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.

Some examples are the following:

Fermi's theory of the weak interaction. The interaction term has a V-A (vector minus axial) form.
The Gross-Neveu model. This is a four-fermi theory of Dirac fermions without chiral symmetry and as such, it may or may not be massive.
The Thirring model. 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 chiral symmetry and as such, it has no bare mass.

A nonrelativistic example is the BCS theory 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.

Four-fermion interactions

BCS theory Fermi's interaction Gross–Neveu model Hubbard model Luttinger liquid Nambu–Jona-Lasinio model Thirring model Top quark condensate

Book:Four-fermion interactions

Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model