arXiv:hep-th/0206184 · Progress of Theoretical Physics

Structure of Cubic Matrix Mechanics

Yoshiharu Kawamura A

Published Jan 1, 2003• 3 citations

Abstract

We study the structure of cubic matrix mechanics based on three-index objects. It is shown that there exists a counterpart of canonical structure in classical mechanics.

DOI: 10.1143/ptp.109.1

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arXiv:hep-th/0206184 · Progress of Theoretical Physics

Structure of Cubic Matrix Mechanics

Yoshiharu Kawamura A

Published Jan 1, 2003• 3 citations

arXiv HTML PDF

Abstract

We study the structure of cubic matrix mechanics based on three-index objects. It is shown that there exists a counterpart of canonical structure in classical mechanics.

DOI: 10.1143/ptp.109.1

Related papers

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Y. Kawamura

arXiv:hep-th/0207054Progress of Theoretical PhysicsFeb 1, 200320 citations

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Yoshiharu Kawamura A

arXiv:hep-th/0203007Progress of Theoretical PhysicsJun 1, 20022 citations

Cubic Matrix, Nambu Mechanics and Beyond

Yoshiharu Kawamura

Jul 6, 200212 citations

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Yoshiharu Kawamura A

arXiv:hep-th/0304149Progress of Theoretical PhysicsSep 1, 200329 citations

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Yong-Yi Huang

Results in PhysicsJan 1, 20221 citation

Reconstructing the Canonical Commutation Relation from a Correlation Matrix: From Three Technical Assumptions to $[X,P]=i\hbar$ Reconstructing the Canonical Commutation Relation from a Correlation Matrix: From Three Technical Assumptions to $[X,P]=i\hbar$

Yunlong Guan

Abstract The canonical commutation relation $[X,P]=i\hbar$ is a fundamental postulate of quantum mechanics. This paper presents a rigorous reconstruction of this relation from an alternative algebraic starting point: a correlation matrix defined on an abstract index set, supplemented by three technical assumptions regarding its spectral inverse and an emergent distance. The system relaxes to a translation-invariant state, where the correlation matrix acquires a Toeplitz structure. The equidistance constraint enforced by uniform correlations selects a uniform lattice in the continuum limit via $\Gamma$ -convergence. On this lattice, the Toeplitz matrix is diagonalized by the discrete Fourier transform, yielding a plane-wave basis. Translation symmetry yields the momentum operator; Fourier duality yields the position operator. The commutator $[X,P]=i\hbar$ is recovered as an algebraic corollary of the Toeplitz geometry and Fourier duality. The existence of a minimal nonzero action scale $\hbar$ arises from the topology of the compactification core ( $S^3$ , Hopf invariant $c_1=1$ ); its precise numerical value is not required for this derivation. The Hilbert space structure itself—the inner product, completeness of the basis, and the dual representation of observables—emerges naturally from the same Toeplitz–Fourier geometry. This paper does not claim a derivation from deeper principles—rather, it demonstrates that, given the three stated assumptions, the CCR can be rigorously recovered without postulating it as an independent axiom of quantum mechanics. ORCID: 0009-0007-1999-0293 Email: 1352533302@qq.comAbstract The canonical commutation relation $[X,P]=i\hbar$ is a fundamental postulate of quantum mechanics. This paper presents a rigorous reconstruction of this relation from an alternative algebraic starting point: a correlation matrix defined on an abstract index set, supplemented by three technical assumptions regarding its spectral inverse and an emergent distance. The system relaxes to a translation-invariant state, where the correlation matrix acquires a Toeplitz structure. The equidistance constraint enforced by uniform correlations selects a uniform lattice in the continuum limit via $\Gamma$-convergence. On this lattice, the Toeplitz matrix is diagonalized by the discrete Fourier transform, yielding a plane-wave basis. Translation symmetry yields the momentum operator; Fourier duality yields the position operator. The commutator $[X,P]=i\hbar$ is recovered as an algebraic corollary of the Toeplitz geometry and Fourier duality. The existence of a minimal nonzero action scale $\hbar$ arises from the topology of the compactification core ($S^3$, Hopf invariant $c_1=1$); its precise numerical value is not required for this derivation. The Hilbert space structure itself—the inner product, completeness of the basis, and the dual representation of observables—emerges naturally from the same Toeplitz–Fourier geometry. This paper does not claim a derivation from deeper principles—rather, it demonstrates that, given the three stated assumptions, the CCR can be rigorously recovered without postulating it as an independent axiom of quantum mechanics. ORCID: 0009-0007-1999-0293 Email: 1352533302@qq.com

Jul 9, 20260 citations

Reconstructing the Canonical Commutation Relation from a Correlation Matrix: From Three Technical Assumptions to

[X,P]=i\hbar

Abstract The canonical commutation relation

[X,P]=i\hbar

is a fundamental postulate of quantum mechanics. This paper presents a rigorous reconstruction of this relation from an alternative algebraic starting point: a correlation matrix defined on an abstract index set, supplemented by three technical assumptions regarding its spectral inverse and an emergent distance. The system relaxes to a translation-invariant state, where the correlation matrix acquires a Toeplitz structure. The equidistance constraint enforced by uniform correlations selects a uniform lattice in the continuum limit via

\Gamma

-convergence. On this lattice, the Toeplitz matrix is diagonalized by the discrete Fourier transform, yielding a plane-wave basis. Translation symmetry yields the momentum operator; Fourier duality yields the position operator. The commutator

[X,P]=i\hbar

is recovered as an algebraic corollary of the Toeplitz geometry and Fourier duality. The existence of a minimal nonzero action scale

\hbar

arises from the topology of the compactification core (

S^3

, Hopf invariant

c_1=1

); its precise numerical value is not required for this derivation. The Hilbert space structure itself—the inner product, completeness of the basis, and the dual representation of observables—emerges naturally from the same Toeplitz–Fourier geometry. This paper does not claim a derivation from deeper principles—rather, it demonstrates that, given the three stated assumptions, the CCR can be rigorously recovered without postulating it as an independent axiom of quantum mechanics. ORCID: 0009-0007-1999-0293 Email: 1352533302@qq.com