Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.23935

A Categorical Framework for Wave-Particle Complementarity: Continuous Observable Structures and Fourier-Pontryagin Duality

Samuel B. Soltau

quant-phSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations
v1 · Aug 25, 2026v2 · Sep 1, 2026v3 · Sep 13, 2026
arXivHTMLPDF

Abstract

We present a formulation wave-particle complementarity and the de Broglie relation $λ=h/p$ within the framework of categorical quantum mechanics on rigged Hilbert spaces. Position and momentum are represented by continuous dagger-Frobenius structures associated with the locally compact abelian group $\mathbb{R}$ and its Pontryagin dual. The Fourier transform is the unitary implementation of Pontryagin duality and relates the two observable structures. We show that a general character pairing $χ_p^{(α)}(x)=\exp(ipx/α)$ yields a one-parameter family of unitarily equivalent Fourier transforms. Pontryagin duality therefore determines the structural form of the complementarity, but not the numerical value of Planck's constant. The identification $α=\hbar$ is a physical input fixed by the Weyl commutation relations. With this input, the spatial period of the character yields $λ=h/p$.

Comments
1 figure

Related papers

Physics of Unitary Transformations

K. Kong Wan

Q23(1) Explain why a coordinate representation space and its corresponding momentum representation space introduced in §18.4.2 give rise to two mathematically different descriptions of the position and momentum which are physically equivalent.

•Aug 27, 2020•0 citations

Quantum Mechanics on Lie Groups: I. Noncommutative Fourier Transforms

Mathieu Beauvillain, Blagoje Oblak, Marios Petropoulos

Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.

quant-pharXiv:2512.19840•Dec 24, 2025

Translational Invariance and Spectral Objectivity: Plancherel-Parseval and Uncertainty under a Unitary Shift Representation

Gustaf Ullman

This working paper examines translational invariance as a minimal model of observer equivariance. Instead of treating translations only as displacements in physical space, it interprets them as abstract shifts between observer perspectives acting on a space of perspectives. The paper separates three layers: standard harmonic analysis, its observer-equivariance interpretation, and the additional physical calibration required to identify the dual spectral variable with momentum. Using a strongly continuous unitary representation of a locally compact abelian shift group, the note recalls the spectral description given by Pontryagin duality, Plancherel--Parseval, and, in the Euclidean case, the Heisenberg--Kennard uncertainty relation. The main interpretive point is that localization in perspective space and localization in the dual structural variable are subject to the usual Fourier trade-off. In this sense, uncertainty is read not merely as a feature of physical position and momentum, but as a structural constraint on perspective localization and the resolution of invariant modes. This version is a technical building block in a broader series of notes on observer equivariance, epistemic structural realism, and structural objectivity. The mathematical content is standard; the contribution is the reinterpretation of translational invariance as invariance under shifts of observer perspective.

•May 23, 2026•0 citations

The ubiquitous XP commutator

Alberto C de la Torre

Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation $[X,P]=i\hbar$. The article is complete and self contained, adequate for didactic use.

arXiv:quant-ph/0509171•European Journal of Physics•Jan 3, 2006•0 citations

Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator

Jean-Pierre Gazeau

We introduce a bi-oriented interlaced matrix product on $2\times 2$ matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley--Dickson doubling. The opposite orientations $X_R$ and $X_L$ coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators $Q,P$ on $L^2(\mathbb{R})$, a matrix $D$ and its companion $\widetilde{D}$ satisfy, on the common Schwartz domain, $\widetilde{D} X_R D =\widetilde{D} X_L D =2H Id_2$, where $H=(P^2+Q^2)/2$. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations $Φ=A+iB$ and $Ψ=A-iB$ decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space $S(\mathbb{R}^2)\subset L^2(\mathbb{R}^2) \subset S'(\mathbb{R}^2)$; the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.

math-phCross-listedarXiv:2609.28511•Sep 22, 2026•0 citations

The ubiquitous XP commutator

Alberto C. De La Torre

Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation $[X,P]=i\\hbar$. The article is complete and self contained, adequate for didactic use.

•Jan 1, 2005•3 citations
We present a formulation wave-particle complementarity and the de Broglie relation λ=h/pλ=h/pλ=h/p within the framework of categorical quantum mechanics on rigged Hilbert spaces. Position and momentum are represented by continuous dagger-Frobenius structures associated with the locally compact abelian group R\mathbb{R}R and its Pontryagin dual. The Fourier transform is the unitary implementation of Pontryagin duality and relates the two observable structures. We show that a general character pairing χp(α)(x)=exp⁡(ipx/α)χ_p^{(α)}(x)=\exp(ipx/α)χp(α)​(x)=exp(ipx/α) yields a one-parameter family of unitarily equivalent Fourier transforms. Pontryagin duality therefore determines the structural form of the complementarity, but not the numerical value of Planck's constant. The identification α=ℏα=\hbarα=ℏ is a physical input fixed by the Weyl commutation relations. With this input, the spatial period of the character yields λ=h/pλ=h/pλ=h/p.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.
This working paper examines translational invariance as a minimal model of observer equivariance. Instead of treating translations only as displacements in physical space, it interprets them as abstract shifts between observer perspectives acting on a space of perspectives. The paper separates three layers: standard harmonic analysis, its observer-equivariance interpretation, and the additional physical calibration required to identify the dual spectral variable with momentum. Using a strongly continuous unitary representation of a locally compact abelian shift group, the note recalls the spectral description given by Pontryagin duality, Plancherel--Parseval, and, in the Euclidean case, the Heisenberg--Kennard uncertainty relation. The main interpretive point is that localization in perspective space and localization in the dual structural variable are subject to the usual Fourier trade-off. In this sense, uncertainty is read not merely as a feature of physical position and momentum, but as a structural constraint on perspective localization and the resolution of invariant modes. This version is a technical building block in a broader series of notes on observer equivariance, epistemic structural realism, and structural objectivity. The mathematical content is standard; the contribution is the reinterpretation of translational invariance as invariance under shifts of observer perspective.
Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ. The article is complete and self contained, adequate for didactic use.
We introduce a bi-oriented interlaced matrix product on 2×22\times 22×2 matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley--Dickson doubling. The opposite orientations XRX_RXR​ and XLX_LXL​ coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators Q,PQ,PQ,P on L2(R)L^2(\mathbb{R})L2(R), a matrix DDD and its companion D~\widetilde{D}D satisfy, on the common Schwartz domain, D~XRD=D~XLD=2HId2\widetilde{D} X_R D =\widetilde{D} X_L D =2H Id_2DXR​D=DXL​D=2HId2​, where H=(P2+Q2)/2H=(P^2+Q^2)/2H=(P2+Q2)/2. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations Φ=A+iBΦ=A+iBΦ=A+iB and Ψ=A−iBΨ=A-iBΨ=A−iB decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space S(R2)⊂L2(R2)⊂S′(R2)S(\mathbb{R}^2)\subset L^2(\mathbb{R}^2) \subset S'(\mathbb{R}^2)S(R2)⊂L2(R2)⊂S′(R2); the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.
Position and momentum of a particle can take any value in a continuous spectrum; these values are independent but their indeterminacies are correlated; momentum and position are mutually the generators of the transformations in each other. It is shown in a concise way, how all these features arise solely from their commutation relation [X,P]=ihbar[X,P]=i\\hbar[X,P]=ihbar. The article is complete and self contained, adequate for didactic use.