Wavefunction collapse induced by gravity in a relativistic Schr{\''o}dinger-Newton model

TL;DR

This paper explores a relativistic extension of the Schrödinger-Newton equation using a novel approach, revealing that gravitational effects can induce wavefunction collapse near the Planck mass.

Contribution

It introduces a relativistic version of the Schrödinger-Newton model based on the Grave de Peralta approach, demonstrating gravity-induced wavefunction collapse at the Planck scale.

Findings

Wavefunction collapse occurs near the Planck mass.

The method is validated with a particle in a box.

Relativistic effects modify the characteristic length scale.

Abstract

A relativistic version of the Schr{\"o}dinger-Newton equation is analyzed within the recently proposed Grave de Peralta approach [L. Grave de Peralta, {\em Results Phys.} {\bf 18} (2020) 103318], which include relativistic effects by a parametrization of the non-relativistic hamiltonian, so as to impose that the average kinetic energy of the system coincide with its relativistic kinetic energy. The reliability of this method is tested for the particle in a box. By applying this method to the Schr{\"o}dinger-Newton equation we shows that the characteristic length of the model [L. Di{\'o}si, {\em Phys. Lett}. {\bf 105A} (1984) 199] goes to zero for a mass of the order of the Planck mass, suggesting a collapse of the wavefuncton, induced by gravity.