 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2014/paper-1/5/a/solution

For the supplied Lagrangian, variation with respect to

ϕ^{*}

\frac{\partial L}{\partial ( \partial _{μ} ϕ ^{*} )} = \partial^{μ} ϕ + i e A^{μ} ϕ, \frac{\partial L}{\partial ϕ ^{*}} = - m^{2} ϕ - i e A_{μ} \partial^{μ} ϕ + f .

Thus

□ ϕ + 2 i e A^{μ} \partial_{μ} ϕ + i e (\partial_{μ} A^{μ}) ϕ + m^{2} ϕ = f

(□ + 2 i e A^{μ} \partial_{μ} + m^{2}) ϕ = f (t) .

For real external fields and driving, the conjugate equation has

- 2 i e A^{μ} \partial_{μ}

ϕ^{*}

. The supplied model has no

e^{2} A^{2} ∣ ϕ ∣^{2}

term; one must not insert such a term from a different minimally coupled Lagrangian.

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