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A Fourier transform and translational invariance

  1. Nov 20, 2016 #1

    ShayanJ

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    Can anyone explain what does the author mean by the statement below?
    page 27 of this paper


    I don't understand the relation between the Fourier transform and translational invariance.

    Thanks
     
  2. jcsd
  3. Nov 20, 2016 #2
    I think the author is referring to the translational invariance of the equation of motion for ##ϕ##.
     
  4. Nov 20, 2016 #3

    ShayanJ

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    What does that have to do with Fourier transforming it?
     
  5. Nov 20, 2016 #4
    The author mentioned Fourier decomposition. This is not the same as a Fourier transform.
     
  6. Nov 20, 2016 #5

    ShayanJ

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    He Fourier transforms the x coordinates and because the derivatives in the equation are only w.r.t. z, you can write the equation for each mode of the transformed field separately.
     
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