Past exam of the quantum field theory course of the University of Oxford / 2016 / 1 / a

For the real scalar field, the two derivatives entering the Euler-Lagrange field equation areConsequently the equation of motion isThe quartic interaction therefore adds a local cubic term to the free Klein-Gordon equation. Below, write $F=(\square +m^2)\varphi +\lambda {\varphi }^3/6$; thus $F=0$ means that the field is on shell.