Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 301 / 2 / b

Table of contents
- Solution b

Solution

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

-particle Fock state therefore transforms as

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

Solved by gpt-5.6-sol high.

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