The Dirac sea is a theoretical model of the electron vacuum as an infinite sea of electrons with negative energy, now called positrons. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons (electrons traveling near the speed of light). The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, before its experimental discovery in 1932.
In hole theory, the solutions with negative time evolution factors are reinterpreted as representing the positron, discovered by Carl Anderson. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum mechanics, but it also implies that the number of particles cannot be conserved.
Dirac sea theory has been displaced by quantum field theory, though they are mathematically compatible.
Origins
Similar ideas on holes in crystals had been developed by Soviet physicist Yakov Frenkel in 1926, but there is no indication the concept was discussed with Dirac when the two met in a Soviet physics congress in the summer of 1928.
The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, an extension of the Schrödinger equation consistent with special relativity, an equation that Dirac had formulated in 1928. Although this equation was extremely successful in describing electron dynamics, it possesses a rather peculiar feature: for each quantum state possessing a positive energy , there is a corresponding state with energy -. This is not a big difficulty when an isolated electron is considered, because its energy is conserved and negative-energy electrons may be left out. However, difficulties arise when effects of the electromagnetic field are considered, 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 ever lower energy states. However, real electrons clearly do not behave in this way.
Dirac's solution to this was to rely on the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom. 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 further 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.
Inelegance of Dirac sea
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 assume that the "bare vacuum" must have an infinite positive charge density which is exactly cancelled by the Dirac sea. Since the absolute energy density is unobservable—the cosmological constant aside—the infinite energy density of the vacuum does not represent a problem. Only changes in the energy density are observable. Geoffrey Landis (author of "Ripples in the Dirac Sea", a hard science fiction short story) also notes that Pauli exclusion does not definitively mean that a filled Dirac sea cannot accept more electrons, since, as Hilbert elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have a chiral anomaly and a gauge instanton.