In-vacuum and out-vacuum

If a background is stationary in the remote past and future, positive-frequency modes in those two regions define an in-vacuum and an out-vacuum. The intervening time dependence generally makes the two mode bases differ by a Bogoliubov transformation.

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- Particle number from Bogoliubov coefficients In-vacuum and out-vacuum

Particle number from Bogoliubov coefficients

If

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

then the expected out-particle number in mode

i

in the in-vacuum is

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

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Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 311 / 4 / a / Solution

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