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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

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

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.

Quantum field theories

Chern–Simons Conformal field theory Ginzburg–Landau Kondo effect Local QFT Noncommutative QFT Quantum Yang–Mills Quartic interaction sine-Gordon String theory Toda field Topological QFT Yang–Mills Yang–Mills–Higgs


Chiral Non-linear sigma Schwinger Standard Model Thirring–Wess Wess–Zumino Wess–Zumino–Witten Yukawa

Four-fermion interactions


BCS theory Fermi's interaction Luttinger liquid Top quark condensate


Gross–Neveu Hubbard Nambu–Jona-Lasinio Thirring Thirring–Wess


History Axiomatic QFT Loop quantum gravity Loop quantum cosmology QFT in curved spacetime Quantum chaos Quantum chromodynamics Quantum dynamics Quantum electrodynamics
links Quantum gravity
links Quantum hadrodynamics Quantum hydrodynamics Quantum information Quantum information science
links Quantum logic Quantum thermodynamics

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