The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. It was invented by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.
The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, a special case of the Schrödinger equation that is consistent with special relativity, that Dirac had formulated in 1928. Although the equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for each quantum state possessing a positive energy E, there is a corresponding state with energy -E. This is not a big difficulty when we are looking at an isolated electron, because its energy is conserved and we can simply choose not to introduce any negative-energy electrons. However, it becomes serious when we start to think about how to include the effects of the electromagnetic field, because a positive-energy electron would be able to shed energy by continuously emitting photons, a process that could continue without limit as the electron descends into lower and lower energy states. Real electrons clearly do not behave in this way.
Dirac's solution to this was to turn to the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle, which means that no two electrons can share a single energy state. Dirac hypothesized that what we think of as the "vacuum" is actually the state in which all the negative-energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy.
Dirac also pointed out that a situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively-charged particle. Initially, Dirac identified this hole as a proton. However, Robert Oppenheimer pointed out that an electron and its hole would be able to annihilate each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atoms would not exist. Hermann Weyl also noted that a hole should act as though it has the same mass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932 when the positron was discovered by Carl Anderson, with all the physical properties predicted for the Dirac hole.
Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite negative electric charge filling all of space. In order to make any sense out of this, one must postulate that the electrons in the sea do not produce any net electric field, or contribute to the total energy or momentum of a system. They also must not interact with one another, or the interactions would be infinitely strong, which would throw into doubt the picture of individual, more or less independent particles that we started out with (the Dirac equation).
The development of quantum field theory in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture is much more convincing, especially since it recaptures all the valid predictions of the Dirac sea, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular, the problem of the vacuum possessing infinite energy is not so much resolved as swept under the carpet.
It is interesting to note that Dirac's idea is absolutely correct in the context of solid state physics, where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge – the charge of the ionic lattice – that cancels out the electric charge of the sea.
It also appears in the sci-fi anime Neon Genesis Evangelion - noted for its high-quality technobabble - as an expanding void that sucks up matter during the attack of the 12th Angel, Leliel, in episode 16. It is mentioned again in episode 17 as a possible reason for the mysterious disappearance of NERV's Nevada branch.