A theoretical preprint argues that the abstract duality behind the position-momentum relationship does not, by itself, determine the physical scale of momentum. The mathematical structure can supply a family of possible labels for the dual coordinate, but an additional physical normalization is needed before one of those labels can be identified with momentum. The authors describe the work as a clarification of logical dependence, not a new derivation of the de Broglie relation.
That distinction is the central question of the note: which mathematical and physical steps underlie the usual position-momentum correspondence? The work is entirely theoretical. It studies the additive real line, its continuous character group, Fourier transforms, translations and modulations, as well as Schwartz functions and their continuous dual. There are no human, animal, clinical or experimental participants.
The scale is not supplied for free
Pontryagin duality enters by pairing the real line with continuous characters, the mathematical objects used to represent the dual side. Those characters can be described using a wave-number coordinate, while a proposed momentum coordinate is allowed to be a rescaled version of it, with a scale factor between them. The key point is not the notation itself: the abstract dual group does not say which value of that scale factor is the physical one. Momentum is therefore calibrated by physics rather than selected by duality alone.
The note makes that calibration explicit through a family of momentum-like differential operators. For a chosen scale, the commutator with the position operator is i times that scale times the identity operator. Imposing the standard position-momentum commutation normalization then fixes the scale to the reduced Planck constant. In the paper's logic, this is the step that identifies the abstract dual coordinate with physical momentum.
Fourier analysis adds another distinction
Fourier analysis does not simply turn every position-space operation into ordinary multiplication in momentum space. The unitary Fourier transform carries pointwise multiplication by position into convolution, an operation that combines functions rather than multiplying them at each point. The diagonal distribution is carried through the addition map for the two dual coordinates, rather than becoming pointwise multiplication by momentum. That distinction is central to the note's formal account.
The authors also limit what can be inferred from the Weyl relation, the representation-theoretic relation between translations and modulations. They keep it separate from categorical strong complementarity, a stronger notion in the paper's framework. The result is a carefully bounded claim: the relation is part of the representation theory, but the note does not establish that it is identical to every stronger or operational meaning of complementarity.
The wavelength claim is narrower than it sounds
Once the physical normalization has been chosen, the ideal character associated with nonzero momentum has a spatial period equal to Planck's constant divided by the absolute value of momentum. This is the de Broglie wavelength relation in the note's calibrated setting. At zero momentum, the character is constant, so its wavelength is interpreted as infinite.
That wording matters. The result concerns a generalized character, or ideal momentum eigenfunction, not every square-integrable wave packet. A generic wave packet need not have one single wavelength under this statement. The paper is therefore not reporting a measured wavelength, and it does not offer experimental validation.
The same caution applies to the distributions used in the formal work. The analysis stays within a standard rigged Hilbert-space setting and uses only distributional operations the note explicitly defines. It does not assume that arbitrary tempered distributions can be multiplied globally. That restriction is part of the result's scope, rather than a claim about all possible distributional constructions.
Taken together, the note draws a line between three stages: duality supplies the abstract character group; Fourier analysis carries position-space operations into their momentum-side forms; and the physical commutation rule selects the momentum scale. The authors' conclusion is modest but precise. The de Broglie period follows after calibration, while Planck's constant is an input to that calibration, not something supplied by Pontryagin duality alone. As a versioned arXiv preprint, the work presents a conceptual clarification rather than empirical evidence about particles or wavelengths.
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
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