Dirac sea
Encyclopedia
The Dirac sea is a theoretical model of the vacuum
as an infinite sea of particles with negative energy. 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
electron
s. 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 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 was used to predict the existence of 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.
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 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 photon
s, 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 fermion
s, 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 atom
s 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.
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
.
, 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 creation operators and annihilation operators, in a schematic notation:
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 that 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:
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. 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 semiconductor
s, 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.
series Neon Genesis Evangelion
as being a component of the angel that attacks in that episode.
The Dirac Sea is an important factor in the visual novel ChaoS;HEAd
. People known as gigalomaniacs have the ability to interfere with the Dirac Sea to induce a "real boot"; in other words, bring delusions to reality. The "real boot" is done by projecting particles of delusions into a person's blind spot, making it real. Likewise, antiparticles of the same form can be projected, the matter and antimatter will destroy each other, and the delusion will be destroyed. One of the heroines of the game, Aoi Sena also mentions that "the world is governed by three numbers: 0, 1, and -1.", which is a reference to the diagram of the Dirac sea. An item within the game, called a Di-Sword, helps to interfere with the Dirac Sea and create the "-1".
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
as an infinite sea of particles with negative energy. It was first postulated by the British
United Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
for relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. The positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
, the antimatter
Antimatter
In particle physics, antimatter is the extension of the concept of the antiparticle to matter, where antimatter is composed of antiparticles in the same way that normal matter is composed of particles...
counterpart of the electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
, was originally conceived of as a hole
Electron hole
An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics, chemistry, and electrical engineering. The concept describes the lack of an electron at a position where one could exist in an atom or atomic lattice...
in the Dirac sea, well before its experimental discovery in 1932.
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 HamiltonianHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the Dirac equation is:
Here the negative solution was used to predict the existence of antimatter
Antimatter
In particle physics, antimatter is the extension of the concept of the antiparticle to matter, where antimatter is composed of antiparticles in the same way that normal matter is composed of particles...
, discovered by Carl Anderson
Carl David Anderson
Carl David Anderson was an American physicist. He is best known for his discovery of the positron in 1932, an achievement for which he received the 1936 Nobel Prize in Physics, and of the muon in 1936.-Biography:...
as the positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
. 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 spectrumHamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, an extension of the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
that is consistent with special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, 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 an isolated electron is considered, because its energy is conserved
Conservation of energy
The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
and negative-energy electrons may be left out. However, difficulties arise when effects of the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
are considered, because a positive-energy electron would be able to shed energy by continuously emitting photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s, 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
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
. Electrons are fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s, and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom (if spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
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
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
. However, Robert Oppenheimer
Robert Oppenheimer
Julius Robert Oppenheimer was an American theoretical physicist and professor of physics at the University of California, Berkeley. Along with Enrico Fermi, he is often called the "father of the atomic bomb" for his role in the Manhattan Project, the World War II project that developed the first...
pointed out that an electron and its hole would be able to annihilate
Annihilation
Annihilation is defined as "total destruction" or "complete obliteration" of an object; having its root in the Latin nihil . A literal translation is "to make into nothing"....
each other, releasing energy on the order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...
s would not exist. Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
also noted that a hole should act as though it has the same mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932 when the positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
was discovered by Carl Anderson
Carl David Anderson
Carl David Anderson was an American physicist. He is best known for his discovery of the positron in 1932, an achievement for which he received the 1936 Nobel Prize in Physics, and of the muon in 1936.-Biography:...
, 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 constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
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
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel is a mathematical veridical paradox about infinite sets presented by German mathematician David Hilbert .-The paradox:...
elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have a chiral anomaly
Chiral anomaly
A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...
and a gauge instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
.
The development of quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
(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
Vacuum energy
Vacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energy-time uncertainty principle...
.
Modern interpretation
The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformationBogoliubov transformation
In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an...
, 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 creation operators and annihilation operators, in a schematic notation:
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 that 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:
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. 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
Valence band
In solids, the valence band is the highest range of electron energies in which electrons are normally present at absolute zero temperature....
in a solid
Solid
Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...
can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s, though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge — the charge of the ionic lattice
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
— that cancels out the electric charge of the sea.
In popular culture
The Dirac Sea is mentioned in Episode 16 of the animeAnime
is the Japanese abbreviated pronunciation of "animation". The definition sometimes changes depending on the context. In English-speaking countries, the term most commonly refers to Japanese animated cartoons....
series Neon Genesis Evangelion
Neon Genesis Evangelion
, commonly referred to as Evangelion, is a commercially and critically successful Japanese anime series that began airing in October 1995. The series was highly influential, and launched the Neon Genesis Evangelion franchise. It garnered several major animation awards...
as being a component of the angel that attacks in that episode.
The Dirac Sea is an important factor in the visual novel ChaoS;HEAd
Chaos;Head
, stylized as ChäoS;HEAd, is a Japanese visual novel developed by 5pb. and Nitroplus that was released on April 25, 2008 for the PC. A port of the game, named , for the Xbox 360 console was released on February 26, 2009. The game is described by the development team as a...
. People known as gigalomaniacs have the ability to interfere with the Dirac Sea to induce a "real boot"; in other words, bring delusions to reality. The "real boot" is done by projecting particles of delusions into a person's blind spot, making it real. Likewise, antiparticles of the same form can be projected, the matter and antimatter will destroy each other, and the delusion will be destroyed. One of the heroines of the game, Aoi Sena also mentions that "the world is governed by three numbers: 0, 1, and -1.", which is a reference to the diagram of the Dirac sea. An item within the game, called a Di-Sword, helps to interfere with the Dirac Sea and create the "-1".
See also
- Fermi sea
- PositroniumPositroniumPositronium is a system consisting of an electron and its anti-particle, a positron, bound together into an "exotic atom". Being unstable, the two particles annihilate each other to produce two gamma ray photons after an average lifetime of 125 ps or three gamma ray photons after 142 ns in...
- Vacuum energyVacuum energyVacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energy-time uncertainty principle...
- Vacuum polarizationVacuum polarizationIn quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic...
- Virtual particleVirtual particleIn physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
External links
Papers:- http://openseti.org/Docs/HotsonPart1.pdf
- http://openseti.org/Docs/HotsonPart2.pdf