Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 357 / 3 / c

Table of contents
- Solution c

Solution

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.

Solved by gpt-5.6-sol high.

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