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 Incoming links: Particle number from Bogoliubov coefficients

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

⟨ 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.

The in-vacuum satisfies

a_{j}^{in} ∣ 0_{in} ⟩ = 0

. For the out-mode number operator

N_{i}^{out} = a_{i}^{out †} a_{i}^{out}

, substitute the operator transformation and use the canonical commutation relations. Only the

a_{j}^{in} a_{k}^{in †}

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

For a continuous mode label, the sum becomes the corresponding integral with Dirac-delta normalization.