Discrete quantum modes of the Dirac field in $AdS_{d+1}$ backgrounds

TL;DR

This paper analytically solves the Dirac equation in $AdS_{d+1}$ backgrounds, deriving energy levels and eigenspinors using a covariant approach that separates variables in spherical coordinates.

Contribution

It introduces a covariant method to solve the Dirac equation in $AdS_{d+1}$, providing explicit formulas for energy levels and eigenspinors in these backgrounds.

Findings

Analytical solutions for Dirac energy levels in $AdS_{d+1}$.

Explicit normalized eigenspinors derived.

Covariant separation of variables in spherical coordinates.

Abstract

It is shown that the free Dirac equation in spherically symmetric static backgrounds of any dimensions can be put in a simple form using a special version of Cartesian gauge in Cartesian coordinates. This is manifestly covariant under the transformations of the isometry group so that the generalized spherical coordinates can be separated in terms of angular spinors like in the flat case, obtaining a pair of radial equations. In this approach the equation of the free field Dirac in $AdS_{d+1}$ backgrounds is analytically solved obtaining the formula of the energy levels and the corresponding normalized eigenspinors.