Dirac sea
It was first postulated by the British physicist Paul Dirac in 1930[1] to explain the anomalous negative-energy quantum states predicted by the relativistically-correct Dirac equation for electrons.
[2] 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.
[nb 1] In hole theory, the solutions with negative time evolution factors[clarification needed] 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.
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.
Although this 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 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.
Dirac's solution to this was to rely on the Pauli exclusion principle.
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.
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.
[4] 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 assume that the "bare vacuum" must have an infinite positive charge density which is exactly cancelled by the Dirac sea.
Geoffrey Landis also notes[citation needed] 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 picture recaptures all the valid predictions of the Dirac sea[citation needed], 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.
for plane wave solutions with 3-momentum p. This is a direct consequence of the relativistic energy-momentum relation
The quantity ε is called the time evolution factor, and its interpretation in similar roles in, for example, the plane wave solutions of the Schrödinger equation, is the energy of the wave (particle).
In that case, the absolute value of ε can be interpreted as the energy of the wave since in the canonical formalism, waves with negative ε actually have positive energy Ep.
[7] The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformation, an identification between the creation and annihilation operators of two different free field theories.
To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states.
The price is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number.
which means that if one replaces N by 1−N for negative energy states, there is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles.
Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors, though they are never referred to as "positrons".
Dirac's original concept of a sea of particles was revived in the theory of causal fermion systems, a recent proposal for a unified physical theory.
In this approach, the problems of the infinite vacuum energy and infinite charge density of the Dirac sea disappear because these divergences drop out of the physical equations formulated via the causal action principle.
[9] These equations do not require a preexisting space-time, making it possible to realize the concept that space-time and all structures therein arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea.
