Gravitationally induced wave-function collapse from dynamical bifurcation

Abstract: We propose an effective non-relativistic framework in which wave-function collapse emerges as a deterministic dynamical instability induced by gravitational self-interaction and regulated by short-distance repulsion. The dynamics is described by a nonlinear Schrödinger equation supplemented by a phenomenological repulsive sector ensuring regularity at high densities. Using a variational Gaussian ansatz, we derive an explicit effective energy functional and show that extended quantum states lose stability beyond a critical mass scale. This loss of stability is associated with a bifurcation in the reduced dynamical system governing the wave-function width, leading to the emergence of stable localized configurations. Within this picture, collapse corresponds to the dynamical selection of one of these localized attractors, driven by infinitesimal asymmetries in the initial state and occurring without stochastic noise or environmental coupling. The mechanism provides a controlled and quantitative realization of gravity-induced localization, extending Schrödinger--Newton-type models while avoiding their pathological short-distance behavior. Possible implications for mesoscopic systems probing the quantum-to-classical transition are briefly discussed.

• The paper demonstrates a deterministic collapse mechanism driven by gravitational self-interaction and bifurcation without relying on stochastic noise.
• It employs a nonlinear Schrödinger equation with a repulsive term to regularize the dynamics and analytically predict a critical mass scale (~10⁻¹⁷ kg).
• The framework offers experimentally accessible predictions for mesoscopic quantum-classical transitions with collapse timescales ranging from microseconds to milliseconds.

Gravitationally Induced Wave-Function Collapse via Dynamical Bifurcation

Overview

The paper "Gravitationally induced wave-function collapse from dynamical bifurcation" (2604.16144) develops a deterministic, nonlinear framework describing the emergence of wave-function collapse as a direct consequence of gravitational self-interaction, dynamically regulated by a phenomenological short-distance repulsive sector. The central thesis is that extended quantum states become unstable beyond a critical mass threshold, leading to spontaneous localization via a controlled bifurcation mechanism without invoking stochastic noise, environmental decoherence, or ad hoc collapse postulates. This construction provides a regularized extension of Schrödinger–Newton-type models, yielding finite, stable, localized configurations, and serves as an effective description for quantum-classical transition phenomena in mesoscopic regimes.

Theoretical Framework

The proposed model is a non-relativistic, nonlinear Schrödinger equation incorporating both gravitational self-attraction and a phenomenological repulsive term:

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 - G m \int d^3r' \frac{|\psi(\mathbf{r'})|^2}{|\mathbf{r}-\mathbf{r'}|} + \lambda |\psi(\mathbf{r})|^2 \right] \psi,$$

where $\lambda > 0$ ensures regularity at high densities.

A variational Gaussian ansatz, $\psi(\mathbf{r}) \propto \exp(-r^2/2\sigma^2)$, enables analytical reduction to an effective energy functional:

$$E(\sigma) = \frac{3\hbar^2}{4m\sigma^2} - \frac{Gm^2}{\sqrt{2\pi}\sigma} + \frac{\lambda}{(2\pi)^{3/2}\sigma^3}$$

Governed by this functional, the quantum state undergoes qualitative change as $m$ increases, with the gravitational term eventually overwhelming quantum kinetic and repulsive effects.

Dynamical Instability and Bifurcation Analysis

For $m \ll m_c$, the system presents a single stable extended state (large $\sigma$ minimum). The critical mass $m_c$ is defined by a saddle-node bifurcation—simultaneous solution to $dE/d\sigma = d^2E/d\sigma^2=0$—where two stationary points of $E(\sigma)$ (minimum and inflection) coalesce. For $\lambda > 0$0, the extended state becomes unstable, and stable localized minima at finite $\lambda > 0$1 appear.

The collapse process is governed by a deterministic reduced dynamical system:

$\lambda > 0$2

In overdamped conditions, $\lambda > 0$3 describes relaxation toward attractors. Collapse is thus interpreted as deterministic but unpredictable localization, with infinitesimal initial asymmetries selecting the final state.

Key Results

• Critical mass scale: $\lambda > 0$4 (up to order unity factors)
• For plausible $\lambda > 0$5 (regularization length $\lambda > 0$6m), $\lambda > 0$7kg—mesoscopic scale accessible to current matter-wave and optomechanical experiments
• Collapse timescale near threshold: $\lambda > 0$8s for parameters relevant to ongoing experiments

Short-Distance Regularization and Physical Interpretation

The short-distance repulsive sector regularizes the singular behavior intrinsic to the pure Schrödinger–Newton equation, which is only attractive and lacks stable, finite energy states. The introduction of $\lambda > 0$9 is justified phenomenologically, analogously to smearing scales in CSL and Diòsi–Penrose models, and avoids conflicts with known UV behavior of gravity (e.g., non-local quantum gravity, infinite-derivative gravity). It does not constitute a new fundamental force, but rather encodes ignorance of short-scale physics within an effective theory.

Comparison with Decoherence and Collapse Models

This framework sharply contrasts with environmental decoherence and stochastic collapse schemes:

• Decoherence: Suppresses density-matrix interference; does not yield true collapse or selection of outcomes.
• Stochastic models: Employ random noise to effect reduction, now tightly bounded by experiment.
• Present model: Achieves localization through a deterministic, dynamically unstable bifurcation in the effective state-space. No environmental (external) or intrinsic stochasticity is required at the phenomenological level; collapse is emergent from intrinsic dynamics and sensitive dependence on initial conditions.

Implications and Future Directions

This regularized, deterministic model for self-gravitating quantum states provides:

• A well-posed, UV-modified extension of Schrödinger–Newton dynamics for mesoscopic mass scales
• Quantitative predictions for critical mass/length/collapse timescales in tabletop experiments
• A platform for future investigations: Multi-particle generalization (collective mass density), exact numerical simulations of collapse transients, and model-based design of quantum-gravity-sensitive experiments

A full theory of quantum measurements is not addressed; rather, the model specifies the onset and dynamics of localization for degrees of freedom carrying sufficient collective mass.

Conclusion

This work establishes an effective, nonlinear, deterministic mechanism for wave-function collapse, induced by gravitational self-interaction and regulated by short-distance repulsion, formalized as a dynamical instability with a finite bifurcation mass. The resulting framework overcomes pathologies of the Schrödinger–Newton equation, does not rely on stochasticity or environment, and is directly relevant to ongoing experimental searches for deviations from linear quantum dynamics at the quantum–classical interface. Future theoretical and experimental work will elucidate its quantitative scope and testability.