Wave-Particle Complementarity from Fourier-Pontryagin Duality: A Categorical Formulation on Schwartz Rigged Hilbert Spaces

Abstract

We formulate position--momentum complementarity in a continuous categorical setting based on Schwartz rigged Hilbert spaces. The position and momentum descriptions are represented by distributional observable structures related by the Fourier transform associated with Pontryagin duality of the additive group $\mathbb{R}$. A positive normalization parameter $\alpha$ is introduced independently of the physical constant $\hbar$, so that the character pairing takes the form $\chi_{p}^{(\alpha)}(x)=\exp(ipx/\alpha)$. We show that the abstract Pontryagin duality fixes the form of the position -- momentum pairing but does not determine its dimensional normalization: the simultaneous rescaling $(p,\alpha)\mapsto(cp,c\alpha)$ leaves the character pairing unchanged. The momentum observable structure is obtained from the position structure by Fourier conjugation. In the continuous setting, we use the Weyl relations as an operational characterization of position -- momentum complementarity rather than asserting an unproved equivalence with the finite-dimensional Hopf-algebraic definition of strong complementarity. Imposing the physical canonical commutation relation $[\hat X,\hat P]=i\hbar I$ selects $\alpha=\hbar$. The character then becomes $\exp(ipx/\hbar)=\exp(ikx)$, and its spatial period yields the de Broglie relation $\lambda=h/p$. The resulting analysis separates the structural content supplied by Fourier--Pontryagin duality from the physical normalization supplied by quantum mechanics.