# Fermion

In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spin-statistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.

In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin statistics relation is in fact a spin statistics-quantum number relation.[1]

As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can also display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles.

Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter.

The name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi.[2]

Enrico Fermi
Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential.

## Elementary fermions

The Standard Model recognizes two types of elementary fermions: quarks and leptons. In all, the model distinguishes 24 different fermions. There are six quarks (up, down, strange, charm, bottom and top quarks), and six leptons (electron, electron neutrino, muon, muon neutrino, tau particle and tau neutrino), along with the corresponding antiparticle of each of these.

Mathematically, fermions come in three types:

Most Standard Model fermions are believed to be Dirac fermions, although it is unknown at this time whether the neutrinos are Dirac or Majorana fermions (or both). Dirac fermions can be treated as a combination of two Weyl fermions.[3]:106 In July 2015, Weyl fermions have been experimentally realized in Weyl semimetals.

## Composite fermions

Composite particles (such as hadrons, nuclei, and atoms) can be bosons or fermions depending on their constituents. More precisely, because of the relation between spin and statistics, a particle containing an odd number of fermions is itself a fermion. It will have half-integer spin.

Examples include the following:

• A baryon, such as the proton or neutron, contains three fermionic quarks and thus it is a fermion.
• The nucleus of a carbon-13 atom contains six protons and seven neutrons and is therefore a fermion.
• The atom helium-3 (3He) is made of two protons, one neutron, and two electrons, and therefore it is a fermion.

The number of bosons within a composite particle made up of simple particles bound with a potential has no effect on whether it is a boson or a fermion.

Fermionic or bosonic behavior of a composite particle (or system) is only seen at large (compared to size of the system) distances. At proximity, where spatial structure begins to be important, a composite particle (or system) behaves according to its constituent makeup.

Fermions can exhibit bosonic behavior when they become loosely bound in pairs. This is the origin of superconductivity and the superfluidity of helium-3: in superconducting materials, electrons interact through the exchange of phonons, forming Cooper pairs, while in helium-3, Cooper pairs are formed via spin fluctuations.

The quasiparticles of the fractional quantum Hall effect are also known as composite fermions, which are electrons with an even number of quantized vortices attached to them.

### Skyrmions

In a quantum field theory, there can be field configurations of bosons which are topologically twisted. These are coherent states (or solitons) which behave like a particle, and they can be fermionic even if all the constituent particles are bosons. This was discovered by Tony Skyrme in the early 1960s, so fermions made of bosons are named skyrmions after him.

Skyrme's original example involved fields which take values on a three-dimensional sphere, the original nonlinear sigma model which describes the large distance behavior of pions. In Skyrme's model, reproduced in the large N or string approximation to quantum chromodynamics (QCD), the proton and neutron are fermionic topological solitons of the pion field.

Whereas Skyrme's example involved pion physics, there is a much more familiar example in quantum electrodynamics with a magnetic monopole. A bosonic monopole with the smallest possible magnetic charge and a bosonic version of the electron will form a fermionic dyon.

The analogy between the Skyrme field and the Higgs field of the electroweak sector has been used[4] to postulate that all fermions are skyrmions. This could explain why all known fermions have baryon or lepton quantum numbers and provide a physical mechanism for the Pauli exclusion principle.

 Anyon, 2D quasiparticles Chirality (physics), left-handed and right-handed Fermionic condensate Weyl semimetal Fermionic field Identical particles Kogut–Susskind fermion, a type of lattice fermion Majorana fermion, each its own antiparticle Parastatistics Boson

## Notes

1. ^ Physical Review D volume 87, page 0550003, year 2013, author Weiner, Richard M., title "Spin-statistics-quantum number connection and supersymmetry" arxiv:1302.0969
2. ^ Notes on Dirac's lecture Developments in Atomic Theory at Le Palais de la Découverte, 6 December 1945, UKNATARCHI Dirac Papers BW83/2/257889. See note 64 on page 331 in "The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom" by Graham Farmelo
3. ^ T. Morii; C. S. Lim; S. N. Mukherjee (1 January 2004). The Physics of the Standard Model and Beyond. World Scientific. ISBN 978-981-279-560-1.
4. ^ Weiner, Richard M. (2010). "The Mysteries of Fermions". International Journal of Theoretical Physics. 49 (5): 1174–1180. arXiv:0901.3816. Bibcode:2010IJTP...49.1174W. doi:10.1007/s10773-010-0292-7.
Causal fermion system

The theory of causal fermion systems is an approach to describe fundamental physics. Its proponents claim it gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory.

Instead of introducing physical objects on a preexisting space-time manifold, the general concept is to derive space-time as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when space-time no longer has a manifold structure on the microscopic scale (like a space-time lattice or other discrete or continuous structures on the Planck scale). As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity.

Causal fermion systems were introduced by Felix Finster and collaborators.

Chargino

In particle physics, the chargino is a hypothetical particle which refers to the mass eigenstates of a charged superpartner, i.e. any new electrically charged fermion (with spin 1/2) predicted by supersymmetry. They are linear combinations of the charged wino and charged higgsinos. There are two charginos that are fermions and are electrically charged, which are typically labeled
C${\displaystyle {\tilde {\chi }}}$±
1
(the lightest) and
C${\displaystyle {\tilde {\chi }}}$±
2
(the heaviest) although sometimes ${\displaystyle {\tilde {\chi }}_{1}^{\pm }}$ and ${\displaystyle {\tilde {\chi }}_{2}^{\pm }}$ is also used to refer to charginos, when ${\displaystyle {\tilde {\chi }}_{i}^{0}}$ is used to refer to neutralinos. The heavier chargino can decay through the neutral Z boson to the lighter chargino. Both can decay through a charged W boson to a neutralino:

C${\displaystyle {\tilde {\chi }}}$±
2

C${\displaystyle {\tilde {\chi }}}$±
1
+
Z0

C${\displaystyle {\tilde {\chi }}}$±
2

0
2
+
W±

C${\displaystyle {\tilde {\chi }}}$±
1

0
1
+
W±
Composite fermion

A composite fermion is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.

Vortices are an example of topological defect, and also occur in other situations. Quantized vortices are found in type II superconductors, called Abrikosov vortices. Classical vortices are relevant to the Berezenskii–Kosterlitz–Thouless transition in two-dimensional XY model.

Degenerate matter

Degenerate matter is a highly dense state of fermionic matter in which particles must occupy high states of kinetic energy in order to satisfy the Pauli exclusion principle. The description applies to matter composed of electrons, protons, neutrons or other fermions. The term is mainly used in astrophysics to refer to dense stellar objects where gravitational pressure is so extreme that quantum mechanical effects are significant. This type of matter is naturally found in stars in their final evolutionary states, like white dwarfs and neutron stars, where thermal pressure alone is not enough to avoid gravitational collapse.

Degenerate matter is usually modelled as an ideal Fermi gas, an ensemble of non-interacting fermions. In a quantum mechanical description, particles limited to a finite volume may take only a discrete set of energies, called quantum states. The Pauli exclusion principle prevents identical fermions from occupying the same quantum state. At lowest total energy (when the thermal energy of the particles is negligible), all the lowest energy quantum states are filled. This state is referred to as full degeneracy. This degeneracy pressure remains non-zero even at absolute zero temperature. Adding particles or reducing the volume forces the particles into higher-energy quantum states. In this situation, a compression force is required, and is made manifest as a resisting pressure. The key feature is that this degeneracy pressure does not depend on the temperature but only on the density of the fermions. Degeneracy pressure keeps dense stars in equilibrium, independent of the thermal structure of the star.

A degenerate mass whose fermions have velocities close to the speed of light (particle energy larger than its rest mass energy) is called relativistic degenerate matter.

The concept of degenerate stars, stellar objects composed of degenerate matter, was originally developed in a joint effort between Arthur Eddington, Ralph Fowler and Arthur Milne. Eddington had suggested that the atoms in Sirius B were almost completely ionised and closely packed. Fowler described white dwarfs as composed of a gas of particles that became degenerate at low temperature. Milne proposed that degenerate matter is found in most of the nucleus of stars, not only in compact stars.

Dirac fermion

In physics, a Dirac fermion is spin ​1⁄2 particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category.

Fermi liquid theory

Fermi liquid theory (also known as Landau–Fermi liquid theory) is a theoretical model of interacting fermions that describes the normal state of most metals at sufficiently low temperatures. The interaction between the particles of the many-body system does not need to be small. The phenomenological theory of Fermi liquids was introduced by the Soviet physicist Lev Davidovich Landau in 1956, and later developed by Alexei Abrikosov and Isaak Khalatnikov using diagrammatic perturbation theory. The theory explains why some of the properties of an interacting fermion system are very similar to those of the ideal Fermi gas (i.e. non-interacting fermions), and why other properties differ.

Important examples of where Fermi liquid theory has been successfully applied are most notably electrons in most metals and Liquid helium-3. Liquid helium-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase). Helium-3 is an isotope of helium, with 2 protons, 1 neutron and 2 electrons per atom. Because there is an odd number of fermions inside the nucleus, the atom itself is also a fermion. The electrons in a normal (non-superconducting) metal also form a Fermi liquid, as do the nucleons (protons and neutrons) in an atomic nucleus. Strontium ruthenate displays some key properties of Fermi liquids, despite being a strongly correlated material, and is compared with high temperature superconductors like cuprates.

Fermionic condensate

A fermionic condensate is a superfluid phase formed by fermionic particles at low temperatures. It is closely related to the Bose–Einstein condensate, a superfluid phase formed by bosonic atoms under similar conditions. The earliest recognized fermionic condensate described the state of electrons in a superconductor; the physics of other examples including recent work with fermionic atoms is analogous. The first atomic fermionic condensate was created by a team led by Deborah S. Jin in 2003.

Fermionic field

In quantum field theory, a fermionic field is a quantum field whose quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relations of bosonic fields.

The most prominent example of a fermionic field is the Dirac field, which describes fermions with spin-1/2: electrons, protons, quarks, etc. The Dirac field can be described as either a 4-component spinor or as a pair of 2-component Weyl spinors. Spin-1/2 Majorana fermions, such as the hypothetical neutralino, can be described as either a dependent 4-component Majorana spinor or a single 2-component Weyl spinor. It is not known whether the neutrino is a Majorana fermion or a Dirac fermion; observing neutrinoless double-beta decay experimentally would settle this question.

Flavour (particle physics)

In particle physics, flavour or flavor refers to the species of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with flavour quantum numbers that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.

Four-fermion interactions

In quantum field theory, fermions are described by anticommuting spinor fields. A four-fermion interaction describes a local interaction between four fermionic fields at a point. Local here means that it all happens at the same spacetime point. This might be an effective field theory or it might be fundamental.

Gravitino

In supergravity theories combining general relativity and supersymmetry, the gravitino (G͂) is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter.

If it exists, it is a fermion of spin 3/2 and therefore obeys the Rarita-Schwinger equation. The gravitino field is conventionally written as ψμα with μ = 0, 1, 2, 3 a four-vector index and α = 1, 2 a spinor index.

For μ = 0 one would get negative norm modes, as with every massless particle of spin 1 or higher. These modes are unphysical, and for consistency there must be a gauge symmetry which cancels these modes: δψμα = ∂μεα, where εα(x) is a spinor function of spacetime. This gauge symmetry is a local supersymmetry transformation, and the resulting theory is supergravity.

Thus the gravitino is the fermion mediating supergravity interactions, just as the photon is mediating electromagnetism, and the graviton is presumably mediating gravitation. Whenever supersymmetry is broken in supergravity theories, it acquires a mass which is determined by the scale at which supersymmetry is broken. This varies greatly between different models of supersymmetry breaking, but if supersymmetry is to solve the hierarchy problem of the Standard Model, the gravitino cannot be more massive than about 1 TeV/c2.

Heavy fermion material

In solid-state physics, heavy fermion materials are a specific type of intermetallic compound, containing elements with 4f or 5f electrons in unfilled electron bands. Electrons are one type of fermion, and when they are found in such materials, they are sometimes referred to as heavy electrons. Heavy fermion materials have a low-temperature specific heat whose linear term is up to 1000 times larger than the value expected from the free electron model. The properties of the heavy fermion compounds often derive from the partly filled f-orbitals of rare-earth or actinide ions, which behave like localized magnetic moments. The name "heavy fermion" comes from the fact that the fermion behaves as if it has an effective mass greater than its rest mass. In the case of electrons, below a characteristic temperature (typically 10 K), the conduction electrons in these metallic compounds behave as if they had an effective mass up to 1000 times the free particle mass. This large effective mass is also reflected in a large contribution to the resistivity from electron-electron scattering via the Kadowaki–Woods ratio. Heavy fermion behavior has been found in a broad variety of states including metallic, superconducting, insulating and magnetic states. Characteristic examples are CeCu6, CeAl3, CeCu2Si2, YbAl3, UBe13 and UPt3.

Heavy fermion superconductor

Heavy fermion superconductors are a type of unconventional superconductor.

The first heavy fermion superconductor, CeCu2Si2, was discovered by Frank Steglich in 1978.Since then over 30 heavy fermion superconductors were found (in materials based on Ce, U), with a critical temperature up to 2.3 K (in CeCoIn5).

Heavy Fermions are intermetallic compounds, containing rare earth or actinide elements. The f-electrons of these atoms hybridize with the normal conduction electrons leading to quasiparticles with an enhanced mass.From specific heat measurements (ΔC/C(TC) one knows that the Cooper pairs in the superconducting state are also formed by the heavy quasiparticles.

In contrast to normal superconductors it cannot be described by BCS-Theory. Due to the large effective mass, the

Fermi velocity is reduced and comparable to the inverse Debye frequency. This leads to the failing of the picture of electrons polarizing the lattice as an attractive force.Some heavy fermion superconductors are candidate materials for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. In particular there has been evidence that CeCoIn5 close to the critical field is in an FFLO state.

Majorana fermion

A Majorana fermion (), also referred to as a Majorana particle, is a fermion that is its own antiparticle. They were hypothesized by Ettore Majorana in 1937. The term is sometimes used in opposition to a Dirac fermion, which describes fermions that are not their own antiparticles.

With the exception of the neutrino, all of the Standard Model fermions are known to behave as Dirac fermions at low energy (after electroweak symmetry breaking), and none are Majorana fermions. The nature of the neutrinos is not settled – they may be either Dirac or Majorana fermions.

In condensed matter physics, bound Majorana fermions can appear as quasiparticle excitations – the collective movement of several individual particles, not a single one, and they are governed by non-abelian statistics.

Photino

A photino is a hypothetical subatomic particle, the fermion WIMP superpartner of the photon predicted by supersymmetry. It is an example of a gaugino. Even though no photino has ever been observed so far, it is one of the candidates for the lightest stable particle in the universe. It is proposed that photinos are produced by sources of ultra-high-energy cosmic rays.

Second quantization

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were developed, most notably, by Vladimir Fock and Pascual Jordan later.In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

Spinor

In geometry and physics, spinors are elements of a (complex) vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight (infinitesimal) rotation. When a sequence of such small rotations is composed (integrated) to form an overall final rotation, however, the resulting spinor transformation depends on which sequence of small rotations was used: unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360° (see picture). This property characterizes spinors. It is also possible to associate a substantially similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles.Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). There are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle (below). These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class. It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3).

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.

Vacuum expectation value

In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by ${\displaystyle \langle O\rangle .}$ One of the most widely used, but controversial, examples of an observable physical effect that results from the vacuum expectation value of an operator is the Casimir effect.

This concept is important for working with correlation functions in quantum field theory. It is also important in spontaneous symmetry breaking. Examples are:

The observed Lorentz invariance of space-time allows only the formation of condensates which are Lorentz scalars and have vanishing charge.[citation needed] Thus fermion condensates must be of the form ${\displaystyle \langle {\overline {\psi }}\psi \rangle }$, where ψ is the fermion field. Similarly a tensor field, Gμν, can only have a scalar expectation value such as ${\displaystyle \langle G_{\mu \nu }G^{\mu \nu }\rangle }$.

In some vacua of string theory, however, non-scalar condensates are found.[which?] If these describe our universe, then Lorentz symmetry violation may be observable.

Weyl semimetal

Weyl fermions are massless chiral fermions that play an important role in quantum field theory and the standard model. They are considered a building block for fermions in quantum field theory, and were predicted from a solution to the Dirac equation derived by Hermann Weyl called the Weyl equation. For example, one-half of a charged Dirac fermion of a definite chirality is a Weyl fermion.They have not been observed as a fundamental particle in nature. Weyl fermions may be realized as emergent quasiparticles in a low-energy condensed matter system. This prediction was first proposed by Conyers Herring in the context of electronic band structures of solid state systems such as electronic crystals.The first (non-electronic) liquid state which is suggested has similarly emergent but neutral excitation and theoretically interpreted superfluid's chiral anomaly as observation of Fermi points is in Helium-3 A liquids. Crystalline tantalum arsenide (TaAs) is the first discovered topological Weyl fermion semimetal exhibiting topological surface Fermi arcs where Weyl fermion is electrically charged along the line of original suggestion by Herring. An electronic Weyl fermion is not only charged but stable at room temperature where there is no such superfluid or liquid state known.

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