The Dirac particle on central backgrounds and the anti-de Sitter oscillator

TL;DR

This paper derives a simplified, covariant Dirac equation for spherically symmetric static backgrounds, solves it analytically for the anti-de Sitter oscillator, and provides explicit energy levels and eigenspinors.

Contribution

It introduces a reduced Dirac equation in Cartesian tetrad gauge that allows separation of variables and analytical solutions in anti-de Sitter backgrounds.

Findings

Analytical energy spectrum of the anti-de Sitter oscillator

Explicit form of eigenspinors for the system

Covariant formulation simplifies solving Dirac equations in curved backgrounds

Abstract

It is shown that, for spherically symmetric static backgrounds, a simple reduced Dirac equation can be obtained by using the Cartesian tetrad gauge in Cartesian holonomic coordinates. This equation is manifestly covariant under rotations so that the spherical coordinates can be separated in terms of angular spinors like in special relativity, obtaining a pair of radial equations and a specific form of the radial scalar product. As an example, we analytically solve the anti-de Sitter oscillator giving the formula of the energy levels and the form of the corresponding eigenspinors.