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

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?

• In classical mechanics position and momentum are independent coordinates with no general relationship between them – By Symmetry Mar 15 '17 at 16:31

## 1 Answer

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.