quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

s are described by anticommuting

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...

. A four-fermion interaction describes a local interaction between four fermionic

at a point in

spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...

. A theory involving such an interaction might be an

effective field theory In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degrees ...

or it might be fundamental. In four spacetime dimensions, such theories are not renormalisable.

Relativistic models

Some examples are the following: *

Fermi's theory In particle physics, Fermi's interaction (also the Fermi theory of beta decay or the Fermi four-fermion interaction) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting w ...

of the

weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...

. The interaction term has a (

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

minus axial) form. *The

Gross–Neveu model The Gross–Neveu model (GN) is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for qu ...

. This is a four-fermi theory of

Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. Description In particle physics, all fermions in the standard model have distinct antipar ...

s without

chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particl ...

and as such, it may or may not be massive. *The

Thirring model The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions. Definition The Thirring model is given by the Lagrangian density : \mathcal= \overline(i\partial\!\!\!/- ...

. This is a four-fermi theory of fermions with a vector coupling. *The

Nambu–Jona-Lasinio model In quantum field theory, the Nambu–Jona-Lasinio model (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, parallel ...

. This is a four-fermi theory of

s with

and as such, it has no

bare mass In quantum field theory, specifically the theory of renormalization, the bare mass of an elementary particle is the limit of its mass as the scale of distance approaches zero or, equivalently, as the energy of a particle collision approaches infini ...

Nonrelativistic models

A nonrelativistic example is the

BCS theory In physics, the Bardeen–Cooper–Schrieffer (BCS) theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory descr ...

at large length scales with the phonons integrated out so that the force between two dressed electrons is approximated by a

contact term Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * C ...

References

* * * * Quantum field theory {{Quantum-stub

Relativistic models

Nonrelativistic models

See also

References