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UCM 47: Canonical Commutation Relations from the Substrate Path Integral

Norbert Prebeck

Published Apr 12, 2026• 0 citations
Publisher

Abstract

We show that the canonical commutation relations of quantum mechanics follow from a structural property of the U-Cell Model substrate: the free action is exactly quadratic, making the substrate a system of coupled harmonic oscillators. Fourier decomposition into normal modes, followed by the Stone-von Neumann theorem, yields unique commutation relations with no free parameters. The anharmonic gravity-matter coupling does not modify the result, because the symplectic structure depends only on the kinetic term. One irreducible postulate remains: that the substrate fluctuates. Given this empirical fact, the structure, scale, and uniqueness of quantum mechanics follow from the substrate without further input. This closes Open Problem 2 of the UCM Mathematical Companion (Part Q).

DOI
10.5281/zenodo.19531780

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We show that the canonical commutation relations of quantum mechanics follow from a structural property of the U-Cell Model substrate: the free action is exactly quadratic, making the substrate a system of coupled harmonic oscillators. Fourier decomposition into normal modes, followed by the Stone-von Neumann theorem, yields unique commutation relations with no free parameters. The anharmonic gravity-matter coupling does not modify the result, because the symplectic structure depends only on the kinetic term. One irreducible postulate remains: that the substrate fluctuates. Given this empirical fact, the structure, scale, and uniqueness of quantum mechanics follow from the substrate without further input. This closes Open Problem 2 of the UCM Mathematical Companion (Part Q).
Quantum entanglement has no explanation in terms of local hidden variables (Bell 1964). The U-Cell Model (UCM) is an elastic lattice substrate — naively a local theory. We show that the UCM is in fact a non-local theory: entangled particles share a global pattern in the substrate network, not a local hidden variable. The transverse substrate modes ψ^a transform as spin-1/2 spinors under spatial rotations, a consequence of the cubic lattice symmetry and the Helmholtz decomposition. From this we derive that the UCM substrate reproduces the quantum-mechanical Bell correlation C(α,β) = −cos(α−β) exactly, including the maximum CHSH violation |S| = 2√2. The physical mechanism is the global substrate pattern maintained through the cell membranes, which propagate correlations instantaneously without carrying a usable signal. The UCM thus satisfies Bell's theorem, the no-communication theorem, and special relativity simultaneously. An open question is whether the membrane stiffness κ_mem/κ_cell is finite or infinite. If finite but very large, the UCM predicts a preferred frame — the substrate rest frame — as a testable departure from strict Lorentz invariance. Analysis code (ucm_bell.py) included for reproducibility. Part of the UCM preprint series (ORCID 0009-0005-5088-4339, records 19387201–19430524).
UCM Paper 36 (Theoretical). Derives the Schrödinger equation from the UCM substrate action in the non-relativistic limit (Theorem 1). The Feynman path integral is identified with the sum over membrane phase paths (Proposition 2). Double-slit interference follows from instantaneous membrane phase propagation (κ_mem >> κ_cell, Paper 31): both slits simultaneously acquire the global phase pattern without hidden variables (Proposition 3). The de Broglie relation follows from the dispersion relation; confirmed against the Davisson-Germer experiment at 0.1%. The substrate quantum mass m₀ = E_Λ/c² is the U-cell rest mass; composite particle masses arise from substrate binding energy. UCM preprint series on Zenodo, ORCID 0009-0005-5088-4339.
We prove the substrate H-theorem for a single Fourier mode of the substrate one-form bμ under stochastic Langevin dynamics derived from the substrate action of Smith (2026). The dissipation and noise are not postulated: they arise from the coupling chain substrate -> electron (via the guidance hypothesis pμ = bμ) → electromagnetic vacuum. The spectral density of this chain is J(ω) ∝ αω⁵, super-Ohmic with n = 5. The substrate temperature is T = Λfₐ², fixed without new free parameters by the MICROSCOPE-constrained WEP parameter fₐ ∼ 3×10⁻¹⁵. The fluctuation-dissipation relation is derived from the Kubo-Martin-Schwinger condition in Lorentzian signature; no Wick rotation is used. The Fokker-Planck equation has a unique fixed point, the von Mises distribution f_eq ∝ exp(Λ/T cosθ) with sharpness Λ/T = 1/fₐ² ~ 10²⁸. The relative entropy H[f] decreases monotonically. Crucially, J(0) = 0 exactly for any super-Ohmic spectral density, making the von Mises fixed point exact for the full non-Markovian dynamics. Via the Madelung transformation and the identification Φ0 = ℏ, substrate equilibrium forces ρ = R²: the Born rule. Quantum probability is substrate thermodynamics. Two open problems remain: mode coupling in the full field theory (Gap 1), and the multi-particle extension (Gap 3).
A formulation of supersymmetric quantum mechanics is given and superunitary and generalized canonical transformations are defined acting in a module. Next it is assumed that there are operators that give an irreducible representation of the canonical commutation and anticommutation relations, respectively, and it is proved that two such representations are connected by a uniquely determined superunitary transformation, under suitable domain assumptions. This extends the well-known uniqueness theorem of von Neumann to canonical (anti−) commutation relations using anticommuting parameters.
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory
Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.