Gravitational Poissonian Spontaneous Localization Model of Hybrid Quantum-Classical Newtonian Gravity: Energy Increase and Experimental Bounds

TL;DR

This paper analyzes the Gravitational Poissonian Spontaneous Localization (GPSL) model, a hybrid classical-quantum theory in which Newtonian gravity arises from stochastic collapses of smeared mass density and back-reacts via gravity. By allowing distinct smearings for measurement ($r_C$) and gravitational feedback ($r_G$), the authors derive a general state-dependent spontaneous heating rate and identify regimes where asymmetric smearings markedly suppress heating, notably when $r_G$ is larger than $r_C$. They demonstrate that, in isolated-particle regimes, a Gaussian $g_{r_C}$ combined with an optimal non-Gaussian $g_{r_G}$ can reduce gravitational heating by many orders of magnitude, while macroscopic cases favor a top-hat $g_{r_G}$ with limited gains. The paper also provides astrophysical bounds from neutron-star heating, yielding both upper and lower limits on the model parameters, and combines these with previous PSL bounds to produce an exclusion plot, showing GPSL remains viable but tightly constrained. Overall, GPSL offers a testable hybrid framework with realistic heating phenomenology and astrophysical bounds that guide viable parameter choices.

Abstract

The Gravitational Poissonian Spontaneous Localization (GPSL) model is a hybrid classical-quantum framework in which Newtonian gravity emerges from stochastic collapses of a smeared mass-density operator. Consistency of the hybrid dynamics entails momentum diffusion and, hence, spontaneous heating. Without smearing, which enters both the collapse (measurement) and gravitational-feedback components of the dynamics, the heating rate would be divergent. Previous work assumed identical smearings for both components. Here, we treat the general case of distinct spatial smearings $g_{r_C} (\mathbf{x})$ and $g_{r_G} (\mathbf{x})$, characterized, respectively, by length scales $r_C$ and $r_G$. We characterize the spontaneous heating rate for arbitrary $g_{r_C} (\mathbf{x})$ and $g_{r_G} (\mathbf{x})$, and then discuss which smearing profiles minimize the spontaneous heating rate in relevant physical situations. Remarkably, there are situations in which, while the measurement noise remains the same, allowing $g_{r_G} (\mathbf{x}) \neq g_{r_C} (\mathbf{x})$ may reduce the feedback-induced spontaneous heating by more than 60 orders of magnitude already for $r_G = 10 r_C$. Finally, we use our results to estimate the spontaneous heating rate of neutron stars and to set new lower bounds on the model's parameters by comparing the theoretical predictions with astronomical data on temperature, radius, and mass of neutron stars.

Figures (8)

• Figure 1: Pictorial representation of the GPSL mechanism. Two quantum objects, green (G) and blue (B), are both in a superposition of two states: left (L) and right (R). Until there is a collapse event, the two objects do not interact gravitationally. Then, in this example, the blue object spontaneously localizes with a flash located at $\mathbf{x}_{\rm Flash}$. Associated to the flash, the unitary operation $U_G (\mathbf{x}_{\rm Flash})$ attracts all masses toward the collapse center.
• Figure 2: The plot shows the ratio $R/r_G$ obtained by solving Eq. \ref{['eq:GPSL_DifferentialEquationBoundRadius']}. The two horizontal lines represents the constant values $\sqrt{5}$ and $\sqrt{3}$.
• Figure 3: This plot compares $g_{r_G}$ [Eq. \ref{['eq:GPSL_OptimalDistribution']}] to $g_{r_C}$ (black continuous line, a Gaussian distribution with variance $r_C^2$), for different values of $r_G$. Here, $r=\abs{\mathbf{x}}$.
• Figure 4: Pictorial representation of the GPSL mechanism for a single particle and the $g_{r_G}$ of Eq. \ref{['eq:GPSL_OptimalDistribution']}. Before the flash, the particle is in a generic quantum state. In the picture, a superposition of states with spread much larger than $r_C$. When the flash occurs, the particle localizes around the flash on a length $r_C$. The gravitational feedback field is generated by a mass density with shape $g_{r_G}$ centered on the flash. Using the $g_{r_G}$ of Eq. \ref{['eq:GPSL_OptimalDistribution']} (see also Fig. \ref{['fig:SmearingFunctionsPlot']} for $r_G=2.1 r_C$), there is basically no back-reaction on the spontaneously localized particle.
• Figure 5: The plot shows the ratio of $I_0^{(G)}$ computed with Gaussian distributions over $I_0^{(G)}$ computed with Gaussian smearing for the measurement part and the optimal distribution for the gravitational feedback. We set $r_C=1$ to make the numerical calculations. This plot confirms that this ratio is always greater than one.