Propagator in Relativistic Theory

We generalise our propagator development of the nonrelativistic theory and apply it to the relativistic electron theory. Our starting point is provided by the picture of the nonrelativistic $G(x^\prime;x)$ in figure 6.2. One may say the interaction at the $i$'th point, or vertex, destroys the particle propagating up to $x_i$ and creates a particle which propagates on to $x_{i+1}$, with $t_{i+1}\ge t_i$.

In the relativistic case, there are not only the scattering processes, but also the pair creation and annihilation processes shown in figure 6.4. Diagram 6.4a shows the production of an electron-positron pair by a potential acting at point 1; the two particles of the pair then propagate to points $x$ and $x^\prime$. Diagram 6.4b shows an electron originating at $x$ and ending up at $x^\prime$. Along the way, a pair is produced by a potential acting at 1; the positron of the pair annihilates an initial electron in the field at 3; the electron of the pair propagates up to point 2, where it is destroyed by the potential. The potential at 2 creates the electron which propagates to $x^\prime$. Diagram 6.4c shows a pair produced at 1, both propagating up to 2, and then being destroyed in the field there.

Figure 6.4: Examples of space-time diagrams in positron theory for (a) pair production, (b) scattering, and (c) a closed loop.

We see that we need not only the amplitude for an electron to be created, to propagate, and to be destroyed as in the nonrelativistic case, but also the amplitude for a positron to be created, to propagate, and to be destroyed. If this positron amplitude is found, we may then attempt to associate a probability amplitude with each process: pair production, scattering, and annihilation, and to construct the total amplitude for any particular process, by summing or integrating, over all the intermediate paths which can contribute to the process.

We must determine the positron amplitude in accordance with the hole theory. Since the existence of a positron is associated with the absence of a negative-energy electron from the filled sea, we may view the destruction of a positron as equivalent to the creation of a negative-energy electron.

In addition to electron paths which zigzag forward and backward in space-time, there is also the possibility of closed loops. Processes such as these may not simply be ignored. The formalism requires them and experiments verify their existence.