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A Measured Decoherence Curve Must Be a L\'evy Exponent

Zongmin Liu

Published Mar 22, 2025• 0 citations
Publisher

Abstract

Complete positivity and translation covariance force the localization rate $F(\Delta)$ of any Markovian open dynamics---environmental decoherence and objective collapse alike---to be a continuous negative definite function of the path separation $\Delta$, i.e.\ a L\'evy exponent. We turn this structural fact into a finite, model-free, and \emph{complete} test on data. Sampling $F$ at commensurate separations $F_k\equiv F(kd)$ reduces the L\'evy--Khintchine representation to a linear feasibility problem whose dual is a Chebyshev positivity cone; the cone is polyhedral only for two points. Three points already give closed-form walls, $(F_2-F_1)^2\le F_1F_3\le F_1(F_1+2F_2)$ together with $F_2\le 4F_1$, saturated respectively by a single momentum quantum and by Gaussian noise---the two extreme rays of the admissible cone. The test detects a $1\%$ super-quadratic admixture from three points, and it is necessary \emph{and} sufficient, so a curve that survives is realized by an actual noise measure. A visibility-versus- separation scan therefore falsifies Markovian translation-covariant evolution with no model and no fitting.

DOI
10.5281/zenodo.21924087

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Standard quantum mechanics relies on an observer-dependent measurement postulate and an artificial linear-nonlinear dichotomy to resolve the superposition of macroscopically distinct states. While objective collapse models---such as Continuous Spontaneous Localization (CSL), Ghirardi-Rimini-Weber (GRW), and Di\'{o}si-Penrose gravitationally induced models---introduce stochastic non-linearities to achieve objective state reduction, they suffer from three fundamental pathologies: continuous energy non-conservation (spontaneous heating), reliance on memoryless white-noise approximations, and unphysical infinite spatial tails. In this work, I present Quantum Wave Function (QWF) Theory, a comprehensive non-perturbative framework that elevates the state functional Ψ\PsiΨ to an exact physical ontology. QWF introduces a non-Markovian colored noise memory kernel K(t−τ)K(t - \tau)K(t−τ) governed by a finite high-frequency cutoff, capturing realistic vacuum and environmental back-action times. To resolve the long-standing spontaneous heating anomaly, I incorporate a momentum-dependent dissipative counter-term D^diss\hat{D}_{\text{diss}}D^diss​ into the stochastic differential equation, strictly enforcing energy conservation ddt⟨H^⟩=0\frac{d}{dt} \langle \hat{H} \rangle = 0dtd​⟨H^⟩=0 in closed isolated systems. Furthermore, I eliminate the infinite tail pathology by introducing a state-dependent stochastic amplitude filtering protocol where the microscopic collapse coupling dynamically increases in the asymptotic regime to enforce exponential tail damping below a physical threshold ϵ\epsilonϵ, without breaking density matrix linearity. Extending beyond non-relativistic domains, I formulate a fully covariant quantum field description on arbitrary spacelike hypersurfaces Σ\SigmaΣ using a Tomonaga-Schwinger stress-energy framework, ensuring causality and preventing superluminal signaling. Finally, I prove via martingale analysis that standard Born-rule statistics (P=∣Ψ∣2P = |\Psi|^2P=∣Ψ∣2) emerge naturally as a dynamical attractor of the stochastic localization process. QWF provides a mathematically rigorous, falsifiable, and complete resolution to the quantum measurement problem with explicit, testable signatures for modern gravitational wave and optomechanical experiments.