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Operational Distances Between Collapse Models: A Quantitative Comparison of CSL and the Schrödinger–Newton Model

Dewan Sajid Islam

Published Sep 4, 2026• 0 citations
Publisher

Abstract

We present a quantitative operational comparison of the Continuous Spontaneous Localization (CSL) model and the deterministic Schrödinger–Newton model within a specific operational context. Using a common operational pseudometric, we compare standard quantum mechanics, CSL, and the Schrödinger–Newton model for a harmonic oscillator with a position measurement. Within the Gaussian-closure and Gaussian-harmonic approximations adopted here, the three models yield Gaussian position distributions. The CSL sector is treated using Gaussian-closure moment dynamics, while the Schrödinger–Newton sector is evaluated numerically within the Gaussian harmonic approximation. We derive the corresponding total-variation distances and prove two structural properties of the operational pseudometric: monotonicity under coarse-graining and a data-processing inequality. We also establish a perturbative scaling result for the quantum-mechanical versus Schrödinger–Newton distance for a specific protocol. For representative parameters, the CSL effect is extremely small, with a QM–CSL distance of approximately 7.6 × 10⁻¹⁸, whereas the Gaussian-harmonic Schrödinger–Newton calculation gives a substantially larger QM–SN distance of approximately 0.75. We investigate the mass dependence of these distances and identify protocol-dependent crossover scales. The results provide a quantitative operational framework for comparing collapse models and for assessing their distinguishability in prospective experiments.

DOI
10.6084/m9.figshare.33438427

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We present a quantitative operational comparison of the Continuous Spontaneous Localization (CSL) model and the deterministic Schrödinger–Newton model within a specific operational context. Using a common operational pseudometric, we compare standard quantum mechanics, CSL, and the Schrödinger–Newton model for a harmonic oscillator with a position measurement. Within the Gaussian-closure and Gaussian-harmonic approximations adopted here, the three models yield Gaussian position distributions. The CSL sector is treated using Gaussian-closure moment dynamics, while the Schrödinger–Newton sector is evaluated numerically within the Gaussian harmonic approximation. We derive the corresponding total-variation distances and prove two structural properties of the operational pseudometric: monotonicity under coarse-graining and a data-processing inequality. We also establish a perturbative scaling result for the quantum-mechanical versus Schrödinger–Newton distance for a specific protocol. For representative parameters, the CSL effect is extremely small, with a QM–CSL distance of approximately 7.6 × 10⁻¹⁸, whereas the Gaussian-harmonic Schrödinger–Newton calculation gives a substantially larger QM–SN distance of approximately 0.75. We investigate the mass dependence of these distances and identify protocol-dependent crossover scales. The results provide a quantitative operational framework for comparing collapse models and for assessing their distinguishability in prospective experiments.
Wavefunction collapse models modify Schrödinger's equation so that it describes the collapse of a superposition of macroscopically distinguishable states as a dynamical process. This provides a basis for the resolution of the quantum measurement problem. An additional generic consequence of the collapse mechanism is that it causes particles to exhibit a tiny random diffusive motion. Here it is shown that for the continuous spontaneous localization (CSL) model—one of the most well developed collapse models—the diffusions of two sufficiently nearby particles are positively correlated. An experimental test of this effect is proposed in which random displacements of pairs of free nanoparticles are measured after they have been simultaneously released from nearby traps. The experiment must be carried out at sufficiently low temperature and pressure in order for the collapse effects to dominate over the ambient environmental noise. It is argued that these constraints can be satisfied by current technologies for a large region of the viable parameter space of the CSL model. The effect disappears as the separation between particles exceeds the CSL length scale. The test therefore provides a means of bounding this length scale.
Wavefunction collapse models modify Schrödinger's equation so that it describes the collapse of a superposition of macroscopically distinguishable states as a dynamical process. This provides a basis for the resolution of the quantum measurement problem. An additional generic consequence of the collapse mechanism is that it causes particles to exhibit a tiny random diffusive motion. Here it is shown that for the continuous spontaneous localization (CSL) model---one of the most well developed collapse models---the diffusions of two sufficiently nearby particles are positively correlated. An experimental test of this effect is proposed in which random displacements of pairs of free nanoparticles are measured after they have been simultaneously released from nearby traps. The experiment must be carried out at sufficiently low temperature and pressure in order for the collapse effects to dominate over the ambient environmental noise. It is argued that these constraints can be satisfied by current technologies for a large region of the viable parameter space of the CSL model. The effect disappears as the separation between particles exceeds the CSL length scale. The test therefore provides a means of bounding this length scale.
We formulate minimal and dissipative Continuous Spontaneous Localization (CSL) dynamics in the functional Schrödinger representation for a non-relativistic bosonic field. In this framework, the Fock-space state is encoded in a wave functional, and fixed particle-number wave functions are obtained by sector projection. For the minimal CSL coupling to the smeared mass density, this projection gives the standard nonlinear stochastic dynamics in each NNN-particle sector, with the collapse operator acting on the total smeared density of the configuration. This makes the amplification mechanism transparent and allows us to discuss sector superpositions, local probability balance, and the status of Bohmian equivariance at the wave-function level. We then consider a dissipative extension in which the collapse operator includes a smeared current contribution. The one-particle sector reproduces the expected dissipative CSL energy balance, while fixed many-body sectors contain additional collective momentum shifts and pair-mixing terms that are not reducible, in general, to independent one-particle contributions. Within a leading compact closure, the collective pair friction produces a non-extensive stationary mean kinetic energy: in three dimensions and for weak dissipation, TNcomp≃2Tβ/NT_N^{\rm comp}\simeq 2T_β/NTNcomp​≃2Tβ​/N, whereas the corresponding dilute energy remains extensive.
The Schrodinger-Newton (S-N) equation—a non-linear modification of quantum mechanics where the wave function experiences its own gravitational self-energy—has long been studied as a deterministic model for gravitational wave function collapse. This work provides a focused analysis of its phenomenology, explicitly treating it as a deterministic alternative to stochastic collapse models like Continuous Spontaneous Localization (CSL). We clarify the emergence of classical particle trajectories within this framework and derive its unique experimental signature: a mass- and separation-dependent shift in matter-wave interference fringes, distinct from the mere visibility loss predicted by decoherence or stochastic models. We provide a detailed roadmap for testing this prediction against competing approaches, critically assessing the experimental challenges and parameter constraints. Our work reframes the S-N equation as a concrete, falsifiable phenomenological model with a clear discriminant for future matter-wave experiments.
Dynamical reduction models propose a solution to the measurement problem in quantum mechanics: the collapse of the wave function becomes a physical process. We compute the predictions to decaying and flavor-oscillating neutral mesons for the two most promising collapse models, the QMUPL (quantum mechanics with universal position localization) model and the mass-proportional CSL (continuous spontaneous localization) model. Our results are showing (i) a strong sensitivity to the very assumptions of the noise field underlying those two collapse models and (ii) under particular assumptions the CSL case allows one even to recover the decay dynamics. This in turn allows one to predict the effective collapse rates solely based on the measured values for the oscillation (mass differences) and the measured values of the decay constants. The four types of neutral mesons (KKK meson, DDD meson, Bd{B}_{d}Bd​ meson, and Bs{B}_{s}Bs​ meson) lead surprisingly to ranges comparable to those put forward by Adler [J. Phys. A: Math. Theor. 40, 2935 (2007)] and Ghirardi, Rimini, and Weber [Phys. Rev. D 34, 470 (1986)]. Our results show that these systems at high energies are very sensitive to possible modifications of the standard quantum theory, making them a very powerful laboratory to rule out certain collapse scenarios and study the detailed physical processes solving the measurement problem.
Models of dynamical wave function collapse consistently describe the breakdown of the quantum superposition with the growing mass of the system by introducing non-linear and stochastic modifications to the standard Schrödinger dynamics. Among them, Continuous Spontaneous Localization (CSL) was extensively investigated both theoretically and experimentally. Measurable consequences of the collapse phenomenon depend on different combinations of the phenomenological parameters of the model-the strength λ and the correlation length rC-and have led, so far, to the exclusion of regions of the admissible (λ-rC) parameters space. We developed a novel approach to disentangle the λ and rC probability density functions, which discloses a more profound statistical insight.