# Four-fermion interactions

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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.

## Relativistic models[edit]

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.

## Nonrelativistic models[edit]

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.

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