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arXiv:2608.05972

Geometry-Only CSL/DP Ratios and the Nonuniqueness of Decoherence Kernels

Randy Davila, Gerard J. Milburn

quant-phgr-qcSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
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Abstract

We study idealized levitated protocols that create spatial superpositions of massive test particles. For each protocol, we compare the dimensionless contrast-loss exponent of mass-proportional continuous spontaneous localization (CSL) with the Diósi--Penrose (DP) self-energy exponent $E_Gτ/\hbar$. We first prove that the point-particle CSL separation kernel has an exact random-unitary realization: Gaussian momentum kicks arriving at Poisson-distributed times produce the same unconditional decay of spatial coherence, although a pure state conditioned on the complete kick record remains pure. The separation kernel alone therefore specifies an operational decoherence law, not the occurrence of objective collapse. We then show that the ratio of the CSL and DP exponents is independent of particle mass and interrogation time. In the point-particle model it depends only on branch separation and an effective distance; for the standard GRW reference parameters, its resolved-superposition crossover is $x_*\approx1.91\,\mathrm{nm}$. For rigid spherical bodies with an arbitrary normalized radial mass profile, total mass, overall density scale, and interrogation time again cancel, leaving a dimensionless geometry factor. The results distinguish three requirements for a decisive experiment: detectable absolute effects, a controlled comparison of CSL and DP scales, and observables capable of discriminating physically different dynamics that share the same ensemble decoherence kernel.

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The Diosi-Penrose model of quantum-classical boundary postulates gravity-related spontaneous wave function collapse of massive degrees of freedom. The decoherence effects of the collapses are in principle detectable if not masked by the overwhelming environmental decoherence. But the DP (or any other, like GRW, CSL) spontaneous collapses are not detectable themselves, they are merely the redundant formalism of spontaneous decoherence. To let DP collapses become testable physics, recently we extended the DP model and proposed that DP collapses are responsible for the emergence of the Newton gravitational force between massive objects. We identified the collapse rate, possibly of the order of 1/ms, with the rate of emergence of the Newton force. A simple heuristic emergence (delay) time was added to the Newton law of gravity. This non-relativistic delay is in peaceful coexistence with Einstein's relativistic theory of gravitation, at least no experimental evidence has so far surfaced against it. We derive new predictions of such a 'lazy' Newton law that will enable decisive laboratory tests with available technologies. The simple equation of 'lazy' Newton law deserves theoretical and experimental studies in itself, independently of the underlying quantum foundational considerations.

arXiv:1312.6404•Journal of Physics Conference Series•Apr 14, 2014•7 citations
We study idealized levitated protocols that create spatial superpositions of massive test particles. For each protocol, we compare the dimensionless contrast-loss exponent of mass-proportional continuous spontaneous localization (CSL) with the Diósi--Penrose (DP) self-energy exponent EGτ/ℏE_Gτ/\hbarEG​τ/ℏ. We first prove that the point-particle CSL separation kernel has an exact random-unitary realization: Gaussian momentum kicks arriving at Poisson-distributed times produce the same unconditional decay of spatial coherence, although a pure state conditioned on the complete kick record remains pure. The separation kernel alone therefore specifies an operational decoherence law, not the occurrence of objective collapse. We then show that the ratio of the CSL and DP exponents is independent of particle mass and interrogation time. In the point-particle model it depends only on branch separation and an effective distance; for the standard GRW reference parameters, its resolved-superposition crossover is x∗≈1.91 nmx_*\approx1.91\,\mathrm{nm}x∗​≈1.91nm. For rigid spherical bodies with an arbitrary normalized radial mass profile, total mass, overall density scale, and interrogation time again cancel, leaving a dimensionless geometry factor. The results distinguish three requirements for a decisive experiment: detectable absolute effects, a controlled comparison of CSL and DP scales, and observables capable of discriminating physically different dynamics that share the same ensemble decoherence kernel.
The Diósi–Penrose (DP) model assigns a spatial superposition the lifetime τ_DP = ħ/ΔE_G, computed from the instantaneous Newtonian self-energy difference of its branches. We ask what local relativistic causality adds to this picture. First, in a relativistic scalar-field model of the branch difference, a finite protocol duration T replaces the logarithmic infrared dependence ln(ξ/σ_eff) of static configurations by ln[min(ξ, cT)/σ_eff]; the result is independent of the branch separation ξ only for ξ > cT, whereas laboratory protocols satisfy cT ≫ ξ. Second, defining the causal self-crossing time ⟨t⟩ of a branch from the retarded Green function, we prove that E_self⟨t⟩ = Gm²/c for every regular mass profile. Consequently τ_DP/⟨t⟩ = (m_H/m)² with m_H = M_P/√2 ≈ 15.39 μg, independently of the profile and of the regularization width: above m_H the DP reduction would be completed before the self-energy that drives it is causally established across the branch. We also show that the switch-on action deficit of the branch difference vanishes identically by mass conservation. What happens at m_H depends on whether the branch phase keeps accumulating beyond ⟨t⟩. If it does (H1), the decoherence time crosses over from the DP law ∝ m⁻² to a causally limited ∝ m^(−2/3) at m_H, for every profile. If it does not (H2), the phase acquired per causal interval is exactly (m/m_H)², and m_H acts as a Heisenberg cut. Both hypotheses predict that a superposition with m ≥ m_H and ξ ≫ σ_eff loses coherence within about 2⟨t⟩, while their predictions below m_H differ. All results are reproduced by openly available code (doi:10.5281/zenodo.23064101).
A model of the vacuum as a face-centered-cubic lattice predicts a two-step quantum–classical threshold, at 13.1 and 22.6 µg, from the hypothesis that a center-of-mass excitation stays coherent only while its reduced Compton wavelength is resolvable by the lattice. Collapse models also limit macroscopic coherence, so the two must be told apart by more than a scale. We derive three discriminators from the model's postulates and compute the counterpart in Diósi–Penrose collapse and in continuous spontaneous localization (CSL). The thresholds depend on total mass alone, while both collapse models depend on density. Below 13.1 µg the model predicts no intrinsic decoherence, where collapse takes 1 s for Diósi–Penrose and about a microsecond for CSL at its experimental bound, for 10⁻¹⁵ kg over 1 µm. The model predicts no measurable heating; read as position localization, its own rate scale would exceed the LISA Pathfinder bound by 66 orders of magnitude, so coherence loss above the threshold must be dephasing in the energy basis. If its effective mode mass is the relevant mass, a 16.2 µg mechanical cat state then bounds the irreversible coupling at 38 orders below its natural scale. A worked levitated-particle example and a public script complete the analysis.
In the Bose–Marletto–Vedral (BMV) class of tabletop gravity experiments, two mesoscopic masses are each placed in a spatial superposition and their Newtonian interaction is used as a witness of gravitationally induced entanglement. Aspelmeyer [5] has sharpened the standard argument: observing entanglement at the 10-15 kg scale already rules out current Diósi–Penrose (DP) collapse models. Here we ask a complementary question: when the source mass itself is the object in superposition, what is the sign of the gravity-induced decoherence *rate* as a function of M/r (mass over characteristic superposition scale), and where in the mass range is that rate actually measurable? We compare three branches: (A) linearized quantum field theory, which gives reversible phase accumulation and Gamma = 0; (B) DP / continuous spontaneous localization (CSL) collapse, for which Gamma > 0 and grows steeply with M/r (positive trend); and (C) a phenomenological strawman of variable vacuum-foam / stochastic-spacetime suppression, Gammafoam = Gammafbg exp(-3 kappa GM/(rc2)), for which the trend is negative. Branch C is not derived from any specific quantum-gravity model; it is introduced only to give the sign question a non-trivial alternative. We solve the rates for a uniform sphere (density rho = 2000 kg/m3, superposition separation d = 10 r) and find: (i) a rate-resolved window M ~ 2×10-16 to 10-14 kg in which the DP rate exceeds a BMV-class environmental floor of 0.05 s-1 but is still measurable — this window lies below the regime where entanglement-survival alone, à la Aspelmeyer, already excludes DP; (ii) the phenomenological strawman foam-suppression branch is observationally degenerate with QFT-zero on Earth, because d ln Gammafoam / d ln M = -2 kappa GM/(rc2) ~ -10-37 to -10-35 across the entire lab range. Discriminating this toy from QFT-zero would require the near-Schwarzschild regime x = GM/(rc2) ~ O(1). Shot-noise-limited run counts are ~6 at M = 10-15 kg, ~104 at M = 10-16 kg, and environment-limited below M = 10-16 kg. All numerical values are illustrative/synthetic order-of-magnitude estimates.
This note evaluates Diósi–Penrose gravitational self-energy decoherence for a MAQRO-like spatial superposition using a rigid homogeneous-sphere mass model. For (M=10^{10},\mathrm{amu}) and (s=100,\mathrm{nm}), the Newtonian DP timescale lies in the range (\sim 9)–(100) minutes across representative densities. A Yukawa correction (V(r)=-(Gm_1m_2/r),[1+\alpha e^{-r/\lambda}]) is incorporated by modifying the DP kernel; for (\lambda \gg {R,s}), the timescale is reduced approximately by a factor (1+\alpha). For the benchmark ((\alpha,\lambda)=(612.5,20.4,\mu\mathrm{m})), the predicted timescale is (\sim 0.9)–(9.7) seconds.
Local branch selection—a frame-independent postulate under which environmental decoherence einselects a branch structure and a local stochastic process realizes one branch with Born probability—has so far been implemented with continuous spontaneous localization (CSL), whose noise is phenomenological. Here we determine which physical noises can play this role. First we prove a kernel-universality theorem: for any set of commuting Hermitian collapse operators driven by Gaussian noises with an arbitrary positive-semidefinite correlation kernel C C C, the einselected branch weights are exact martingales, so absorption occurs with Born probability for every kernel; the kernel controls only the selection rate Γ=ΔTC Δ \Gamma=\Delta^{T}C\,\Delta Γ=ΔTCΔ, where Δ \Delta Δ is the vector of branch eigenvalue differences. For the Diósi–Penrose (DP) kernel G/ℏ∣x−x′∣ G/\hbar|\mathbf{x}-\mathbf{x}'| G/ℏ∣x−x′∣ acting on mass densities, the rate is Γ=2EG/ℏ \Gamma=2EG/\hbar Γ=2EG/ℏ, with EG EG EG the gravitational self-energy of the branch mass-density difference: Penrose's collapse criterion emerges as the nondegeneracy condition of the Born martingale, and gravitationally induced collapse becomes a parameter-free physical implementation of the selection postulate, with τsel∼3×10−16  \tau_{\rm sel}\sim3\times10^{-16}\, τsel​∼3×10−16s for a milligram pointer. Second, we prove a no-selection theorem: noise coupled as a random Hamiltonian (classical stochastic spacetime fluctuations included) leaves every branch weight exactly constant along every realization while dephasing the ensemble—decoherence without selection—so classical metric-fluctuation dephasing cannot implement the postulate; selection requires measurement-type (norm-changing) noise. Third, for the symmetric EPR configuration we show that gravitational noise cross-correlation between the wings preserves no-signaling exactly (the cross terms of the correlated-kernel Lindblad dissipator vanish under the local partial trace), shifts only the selection rate, and is of relative size ∼R δ2/L3≈4×10−9 \sim R\,\delta^{2}/L^{3}\approx4\times10^{-9} ∼Rδ2/L3≈4×10−9 at one-meter separation; a common-mode stall at perfect correlation is exhibited and verified numerically. Stochastic simulations confirm Born statistics across kernels to sampling precision, machine-zero weight drift for random-unitary noise, and the rate law across the full correlation range. The results upgrade the selection postulate from “CSL-compatible” to “gravitationally implementable,” identify exactly which proposed mechanisms can and cannot supply its randomness, and inherit the active experimental program constraining the DP model.
The Diosi-Penrose model of quantum-classical boundary postulates gravity-related spontaneous wave function collapse of massive degrees of freedom. The decoherence effects of the collapses are in principle detectable if not masked by the overwhelming environmental decoherence. But the DP (or any other, like GRW, CSL) spontaneous collapses are not detectable themselves, they are merely the redundant formalism of spontaneous decoherence. To let DP collapses become testable physics, recently we extended the DP model and proposed that DP collapses are responsible for the emergence of the Newton gravitational force between massive objects. We identified the collapse rate, possibly of the order of 1/ms, with the rate of emergence of the Newton force. A simple heuristic emergence (delay) time was added to the Newton law of gravity. This non-relativistic delay is in peaceful coexistence with Einstein's relativistic theory of gravitation, at least no experimental evidence has so far surfaced against it. We derive new predictions of such a 'lazy' Newton law that will enable decisive laboratory tests with available technologies. The simple equation of 'lazy' Newton law deserves theoretical and experimental studies in itself, independently of the underlying quantum foundational considerations.