φ^{4}(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.

φ^{4} theory got its name from the φ^{4} interaction term in the Lagrangian.

**Fields**
One real, scalar field φ≡φ(`x`).

**Lagrangian**
L(φ, ∂_{μ}φ) = 1/2 (∂_{μ}φ)(∂^{μ}φ) - 1/2 m^{2}φ^{2} - g/4! φ^{4}

**Free parameters**
**Symmetries**
**Equation of motion**
∂_{μ}∂^{μ}φ + m^{2}φ + g/3! φ^{3} = 0.

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

**Feynman rules**
- Free propagator of the field φ: Δ
_{F}(`p`) = (p^{2} - m^{2} + iε)^{-1}
- Four-point vertex: -i
`g`