Interaction between the positrons in the Dirac theory

Question

It seems that Dirac did not consider the interaction between the positrons, right?

How could he ignore them?

Answer 1

The Dirac equation is the equation through which the physicists (Dirac, in particular) discovered positrons – the first particles of antimatter – and Dirac got a well-deserved Nobel prize for that breakthrough. According to this theory, positrons are in some way related to electrons with the opposite sign of the energy or moving backwards in time.

More rigorously, there is a precise symmetry between positrons and electrons according to the Dirac equation, the charge conjugation symmetry $C$, that Dirac was very well aware of. Because Dirac knew about the interactions between the electrons and he was the first man who found the symmetry between matter and antimatter (not to mention antimatter itself), we may be sure that he knew about the interactions between the positrons, too.

In particular, the original Dirac equation describes a single electron interacting with the (classical) external electromagnetic field and the same equation also describes a single positron interacting with the external electromagnetic field. The Dirac field used in a quantum field theory describes many electrons and positrons (and other particles) and they interact the way they should.

Maybe the question is about a possible interaction between the positrons in the Dirac sea, the set of fully occupied negative-energy one-electron states of the quantum field. First of all, if we describe the quantized Dirac field in this way, the particles in the Dirac sea should be called electrons, not positrons.

Second, these negative-energy electrons could be said to repel each other and contribute a positive electrostatic potential energy from the repulsion. However, the actual charge density of the Dirac sea has to be renormalized to $dQ/dV = 0$. In terms of the Dirac sea, one must modify the formula by adding a "background charge" that exactly cancels the Dirac sea electrons' charge. Because it's the total charge density that enters the formula for the electrostatic potential energy, the latter is zero, too.

So this potential energy is cancelled in the ground state. It actually has to be done e.g. to preserve the electromagnetic $U(1)$ gauge symmetry (which is needed for the consistency of QED or any quantum field theory with a gauge symmetry). Also, the hypothetical repulsion doesn't affect the individual excited electrons or positrons. Because the theory including the Dirac sea is Lorentz-covariant, it follows that in an electron's or positron's rest frame, the total force from all the electrons in the Dirac sea has to be $\vec F=0$.

At most, the contribution from the repulsion could contribute a vector $p^{\prime\mu}$ that is parallel to the original $p^\mu$ of the electron or the positron. This is equivalent to a change of the length of the vector, i.e. a renormalization of the mass. In quantum field theory, such effects are represented by corrections to the electron-positron Dirac propagator. The corrections yield various terms, mostly divergent terms, and by the process of renormalization (that Dirac didn't like but is surely needed for any precision calculations), we may impose the experimentally measured mass of the electron-positron and guarantee that all these possible corrections are inconsequantial.

The correct treatment of these higher-order effects is difficult but it justifies the well-definedness of the Dirac free field theory starting point.