Relativistic generalization of the Schrodinger-Newton model for the wavefunction reduction

TL;DR

This paper extends the Schrödinger-Newton model to a relativistic framework using coupled Dirac and Maxwell-like equations, ensuring positive gravitational masses and high-precision equivalence principle adherence, with implications for wavefunction localization.

Contribution

It introduces a relativistic generalization of the Schrödinger-Newton model using Dirac-Maxwell equations, maintaining positive masses and high-precision equivalence principle compliance.

Findings

Solutions form a discrete spectrum with positive gravitational masses.

Masses are approximately equal to inertial masses and quanta mass m.

Wavefunction widths are larger than the Compton length by a factor of α.

Abstract

We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of the Schrodinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the "active" gravitational masses are always positive, and approximately equal to inertial masses and to the mass $m$ of the quanta of Dirac field up to the corrections of order $\alpha^2$. Here $\alpha=(m/M_{pl})^2$ is the gravitational analogue of the fine structure constant negligibly small for nucleons. In the limit $\alpha \to 0$ the model reduces back to the nonrelativistic Schrodinger-Newton one. The equivalence principle is fulfilled with an extremely high precision. The above solutions correspond to various states of the same (free) particle rather than to different particles. These states possess a negligibly small difference in characteristics but essentially differ in the widths of the wavefunctions. For the ground state the latter is $\alpha$ times larger the Compton length, so that a nucleon cannot be sufficiently localized to model the reduction process