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Gravitational Time Dilation Decoheres Without a Rate

Zongmin Liu

Published Dec 2, 2023• 0 citations
Publisher

Abstract

A localization rate $F(\Delta)$ is producible by some Markovian translation-covariant completely positive dynamics if and only if it is a continuous negative definite function of the separation---a L\'evy exponent. We close this question for every standard decoherence mechanism, and the two sides close differently. \emph{Compatible, by construction}: every mechanism whose rate is a self-minus-cross pairing with a positive Fourier symbol is a L\'evy exponent with an explicit positive L\'evy measure. Di\'osi--Penrose is one, with measure $|\tilde\mu(k)|^{2}/k^{2}$; so are collisional decoherence, thermal photon scattering, and CSL. \emph{Anomalous}: gravitational time dilation does not couple to position but to internal energy, giving $F(\Delta)\,t=-\log|\phi_{H}(gt\Delta/\hbar c^{2})|$ with $\phi_H$ the characteristic function of the internal energy. This is a L\'evy exponent if and only if that energy distribution is infinitely divisible. An unbounded thermal ladder is, with L\'evy measure $c_n=q^{n}/n$ verified to $10^{-9}$; \emph{every bounded internal spectrum is not}, so no Markovian translation-covariant completely positive master equation reproduces its separation profile. For a two-level internal system the coefficients alternate, $c_n=(-1)^{n+1}r^{n}/n$ with $r=p/(1-p)$; for a thermal ladder truncated at $M$ levels the first negative coefficient sits at $n=M$ exactly, with depth $q^{M}$. The obstruction is therefore sharp in principle and geometrically soft in practice, and it is invisible in the leading L\'evy coefficients, which reproduce the untruncated answer to six figures.

DOI
10.5281/zenodo.21924100

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Complete positivity and translation covariance force the localization rate F(Δ)F(\Delta)F(Δ) of any Markovian open dynamics---environmental decoherence and objective collapse alike---to be a continuous negative definite function of the path separation Δ\DeltaΔ, i.e.\ a L\'evy exponent. We turn this structural fact into a finite, model-free, and \emph{complete} test on data. Sampling FFF at commensurate separations Fk≡F(kd)F_k\equiv F(kd)Fk​≡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, (F2−F1)2≤F1F3≤F1(F1+2F2)(F_2-F_1)^2\le F_1F_3\le F_1(F_1+2F_2)(F2​−F1​)2≤F1​F3​≤F1​(F1​+2F2​) together with F2≤4F1F_2\le 4F_1F2​≤4F1​, saturated respectively by a single momentum quantum and by Gaussian noise---the two extreme rays of the admissible cone. The test detects a 1%1\%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.
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