A Clarifying Note on the Position-Momentum Correspondence: Pontryagin Duality, Fourier Transport, and Physical Normalization

Abstract

The position--momentum correspondence combines several mathematically distinct identifications that are often conflated. For the additive group $(\mathbb R,+)$, every continuous character has the form $x\mapsto \mathrm e^{ikx}$. After choosing a coordinate $p=αk$ on the dual group, one obtains $\mathrm e^{ipx/α}$, so Pontryagin duality alone does not identify a particular dual coordinate with physical momentum. For the corresponding unitary Fourier transform, multiplication in position space is transported to convolution in momentum space, while the diagonal distribution is transported to the specific composition with the addition map $(p,q)\mapsto p+q$; it is not transported to pointwise multiplication in momentum space. The family $\hat P_α=-iα\,d/dx$ has commutator $[\hat X,\hat P_α]=iαI$ on $\mathcal S(\mathbb R)$, and the standard quantum-mechanical normalization $[\hat X,\hat P]=i\hbar I$ therefore selects $α=\hbar$ within this family. Equivalently, the standard normalization of the translation generator gives $\hat P=-i\hbar\,d/dx$. The resulting characters $\mathrm e^{ipx/\hbar}$ have spatial period $h/|p|$ for $p\neq0$. All distributional statements are formulated in the Schwartz rigging and no product or pullback of arbitrary tempered distributions is used.