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The Canonical Commutation Relation is Unitary Due to Scaling Between Complementary Variables

Steve Faulkner

Published Dec 1, 2017• 0 citations
OpenAlex

Abstract

Abstract Abstract Textbook theory says that the Canonical Commutation Relation derives from the homogeneity of space. This paper shows that the Canonical Commutation Relation does not derive from homogeneity of space or the homogeneity symmetry itself, but derives from a duality viewpoint of homogeneity, seen both from the viewpoint of position space, and from the viewpoint of momentum space, combined. Additionally, a specific particular fixed scale factor, relating position space with momentum space is necessary. It is this additional scaling information which enables complementarity between the system variables and makes the system unitary. Without this particular scaling, the Canonical Commutation Relation is left non-unitary and broken. Indeed, unitarity is separate information, unconnected and logically independent of the quantum system's underlying symmetry. This single counter-example contradicts the current consensus that foundational symmetries, underlying quantum systems, are ontologically, intrinsically and unavoidably unitary. And thus removes ‘unitary ontology’, as reason, for axiomatically imposing unitarity (or self-adjointness) — by Postulate — on quantum mechanical systems. Keywords foundations of quantum theory, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary.

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

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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

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The Dual Dynamical Foundation of Orthodox Quantum Mechanics

Diana Taschetto, Ricardo Correa da Silva

This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the canonical commutation relations (CCR) from the tenets of Matrix Mechanics; (ii) A discussion of the meaning of Schrödinger's first derivation of the wave equation that not only improves on Joas and Lehner's 2009 investigation on the subject, but also demonstrates that the CCR follow of necessity from Schrödinger's first derivation of the wave equation, thus correcting the common misconception that the CCR were only posited by Schrödinger to pursue equivalence with Matrix Mechanics; (iii) A discussion of the mathematical facts and requirements involved in the equivalence of Matrix and Wave Mechanics that improves on F. A. Muller's classical treatment of the subject; (iv) A proof that the equivalence of Matrix and Wave Mechanics is necessitated by the formal requirements of a dual action functional from which both the dynamical postulates of orthodox quantum mechanics, von Neumann's process 1 and process 2, follow; (v) A critical assessment, based on (iii) and (iv), of von Neumann's construction of unified quantum mechanics over Hilbert space. Point (iv) is our main result. It brings to the open the important, but hitherto ignored, fact that orthodox quantum mechanics is no exception to the golden rule of physics that the dynamics of a physical theory must follow from the action functional. If orthodox quantum mechanics, based as it is on the assumption of the equivalence of Matrix and Wave Mechanics, has this "peculiar dual dynamics," as von Neumann called it, then this is so because by assuming the equivalence one has been assuming a peculiar dual action.

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On the principles of quantum mechanics

Eijiro Sakai

We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.

arXiv:quant-ph/0405069•May 13, 2004•15 citations
Abstract
Abstract Textbook theory says that the Canonical Commutation Relation derives from the homogeneity of space. This paper shows that the Canonical Commutation Relation does not derive from homogeneity of space or the homogeneity symmetry itself, but derives from a duality viewpoint of homogeneity, seen both from the viewpoint of position space, and from the viewpoint of momentum space, combined. Additionally, a specific particular fixed scale factor, relating position space with momentum space is necessary. It is this additional scaling information which enables complementarity between the system variables and makes the system unitary. Without this particular scaling, the Canonical Commutation Relation is left non-unitary and broken. Indeed, unitarity is separate information, unconnected and logically independent of the quantum system's underlying symmetry. This single counter-example contradicts the current consensus that foundational symmetries, underlying quantum systems, are ontologically, intrinsically and unavoidably unitary. And thus removes ‘unitary ontology’, as reason, for axiomatically imposing unitarity (or self-adjointness) — by Postulate — on quantum mechanical systems.
Keywords
foundations of quantum theory, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary.
Abstract The homogeneity symmetry is re-examined and shown to be non-unitary, with no requirement for the imaginary unit. This removes symmetry, as reason, for imposing unitarity (or self-adjointness) -- by Postulate. The work here is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a full and complete theory of quantum randomness. Keywords foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
The homogeneity symmetry is re-examined and shown to be non-unitary. This is motivated by the prospect that logical independence in elementary algebra, entering quantum mathematics, will constitute the basis for a theory explaining quantum randomness. Keywords: foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
Reconstructing the Canonical Commutation Relation from a Correlation Matrix: From Three Technical Assumptions to [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ
Abstract The canonical commutation relation [X,P]=iℏ[X,P]=i\hbar[X,P]=iℏ 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 Γ\GammaΓ-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]=iℏ[X,P]=i\hbar[X,P]=iℏ is recovered as an algebraic corollary of the Toeplitz geometry and Fourier duality. The existence of a minimal nonzero action scale ℏ\hbarℏ arises from the topology of the compactification core (S3S^3S3, Hopf invariant c1=1c_1=1c1​=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
This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the canonical commutation relations (CCR) from the tenets of Matrix Mechanics; (ii) A discussion of the meaning of Schrödinger's first derivation of the wave equation that not only improves on Joas and Lehner's 2009 investigation on the subject, but also demonstrates that the CCR follow of necessity from Schrödinger's first derivation of the wave equation, thus correcting the common misconception that the CCR were only posited by Schrödinger to pursue equivalence with Matrix Mechanics; (iii) A discussion of the mathematical facts and requirements involved in the equivalence of Matrix and Wave Mechanics that improves on F. A. Muller's classical treatment of the subject; (iv) A proof that the equivalence of Matrix and Wave Mechanics is necessitated by the formal requirements of a dual action functional from which both the dynamical postulates of orthodox quantum mechanics, von Neumann's process 1 and process 2, follow; (v) A critical assessment, based on (iii) and (iv), of von Neumann's construction of unified quantum mechanics over Hilbert space. Point (iv) is our main result. It brings to the open the important, but hitherto ignored, fact that orthodox quantum mechanics is no exception to the golden rule of physics that the dynamics of a physical theory must follow from the action functional. If orthodox quantum mechanics, based as it is on the assumption of the equivalence of Matrix and Wave Mechanics, has this "peculiar dual dynamics," as von Neumann called it, then this is so because by assuming the equivalence one has been assuming a peculiar dual action.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.