Quantifying superluminal signalling in Schrödinger-Newton model

TL;DR

This work quantifies the potential for superluminal signalling in the Schrödinger–Newton model by embedding quantum dynamics in a relativistic, measure‑theoretic framework on spacetime. It contrasts the nonlinear SN equation with the fully relativistic Einstein–Dirac system, proving causal evolution for the latter and providing a quantitative probe of causality for the former. Numerical 1+1D simulations show that gravitational self‑interaction can suppress superluminal leakage for moderate coupling and larger systems, though extreme coupling can reintroduce leakage via high‑momentum modes. The results argue that SN dynamics need not be deemed fundamentally incompatible with no‑signalling, and they establish a pragmatic framework for assessing nonlinear quantum models in a relativistic setting, with implications for future multi‑particle theories and experimental analogues.

Abstract

The Schrödinger-Newton equation aims at describing the dynamics of massive quantum systems subject to the gravitational self-interaction. As a deterministic nonlinear quantum wave equation, it is generally believed to conflict with the relativistic no-signalling principle. Here we challenge this viewpoint and show that it is of key importance to study the quantitative and operational character of the superluminal effects. To this end we employ a rigorous formalism of probability measures on spacetime and quantify the probability of a successful superluminal bit transfer via the single-particle Schrödinger-Newton equation. We demonstrate that such a quantity decreases with the increasing size and mass of the system. Furthermore, we prove that the Einstein-Dirac system, which yields the Schrödinger-Newton equation in the non-relativistic limit, is perfectly compatible with the relativistic causal structure. Our study demonstrates that the Schrödinger-Newton equation, which is by construction non-relativistic, is in fact `more compatible' with the no-signalling principle than the ordinary free Schrödinger equation.

Figures (10)

• Figure 1: Visualization of the superluminal signalling protocol exploiting the violation of condition \ref{['CE']}. Alice prepares a system trapped in region $K$ and decides whether to release it or not. Bob gathers information from the detection statistic in region $C$. If condition \ref{['CE']} is violated, then Bob can infer (with some probability) Alice's decision.
• Figure 2: Time evolution of probability density $\rho (t,x)$ for the Gaussian initial state.
• Figure 3: The wavepacket spread \ref{['spread']} for the initial Gaussian state in function of time and the self-coupling strength $\kappa$. For $\kappa_c \gtrsim 0.4$ the wavepacket tends to localise.
• Figure 4: Broader perspective on time evolution of probability density $\rho (t,x)$ for Gaussian initial state. Dashed white lines depict the boundary of the light cone starting starting at $x \in [-5,5]$ and $t=0$.
• Figure 5: The plots present the amount of probability density leaking out of the future light cone of the interval $[-R,R]$ at a given time $t$, for a Gaussian initial state evolving under the Schrödinger--Newton equation the coupling constant $\kappa$.

Theorems & Definitions (5)

• Definition 1: AHP2017
• Theorem 1: AHP2017
• Definition 2: PRA2017
• Theorem 2
• proof