Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 4 / b / i

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- Solution i

Solution

The early and late stationary regions define distinct positive-frequency mode bases and hence an in-vacuum and out-vacuum. Write their Bogoliubov transformation as

u_{i}^{out} = \sum_{j} (α_{ij} u_{j}^{in} + β_{ij} u_{j}^{in *}) .

The corresponding operators satisfy

b_{i}^{out} = \sum_{j} (α_{ij}^{*} a_{j}^{in} - β_{ij}^{*} a_{j}^{in †}) .

Using

a_{j}^{in} ∣ 0_{in} ⟩ = 0

and the canonical commutator gives the particle number from Bogoliubov coefficients:

⟨ 0_{in} ∣ N_{i}^{out} ∣ 0_{in} ⟩ = j \sum ∣ β_{ij} ∣^{2} .

Time dependence in the sandwich region can make

β_{ij} \neq = 0

, so the in-vacuum contains out-particles.

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