 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-301/2/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Insert the mode expansion of a free field into the parity law and change integration variable

k \mapsto - k

. Independence of the plane waves gives

P a_{k} P^{- 1} = γ_{p} a_{- k}, P a_{k}^{†} P^{- 1} = γ_{p} a_{- k}^{†} .

Since

P ∣0 ⟩ = ∣0 ⟩

, an

n

P ∣ k_{1}, \dots, k_{n} ⟩ = (γ_{p})^{n} ∣ - k_{1}, \dots, - k_{n} ⟩ .

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