I am working with my director, Nic Stacey, and exec producer, Paul Sen, on a way of visualising on television the way combining quantum mechanics and special relativity necessitates the existence of anti-particles. I know how the textbooks do it, but not very visual. Here is a discussion of two of the ways this stuff can be explained.
By 1927, quantum mechanics was developing along two separate lines. On the one hand, Erwin Schrödinger had come up with a wave theory to describe quantum entities like electrons, describing them not as tiny particles but as waves of pure energy. At the same time, Werner Heisenberg was working with abstract mathematics and had developed a theory based on matrices (basically arrays of numbers). But it was Paul Dirac who saw how to combine these two approaches: Schrödinger’s waves and Heisenberg’s mathematical matrices. But his real motives were different.
You see all this was just a couple of decades after Einstein’s work on relativity theory in which he had shown than when a body moves at close to the speed of light, the rules of the game change. And the only way to describe such a fast moving object properly was to unify the three dimensions of space with the one dimension of time into what is called 4-D space-time.
So, we have the Schrödinger and Heisenberg who could to describe how subatomic entities like electrons move and behave and interact, and we have Einstein’s relativity theory to describe what happens when objects move very fast. BUT… what if it is an electron that is moving very fast? Dirac knew he needed to combine quantum mechanics with relativity. What we say is that he was looking for a relativistic quantum theory. In early 1928, Dirac published his relativistically invariant equation for the electron. What he did was unify If we ignore relativity, then quantum mechanics is described by the Schrödinger equation. Here the energy (E) of a free particle (one just plodding along minding its own business and not under the influence of any force) is related to its momentum via the relation: energy equals the square of momentum divided by twice the mass (E=p2/2m). But Dirac wanted to describe what happens if a particle like an electron moved very fast, close to the speed of light. It is here that we have to take Special Relativity into account. According to Einstein, the total energy of the electron is now related to its momentum in a more complicated way, because we also have to consider the energy frozen in the electron as its mass (the famous mc2 bit). And the awkwardness is that the total energy is not just the sum of the old expression and this mc2, but comes in as the sum of their squares, namely: E2 = (mc2)2 + (pc)2. If you can remember school geometry then this is similar to Pythagoras’ theorem whereby the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides. So, why is this so awkward?
Feynman said that if a particle only has positive energies (as it propagates from the first point to the second) then the QM formalism insists on the possibility of ftl travel. Of course we cannot really talk about a particle moving ftl. Instead, as Feynman does, we say there is a non-zero prob that the particle could find itself at a space-like separated point. I must say I don’t like this violation of relativity explanation in the least.
OK, the reason Feynman does this is because he can use this business about another frame of reference in which it looks as though the second event happens first. But rather than a particle arriving at this point then moving on, it seems like a particle and antiparticle spontaneously appear at this point (pair creation). The antiparticle is basically the same as a particle travelling backwards in time. The anti-particle moves forward in time until it reaches the later time (that was the earlier time in first frame), whereby it annihilates the original particle. So, with this argument he shows necessity of existence of antiparticles: they are just particles moving faster than light as viewed from different inertial reference frames.
All this is fine, but the connection with the Dirac equation is lost. Basically, Feynman does use the relativistic energy equation E2 = (mc2)2 + (pc)2, but he then says he insists on positive energies only and that this is what gives rise to antiparticles. Whereas, for Dirac it was the insistence on retaining negative energies that suggested need for antiparticles. I find this a little puzzling to reconcile and welcome comments.