phi-four theory (idea)

φ⁴(phi-four) theory is a simple example of a quantum field theory (QFT) that exhibits many of the general properties of such theories, and is thus often used as a "toy model" for teaching QFT basics or for theoretical studies. It is renormalizable but it is not a gauge theory.

φ⁴ theory got its name from the φ⁴ interaction term in the Lagrangian.

One real, scalar field φ≡φ(x).

L(φ, ∂_μφ) = 1/2 (∂_μφ)(∂^μφ) - 1/2 m²φ² - g/4! φ⁴

• m: mass of the field φ
• g: coupling constant of the interaction

• Lorentz invariance
• symmetric under the transformation φ → -φ

∂_μ∂^μφ + m²φ + g/3! φ³ = 0.

For a free field (g = 0), this is the Klein-Gordon equation.

• Free propagator of the field φ: Δ_F(p) = (p² - m² + iε)^-1
• Four-point vertex: -ig