Are position and momentum spaces of a particle in classical mechanics related by Fourier transform?

Question

Please don't be so harsh on me and correct me if I'm wrong, since physics is not my major and I am not native English speaker.

The article position and momentum space (https://en.wikipedia.org/wiki/Position_and_momentum_space) on Wikipedia stated that in quantum mechanics, the position and momentum spaces are related by Fourier transform since they are Pontryagin dual.

I am not clear if this conjugation applies in quantum mechanics only or if it applies for position and momentum spaces of a particle in general, including classical, Lagrangian and Hamiltonian mechanics also? If yes, how is it formulated?

Answer 1

In classical mechanics, position and momentum are independent variables on phase space and are not related by Fourier transformation at all.

Their Fourier relation in quantum mechanics arises from the Stone-von Neumann theorem, saying that all unitary representations of the canonical commutation relations $[x,p] = \mathrm{i}\hbar$ (which are, apart from the appearance of the $\hbar$, essentially classical, since $\{x,p\} = 1$ for the Poisson bracket on phase space) are unitarily equivalent to representing $x$ as the multiplication and $p$ as the differentiation operator on $L^2(\mathbb{R})$. Now, the Fourier transformation famously interchanges multiplication and differentiation and therefore relates position and momentum representations in quantum mechanics.