Quartic interaction

Example

The quartic interaction, or $\phi^4$ theory, is the quantum field theory of one real scalar field $\phi$ on Minkowski space $\R^{1,d-1}$. Its Lagrangian density, with mass $m \ge 0$ and coupling $\lambda > 0$, is $$\mathcal{L} = \tfrac12 \partial_\mu \phi \, \partial^\mu \phi - \tfrac12 m^2 \phi^2 - \tfrac{\lambda}{4!} \phi^4.$$ The Euler–Lagrange equation is the nonlinear Klein–Gordon equation $(\partial_\mu \partial^\mu + m^2)\phi + \tfrac{\lambda}{3!}\phi^3 = 0$. For $\lambda = 0$ it is the free field. In Euclidean form, the theory is the formal measure proportional to $$\exp\left(-\int_{\R^d} \left( \tfrac12 \abs{\nabla\phi}^2 + \tfrac12 m^2\phi^2 + \tfrac{\lambda}{4!}\phi^4 \right) dx\right) \mathcal{D}\phi.$$ Its Gaussian part is the Gaussian free field, and $\lambda\phi^4$ is the interaction.