Квартичное взаимодействие в квантовой теории поля

Quantum field theory with four point interactions
In quantum field theory, a quartic interaction is a type of self interaction in a scalar field. Other types of quartic interactions may be found under the topic of four fermion interactions. A classical free scalar field satisfies the Klein–Gordon equation. If a scalar field is denoted , a quartic interaction is represented by adding a potential energy term to the Lagrangian density. The coupling constant is dimensionless in 4 dimensional spacetime. This article uses the metric signature for Minkowski space.

The integrals over unconstrained momenta, called "loop integrals", in the Feynman graphs typically diverge. This is normally handled by renormalization, which is a procedure of adding divergent counter terms to the Lagrangian in such a way that the diagrams constructed from the original Lagrangian and counterterms are finite. A renormalization scale must be introduced in the process, and the coupling constant and mass become dependent upon it. It is this dependence that leads to the Landau pole mentioned earlier, and requires that the cutoff be kept finite. Alternatively, if the cutoff is allowed to go to infinity, the Landau pole can be avoided only if the renormalized coupling runs to zero, rendering the theory trivial.

An interesting feature can occur if m2 turns negative, but with λ still positive. In this case, the vacuum consists of two lowest energy states, each of which spontaneously breaks the Z2 global symmetry of the original theory. This leads to the appearance of interesting collective states like domain walls. In the O(2) theory, the vacua would lie on a circle, and the choice of one would spontaneously break the O(2) symmetry. A continuous broken symmetry leads to a Goldstone boson. This type of spontaneous symmetry breaking is the essential component of the Higgs mechanism.