The paper draws a line between two parts of the position-momentum link: mathematical duality determines the form of the pairing, while the physical canonical commutation relation fixes its scale. It presents the distinction as the route from an abstract position-momentum relationship to a physically normalized one. The result is a claim about the architecture of a mathematical model, not a measured effect in a population.
The document is an arXiv preprint, version 2, dated 1 September 2026. It is a theoretical construction, not an empirical study. It uses Schwartz rigged Hilbert spaces and distributional observable structures to formulate continuous position and momentum, and it reports no participants, specimens or empirical data source.
The shape of the link
At the center is a pairing between position and momentum. The momentum observable structure is obtained from the position structure through Fourier conjugation, so the Fourier transform links the two descriptions. The paper also uses the Weyl relation as a practical way to characterize continuous position-momentum complementarity. The emphasis is on how the two observable structures fit together.
At first, however, that pairing has no fixed physical scale. The construction gives it an exponential form in which momentum and position appear together with a positive normalization parameter, called alpha. The same pairing survives if momentum and alpha are rescaled by the same positive factor. In this framework, Pontryagin duality, the mathematical rule used to pair the two descriptions, fixes the structure of the link but does not by itself decide its physical scale.
Before physical normalization, the paper defines a one-parameter family of momentum operators. The parameter is positive, and the commutator between position and the corresponding momentum operator changes with that parameter. Changing the parameter therefore changes the mathematical operator family even though the abstract pairing can be preserved under the matching rescaling.
Where physics enters
The paper then imposes the physical canonical commutation relation and identifies alpha with the reduced Planck constant. This selects the physically normalized member of the family.
With that normalization in place, the pairing can be written as a plane wave. Identifying it with the standard wave form gives the link between momentum and wave number, and the wave's spatial period gives the de Broglie relation: wavelength equals Planck's constant divided by momentum. The paper does not attribute that relation to Pontryagin duality alone; it depends on the additional physical normalization.
How far the claim goes
The construction is narrower than a universal theory of continuous observables. It uses a restricted Schwartz-based kernel framework in which only kernels with defined contractions and compositions are admitted. It does not claim a general category covering arbitrary rigged Hilbert spaces with unrestricted distributional composition. The restriction concerns the analytic setting in which the construction is defined.
Nor does the paper claim that its Weyl characterization is equivalent to the finite-dimensional Hopf-algebraic definition of strong complementarity. No empirical measurements or experimental test are reported for the de Broglie relation, so the evidence remains mathematical and conceptual within the specified framework. That boundary is part of how the result should be read.
Declarations report no organizational support for the submitted work and no relevant financial or non-financial interests.
Paper data and sources
Original title: A Categorical Framework for Wave-Particle Complementarity: Continuous Observable Structures and Fourier-Pontryagin Duality
Authors: Samuel B. Soltau
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text