 Solution

ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2016/1/a/solution

\frac{\partial L}{\partial ( \partial _{μ} ϕ )} = \partial^{μ} ϕ, \frac{\partial L}{\partial ϕ} = - m^{2} ϕ - \frac{λ}{6} ϕ^{3} .

Consequently the equation of motion is

(□ + m^{2}) ϕ + \frac{λ}{6} ϕ^{3} = 0.

The quartic interaction therefore adds a local cubic term to the free Klein-Gordon equation. Below, write

F = (□ + m^{2}) ϕ + λ ϕ^{3} /6

; thus

F = 0

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