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Dirac sea
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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. Dirac, Einstein and others recognised that it is related to the 'metaphysical' aether :
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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. Dirac, Einstein and others recognised that it is related to the 'metaphysical' aether :
- ... with the new theory of electrodynamics we are rather forced to have an aether.
– P.A.M. Dirac, ‘Is There An Aether?,’ Nature, v.168, 1951, p.906.
The equation relating energy, mass and momentum in special relativity is:
,
In the special case of a particle at rest (i.e. p = 0), the above equation reduces to , which is usually quoted as the familiar . However, this is a simplification because, while , we can also see that . Therefore, the correct equation to use to relate energy and mass in the Hamiltonian of the Dirac equation is:
Here the negative solution is antimatter, discovered by Carl Anderson as the positron. The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum field theory, but it also implies that the number of particles cannot be conserved .
Origins
The origins of the Dirac sea lie in the energy spectrum of the Dirac equation, an extension 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 within an atom (if spin is ignored). 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.
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. Landis 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.
The development of quantum field theory (QFT) 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.
Modern interpretation
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. In the modern interpretation, the field operator for a Dirac spinor is a sum of a creation operators and annihilation operators, in a schematic notation:
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An operator with negative frequency lowers the energy of any state by an amount equal to the frequency, while operators with positive frequency raises the energy of any state.
In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For a Fermionic field, the creation operator gives zero when the state with momentum k is already filled, while the annihilation operator gives zero when the state with momentum k is empty.
But then it is possible to reinterpret the annihilation operator as a creation operator for a negative energy particle. It still lowers the energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. 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. Instead of being states with no antiparticle, these are states which are already filled with a negative energy particle.
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. The number operator for a Fermi field is
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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 which count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum is Lorentz invariant, but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation.
Still, Dirac's idea is completely 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.
In fiction
- While Dirac's name is never used, it is obvious that the description of E. E. Smith's negasphere in Gray Lensman is a fictional interpretation of the Dirac sea.
- The term Dirac sea also appears in the anime series Neon Genesis Evangelion twice. The twelfth Angel Leliel, which absorbs the Eva Unit 01, is believed to be composed of a Dirac sea. Also, after Unit 04 was tested using an S² engine, it supposedly generated a Dirac sea. This resulted in the ambiguous disappearance of everything within an 89km radius, including the Second Branch of NERV in Nevada. Ritsuko Akagi speculates that the missing landmass had entered a Dirac sea.
- Dirac Shore is the name of a track in the Half-Life soundtrack.
- The Dirac sea plays a critical role in the visual novel Chaos;Head, being the conduit through which the main characters channel energy to project their own delusions into reality.
See also
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