On solutions of the Pauli equation in non-static de Sitter metrics

TL;DR

This paper investigates solutions to the Pauli equation for spin-1/2 particles in expanding and oscillating de Sitter cosmological models, deriving exact non-relativistic solutions and analyzing their properties.

Contribution

It introduces a method to transition from the covariant Dirac equation to the Pauli approximation in non-static de Sitter metrics, providing exact solutions in these cosmological models.

Findings

Exact non-relativistic solutions in expanding and oscillating de Sitter spaces.

Wave functions are non-stationary but probability density remains time-independent.

Demonstrates existence of non-relativistic limit in these curved spacetime geometries.

Abstract

A particle with spin 1/2 is investigated both in expanding and oscillating cosmological de Sitter models. It is shown that these space-time geometries admit existence of the non-relativistic limit in the covariant Dirac equation. Procedure for transition to the Pauli approximation is conducted in the equations in the variables $(t, r)$, obtained after separating the angular dependence of $(\theta, \phi)$ from the wave function. The non-relativistic systems of equations in the variables $(t, r)$ is solved exactly in both models. The constructed wave functions do not represent stationary states with fixed energy, however the corresponding probability density does not depend on the time.