The discussion centers around the relationship between Fourier transforms and translational invariance in the context of scalar fields in d dimensions, as referenced in a specific paper. Participants seek to clarify the meaning of a statement regarding Fourier decomposition and its connection to translational invariance.
There is no consensus on the exact relationship between Fourier transforms and translational invariance, as participants express differing interpretations and raise questions about the terminology used.
Participants highlight potential confusion between Fourier decomposition and Fourier transform, indicating a need for clarity in definitions and concepts related to the equations of motion.
page 27 of this paperI don't understand the relation between the Fourier transform and translational invariance.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) ##
What does that have to do with Fourier transforming it?The Bill said:I think the author is referring to the translational invariance of the equation of motion for ##ϕ##.
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.The Bill said:The author mentioned Fourier decomposition. This is not the same as a Fourier transform.