 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-357/3/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Because

n^{μ} \partial_{μ} ϕ = - 2Π

Q = - n_{μ} J^{μ} = i (\overset{ˉ}{ϕ} Π - ϕ \overset{ˉ}{Π}) .

ϕ = ϕ_{R} + i ϕ_{I}

and

Π = Π_{R} + i Π_{I}

gives

Q = 2 (ϕ_{I} Π_{R} - ϕ_{R} Π_{I}) .

Every quantity on the right is real, so the Noether charge density is explicitly real valued.

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