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Contents

Idea

A Lagrangian field theory is interacting if it is not a free field theory. Just as in the discussion at free field theory, there is some room for making this precise, but at the very least it should mean that the Euler-Lagrange equations of motion are not linear.

In perturbative quantum field theory the algebras of observables of interacting field theories are quantized as formal power series in Planck's constant and in the coupling constant (interacting field algebra), as a formal expansion around the quantizaton of a free field theory (Wick algebra).

To date perturbative quantum field theory is the only way known to approach the quantization of interacting field theories in spacetime dimension $\geq 4$. In constructive field theory examples of non-perturbative interacting quantum field theores in dimension 3 have been constructed and examples in dimension 2 are common, such as 2d CFTs.

Examples

Key examples of interacting field theories are phi^n theory (scalar field theory with interaction term given by the power $(\mathbf{\Phi}(x))$ of the field observable), quantum electrodynamics (with its electron-photon interaction), pure Yang-Mills theory for nonabelian gauge group, quantum chromodynamics, and gravity.

The quantization of Yang-Mills theory, as a non-perturbative quantum field theory is a famous open problem. Yang-Mills theory with abelian gauge group the circle group is a free field theory, but coupled to a Dirac field there is a canonical interaction term that makes this the interacting field theory called quantum electrodynamics (QED). Similarly quantum chromodynamics (QCD) is an interacting field theories. These interacting theories (QED, QCD with Higgs field-coupling) make up the standard model of particle physics.

Related concepts

• interaction

• interacting field algebra

• interaction picture