Dirac electronic sea

When he was 26 years old, Dirac made the absurd assumption that what people call a vacuum is in reality a sea of electrons occupying the negative energy continuum (known as the Dirac electronic sea). The sea was supposed to consist of an infinite number of electrons, which had to imply catastrophic consequences concerning, for example, the infinite mass of the Universe, but Dirac did not feel any doubt about his notion. " We see only those electrons that have positive energy " said Dirac. Why he was so determined Well, Dirac s concept of the sea was proposed to convince us that due to the Pauli exclusion principle, the doubly occupied sea electronic states... 

None of the many fathers of the Fock-Klein-Gordon equation dared to take into account another possibility, the one with the negative square root in Eq. (3.40), a step made by Paul Dirac. In this case the Dirac s argument about the electron sea and the Pauli exclusion principle would not work, since we have to do with the bosons We would have an abyss of negative energies, a disaster for the theory. 

We focus here on the non-relaiivistic case (Eq. (2.1)), where the lowest eigenvalue of H is bound from below (> —oo). As we remember from Chapter 3, this is not fulfilled in the relativistic case (Dirac s electronic sea), and may lead to serious difficulties in applying the variational method. 

A detailed study of the Dirac equation and its solutions will not be required it will simply be assumed, as already indicated, that the S3rstem of N electrons above the negative-energy sea may be described using a wavefunction constructed from antisymraetrized products of (positive energy) spin-orbitals of type (29). It is, however, necessary to know the basic properties of the operators Q/i, which appear in the Dirac equation... 

In a non-relativistic theory we would now continue by adding a second quantized operator for two-body interactions. In the relativistic case we need to step back and first consider the interpretation of the eigenvalues of the Hamiltonian. Dirac stated that positrons could be considered as holes in an infinite sea of electrons . In this interpretation the reference state for a system with neither positrons nor electrons is the state in which all negative energy levels are filled with electrons. This vacuum state... 

It looks like the redefinition or renormalization, as the procedure is usually called solves most of the problems associated with the Dirac sea of electrons formalism. One problem still remains, however. For any reasonable choice of one may unambiguously assign a spinor to either the positive or the negative energy subspace. Such an assignment is, however, not universally valid as can... 

---