the electron isn’t travelling too fast,5 it will typically travel around in a circle. That electrons can be deflected by a magnet is, as we have said before, the basic idea behind the construction of old-fashioned CRT television sets and, more glamorously, particle accelerators, including the Large Hadron Collider. Now imagine that we take the video footage and play it backwards. This is what ‘an electron going backwards in time’ would look like from our ‘forwards in time’ perspective. We’d now see the ‘backwards in time electron’ circle in the opposite direction as the movie advances. From a physicist’s perspective, the backwards in time video will look exactly like a forwards in time video shot using a particle which is in every way identical to an electron except that the particle appears to carry positive electric charge. Now we have the answer to our question: electrons travelling backwards in time would appear, to us, as ‘electrons of positive charge’. Thus, if electrons do actually travel back in time then we expect to encounter them as ‘electrons of positive charge’.

Figure 10.4. An electron, circling near a magnet.

Such particles do exist and they are called ‘positrons’. They were introduced by Dirac in early 1931 to solve a problem with his quantum mechanical equation for the electron – namely that the equation appeared to predict the existence of particles with negative energy. Later, Dirac gave a wonderful insight into his way of thinking, and in particular his strong conviction in the correctness of his mathematics: ‘I was reconciled to the fact that the negative energy states could not be excluded from the mathematical theory, and so I thought, let us try to find a physical explanation for them.’

Just over a year later, and apparently unaware of Dirac’s prediction, Carl Anderson saw some strange tracks in his experimental apparatus while observing cosmic ray particles. His conclusion was that, ‘It seems necessary to call upon a positively charged particle having a mass comparable with that of an electron.’ Once again, this illustrates the wonderful power of mathematical reasoning. In order to make sense of a piece of mathematics, Dirac introduced the concept of a new particle – the positron – and a few months later it was found, produced in high-energy cosmic ray collisions. The positron is our first encounter with that staple of science fiction, anti-matter.

Armed with this interpretation of time-travelling electrons as positrons, we can finish off the job of explaining Figure 10.3. We are to say that when the photon reaches Y at time T2 it splits into an electron and a positron. Each head forwards in time until time T3 when the positron from Y reaches X, whereupon it fuses with the original upper electron to produce a second photon. This photon propagates to time T4, when it gets absorbed by the lower electron.

This might all sound a little far fetched: anti-particles have emerged from our theory because we are permitting particles to travel backwards in time. Our hopping and branching rules allow particles to hop both forwards and backwards in time, and despite our possible prejudice that this must be disallowed, it turns out that we do not, indeed must not, prevent them from doing so. Quite ironically, it turns out that if we did not allow particles to hop back in time then we would have a violation of the law of cause and effect. This is odd, because it seems as if things ought to be the other way around.

That things work out just fine is not an accident and it hints at a deeper mathematical structure. In fact, you may have got the feeling on reading this chapter that the branching and hopping rules all seem rather arbitrary. Could we make up some new branching rules and tweak the hopping rules then explore the consequences? Well, if we did that we would almost certainly build a bad theory – one that would violate the law of cause and effect, for example. Quantum Field Theory (QFT) is the name for the deeper mathematical structure that underpins the hopping