 Solution

ID: past-exam-of-the-advanced-quantum-field-theory-for-particle-physics-course-of-the-university-of-oxford/2025/1/b/solution

Vary the action, taking the field variation to vanish at the boundary. Integration by parts gives

δ S_{0} = \int d^{4} x [\partial^{μ} ϕ \partial_{μ} δ ϕ - m^{2} ϕ δ ϕ - \frac{λ}{6} ϕ^{3} δ ϕ] = - \int d^{4} x (□ ϕ + m^{2} ϕ + \frac{λ}{6} ϕ^{3}) δ ϕ .

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

E (ϕ) = δ S_{0} / δ ϕ = - (□ ϕ + m^{2} ϕ + λ ϕ^{3} /6)

. The sign does not affect the equation of motion, but it fixes the sign of an added redundant operator.

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