 Solution

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2014/paper-1/4/b/solution

An infinitesimal constant phase change has

δ ϕ = - i q α ϕ

and

δ ϕ^{*} = i q α ϕ^{*}

. Both the derivative product and the mass term are invariant. Noether's theorem gives, per unit

α

j^{μ} = (\partial^{μ} ϕ^{*}) (- i qϕ) + (\partial^{μ} ϕ) (i q ϕ^{*}) = i q (ϕ^{*} \partial^{μ} ϕ - ϕ \partial^{μ} ϕ^{*}) .

Thus the global phase symmetry supplies the conserved current in part a.

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