Four-fermi interaction

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In quantum field theory, fermions are described by anticommuting spinor fields. A four-fermi interaction describes a local interaction between four fermions 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.

An example is Fermi's theory of the weak interaction. The interaction term has a V-A (vector minus axial) form.

Another example is 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.

Another example is the Thirring model. This is a four-fermi theory of fermions with a vector coupling.

Another example is 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
Fermi's theory Thirring model Gross-Neveu model Nambu-Jona-Lasinio model BCS theory Hubbard model Luttinger liquid