1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bogoliubov transformations in QFT

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant equations
    <0|N/V|0>=1/(2∏)^3 ∫d^3 k (a[/r](η,r)a^\dagger[/r](η,r))

    3. 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
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?