Stochastic extensions of the regularized Schr\"odinger-Newton equation

TL;DR

This paper introduces stochastic modifications to the Schr"odinger-Newton equation, ensuring compatibility with no-signaling principles by regularizing divergences and incorporating noise, resulting in a linear, invariant evolution of quantum states.

Contribution

It proposes a novel stochastic extension of the Schr"odinger-Newton equation that preserves physical principles and resolves divergence issues in the nonlinear model.

Findings

Stochastic extensions eliminate superluminal signaling.

Regularization prevents divergence in energy corrections.

The resulting evolution is linear and Galilean invariant.

Abstract

We show that the Schr\"{o}dinger-Newton equation, which describes the nonlinear time evolution of self-gravitating quantum matter, can be made compatible with the no-signaling requirement by elevating it to a stochastic differential equation. In the deterministic form of the equation, as studied so far, the nonlinearity would lead to diverging energy corrections for localized wave packets and would create observable correlations admitting faster-than-light communication. By regularizing the divergencies and adding specific random jumps or a specific Brownian noise process, the effect of the nonlinearity vanishes in the stochastic average and gives rise to a linear and Galilean invariant evolution of the density operator.