In quantum field theory, fermions are described by

anticommuting In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...

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 sligh ...

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 b ...

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

at a point. Local here means that it all happens at the same

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...

point. This might be an effective field theory or it might be fundamental.

Relativistic models

Some examples are the following: * Fermi's theory of the

weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...

. The interaction term has a (

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

minus axial) form. *The

Gross–Neveu model The Gross–Neveu (GN) model 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 quantum ...

. This is a four-fermi theory of Dirac fermions 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 particle, ...

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. This is a four-fermi theory of Dirac fermions with

and as such, it has no bare mass.

Nonrelativistic models

nonrelativistic The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

example is the

BCS theory BCS theory or Bardeen–Cooper–Schrieffer 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 describes sup ...

at large length scales with the phonons integrated out so that the force between two dressed electrons is approximated by a contact term. In four space-time dimensions, such theories are not renormalisable.

References

Quantum field theory {{Quantum-stub