Fermion

The Standard Model of elementary particles, with the fermions in the first three columns

In particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In contrast to bosons, which have Bose-Einstein statistics, only one fermion can occupy a quantum state at a given time; this is the Pauli Exclusion Principle.

Thus, if more than one fermion occupies the same place in space, the properties of each fermion (e.g. its spin) must be different from the rest. Therefore, fermions are usually associated with matter while bosons are often force carrier particles, though the distinction between the two concepts in quantum physics is unclear.

Fermions can be elementary, like the electron, or composite, like the proton. All observed fermions have half-integer spin, as opposed to bosons, which have integer spin. This is in accordance with the spin-statistics theorem which states that in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.

In the Standard Model there are two types of elementary fermions: quarks and leptons. In total, there are 24 different fermions: being 6 quarks and 6 leptons, each with a corresponding antiparticle:

• 12 quarks - 6 particles (
  u
  ·
  d
  ·
  s
  ·
  c
  ·
  b
  ·
  t
  ) with 6 corresponding antiparticles (
  u
  ·
  d
  ·
  s
  ·
  c
  ·
  b
  ·
  t
  );

• 12 leptons - 6 particles (
  e⁻
  ·
  μ⁻
  ·
  τ⁻
  ·
  ν
  _e ·
  ν
  _μ ·
  ν
  _τ) with 6 corresponding antiparticles (
  e⁺
  ·
  μ⁺
  ·
  τ⁺
  ·
  ν
  _e ·
  ν
  _μ ·
  ν
  _τ).

Composite fermions, such as protons and neutrons, are essential building blocks of matter. Weakly interacting fermions can also display bosonic behaviour, as in superconductivity.

Definition and basic properties

By definition, fermions are particles which obey Fermi-Dirac statistics: when one swaps two fermions, the wavefunction of the system changes sign. This "antisymmetric wavefunction" behavior implies that fermions are subject to the Pauli exclusion principle — no two fermions can occupy the same quantum state at the same time. This results in "rigidity" or "stiffness" of states which include fermions (atomic nuclei, atoms, molecules, etc.), so fermions are sometimes said to be the constituents of matter, while bosons are said to be the particles that transmit interactions (force carriers), or the constituents of radiation. The quantum fields of fermions are fermionic fields, obeying canonical anticommutation relations.

The Pauli exclusion principle for fermions and the associated rigidity of matter is responsible for the stability of the electron shells of atoms (thus for stability of atomic matter) and the complexity of atoms (making it impossible for all atomic electrons to occupy the same energy level), thus making complex chemistry possible. It is also responsible for the pressure within degenerate matter which largely governs the equilibrium state of white dwarfs and neutron stars. On a more everyday scale, the Pauli exclusion principle is a major contributor to the Young modulus of matter.

All known fermions are particles with half-integer spin: as an observer circles a fermion (or as the fermion rotates 360° about its axis) the wavefunction of the fermion changes sign. In the framework of nonrelativistic quantum mechanics, this is a purely empirical observation. However, in relativistic quantum field theory, the spin-statistics theorem shows that half-integer spin particles cannot be bosons and integer spin particles cannot be fermions.

In large systems, the difference between bosonic and fermionic statistics is only apparent at large densities when their wave functions overlap. At low densities, both types of statistics are well approximated by Maxwell-Boltzmann statistics, which is described by classical mechanics.

Elementary fermions

See also: List of particles § Fermions

All observed elementary particles are either fermions or bosons. The known elementary fermions are divided into two groups: quarks and leptons.

• Quarks make up protons, neutrons and other baryons, which are composite fermions; they also comprise mesons, which are composite bosons.
• Leptons include the electron and similar, heavier particles (the muon and tauon); they also include neutrinos.

The known fermions of left-handed helicity experience weak interactions while the known right-handed fermions do not. Or put another way, only left-handed fermions and right-handed antifermions interact with the W boson.

Composite fermions

See also: List of particles § Composite particles

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 is therefore a fermion;
• The nucleus of a carbon-13 atom contains 6 protons and 7 neutrons and is therefore a fermion;
• The atom helium-3 (³He) is made of 2 protons, a neutron and 2 electrons and is therefore 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) distance. 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 fundamental building blocks of the fractional quantum Hall effect are also particles 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 elementary 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 involves fields which take values on a three dimensional sphere, the original nonlinear sigma model that describes the large distance behavior of pions. In Skyrme's model, which is reproduced in the large N or string approximation to QCD, the proton and neutron are fermionic topological solitons of the pion field. While Skyrme's example involves 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 would form a fermionic dyon.

See also

• Fermionic field
• Identical particles
• Parastatistics
• Anyon
• Fermionic condensate
• Superconductivity
• Fractional quantum Hall effect

References

• Sakurai, J.J. (1994). Modern Quantum Mechanics (Revised Edition), pp 361-363. Addison-Wesley Publishing Company, ISBN 0-201-53929-2.
• Srednicki, Mark (2007). Quantum Field Theory, Cambridge University Press, ISBN 978-0521864497.