Fourier transform and translational invariance

[ShayanJ]

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

 

[The Bill]

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

 

[ShayanJ]

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 Bill]

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

 

[ShayanJ]

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.