When Paul Dirac developed his relativistic theory of the electron, there were certain startling features of it that he had a difficult time reconciling with the known laws of physics. Chief among these is that the Dirac equation, unlike the non-relativistic Schrodinger equation, admits solutions with both positive and negative energy. Moreover, for every positive energy state there is a corresponding negative energy state. The existence of arbitrary negative energy states for the electron is troublesome, as it permits an electron to emit an infinite amount of energy while travelling to the lowest energy state, at minus infinity, which certainly does not occur in reality.

Dirac's solution to this paradox was to invent the Dirac sea. The Pauli exclusion principle prevents two electrons from occupying the same state at the same time, thus it is possible to assume that all the negative energy states predicted by the Dirac equation are already occupied, redefining the vacuum to include an infinite number of negative-energy electrons. Since all of the negative energy states are full, a positive energy electron is prevented from dropping below zero energy by Pauli exclusion, re-establishing the behaviour observed in reality.

The Dirac sea hypothesis does not limit itself to reconciling theory with reality, though. Dirac went on to consider the case where one of the negative energy electrons in the 'sea' is excited into a positive energy state. This produces a vacancy in the Dirac sea which then appears to be a positively-charged particle. Dirac had originally hoped that this particle corresponded to the single positively-charged particle known at the time, the proton. However, when this vacancy encounters another electron, the electron would take the opportunity to decay into a negative energy state and the two particles would appear to annihilate, which was not something protons and electrons were observed to do.

The vacancy in the Dirac sea was then shown to have the same apparent mass as the electron, and this 'positron' was then discovered experimentally in 1932. While this result showed the general correctness of the Dirac theory, the infinite negative charge density of the Dirac sea was deeply uncomfortable to many physicists. When the first quantum field theories were developed in the 1930s, the Dirac theory was reformulated using the tools provided by quantum field theory (QFT). A QFT-based Dirac equation is best written in terms of two separate, but related, particle types, corresponding to the positive- and negative-energy solutions to the original Dirac equation. In this situation, no Dirac sea is necessary and the concept fell into disuse.

A very similar concept to the Dirac sea is used in solid-state physics, especially in the theory of semiconductors. The electrons in a solid act like a finite Dirac sea, and when an electron is excited into a higher-energy state it leaves behind a hole which acts as a positively-charged counterpart to the electron that was excited. In a semiconductor electron-hole pairs are made quite readily, and in many semiconductor applications the conduction due to the motion of holes is at least as important as that due to the motion of electrons.

While the Dirac sea is now considered an obsolete idea in particle physics, it was very useful for making sense of the Dirac theory of the electron, and for its prediction of the existence of antimatter. It is still useful as a teaching tool, as it is a more straightforward qualitative explanation of antimatter than the full QFT treatment.

This writeup is copyright 2008 D.G. Roberge and is released under the Creative Commons Attribution-NonCommercial-ShareAlike licence. Details can be found at http://creativecommons.org/licenses/by-nc-sa/3.0/ .

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