# A Fourier transform and translational invariance

1. Nov 20, 2016

### ShayanJ

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. Nov 20, 2016

### The Bill

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

3. Nov 20, 2016

### ShayanJ

What does that have to do with Fourier transforming it?

4. Nov 20, 2016

### The Bill

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

5. Nov 20, 2016

### ShayanJ

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.