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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