WHY WAVES? The Schrödinger Equation, Wave-Particle Duality, the Uncertainty Principle, and Quantum Measurement from One Algebra — A Visual Guide from A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ)

Daland Montgomery

Published Mar 16, 2026• 0 citations

Abstract

Abstract: We derive the Schrödinger equation, wave-particle duality, the uncertainty principle, the double slit experiment, and quantum measurement from the ascending superoperator of A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) at KMS inverse temperature β = 2π. The crystal's four eigenvalues {1, ½, ⅓, ⅙} are the four frequency bands of the vacuum's wavelet decomposition. The MERA tensor network is a discrete wavelet transform (confirmed: Alves 2024, wMERA = MERA + DWT). Wave-particle duality is the Fourier transform between a single MERA layer (particle) and the full tower (wave). The uncertainty principle arises from the finite bond dimension χ = 6. Measurement is the ascending superoperator destroying the kernel (24 mixed modes). The Schrödinger equation is the continuous limit of the MERA circuit, with H = generator of the ascending superoperator and i = Wick rotation from imaginary to real time. The vacuum oscillates at β = 2π (Bisognano-Wichmann theorem); the crystal decomposes this oscillation into four harmonics; those harmonics are the four forces. Nine independent research traditions confirm this synthesis: Feynman, Wilson, Bisognano-Wichmann, Daubechies, Vidal, Swingle, Harlow-Preskill, Evenbly-Vidal, and Alves. Five figures, an interactive 3D HTML demonstration, and a Python verification script are included. Zero free parameters. Copyright © 2026 Daland Montgomery. This work is licensed under CC BY-SA 4.0. COPYLEFT NOTICE: Any work, derivation, or industrial application incorporating this material must be distributed under the same Open Source license. Commercial use without public disclosure of derivative works is prohibited. For a private, proprietary license (exempt from ShareAlike requirements), contact: quidbit@icloud.com Software Implementation: The formulas and constants derived in this work are implemented in the CrystalAgent engine, available under the AGPL-3.0 license at: https://github.com/CrystalToe/CrystalAgent.

DOI: 10.5281/zenodo.19053594

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