The Dirac field in Taub-NUT background

TL;DR

This paper explores the quantum behavior of Dirac fermions in the Taub-NUT gravitational background, revealing connections to scalar theories and explicit solutions for energy eigenspinors.

Contribution

It demonstrates how Dirac fermion modes can be derived from scalar equations and identifies new observables related to Killing-Yano tensors in the Taub-NUT space.

Findings

Quantum modes recoverable from scalar Klein-Gordon equations

Explicit energy eigenspinors in closed form

Identification of observables from scalar theory

Abstract

We investigate the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole, pointing out that the quantum modes can be recovered from a Klein-Gordon equation analogous to the Schr\" odinger equation in the Taub-NUT background. Moreover, we show that there is a large collection of observables that can be directly derived from those of the scalar theory. These offer many possibilities of choosing complete sets of commuting operators which determine the quantum modes. In addition there are some spin- like and Dirac-type operators involving the covariantly constant Killing-Yano tensors of the hyper-K\" ahler Taub-NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form.