# Fourier transform and translational invariance

• A
Gold Member
Can anyone explain what does the author mean by the statement below?
Let’s take advantage of translational invariance in d dimensions, ##x_μ → x_μ + a_μ## , to Fourier decompose the scalar field:

## \phi(z,x^\mu)=e^{ik_\mu x^\mu} f_k(z) ##
page 27 of this paper

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

Thanks

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I think the author is referring to the translational invariance of the equation of motion for ##ϕ##.

Gold Member
I think the author is referring to the translational invariance of the equation of motion for ##ϕ##.
What does that have to do with Fourier transforming it?

The author mentioned Fourier decomposition. This is not the same as a Fourier transform.

Gold Member
The author mentioned Fourier decomposition. This is not the same as a Fourier transform.
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.