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pleasehelpmeno

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## Homework Statement

I am trying to teach myself QFT and reproduce cosmological equations from papers.

Given the bogoliubov transformations:

i) a(conformal time η, k) = α

_{[/k](η)a(k)+β[/k](η)b^\dagger ii) b[\dagger] = -β*[/k](η)a(k)+α*[/k](η)b^\dagger find the ground state number density N/V, where a|0> = b|0> = 0. Homework Equations <0|N/V|0>=1/(2∏)^3 ∫d^3 k (a[/r](η,r)a^\dagger[/r](η,r)) The Attempt at a Solution I have then taken the complex conjugate of i) and done i) X i)* then made us of a|0> = 0 and thus therefore <0|a^\dagger = 0. I then get 1\(2∏)^3 ∫ (b|β|^2 b^\dagger d^3 k but am supposed to find it equal to 1\(∏)^2 ∫ (|β|^2 k^2 dk. I am not sure why it becomes ∏^2 or where the creation/annihilation operators disappear. I am not sure if i am missing something obvious like a table integral or am miss understanding basic principles. Please help }