Schr\"odinger--Newton equation with spontaneous wave function collapse

TL;DR

This paper introduces a stochastic Schr"odinger--Newton equation combining two gravity-related modifications to explain wave function collapse in macroscopic objects, ensuring localized solitons with conserved momentum and energy.

Contribution

It proposes a new stochastic Schr"odinger--Newton equation that integrates previous models, enabling wave function collapse without energy diffusion in macroscopic systems.

Findings

Wave functions collapse to localized solitons

Conservation of momentum and energy achieved

Collapse occurs gradually and stochastically

Abstract

Based on the assumption that the standard Schr\"odinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals has survived from the nineteen-eighties. The Schr\"odinger--Newton equation (1984) provides well-localized solitons for free macro-objects but lacks the mechanism how extended wave functions collapse on solitons. The gravity-related stochastic Schr\"odinger equation (1989) provides the spontaneous collapse but the resulting solitons undergo a tiny diffusion leading to an inconvenient steady increase of the kinetic energy. We propose the stochastic Schr\"odinger--Newton equation which contains the above two gravity-related modifications together. Then the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons which perform inertial motion without the momentum diffusion: conservation of momentum and energy is restored.