Runge-Lenz operator for Dirac field in Taub-NUT background

TL;DR

This paper explores fermions in the Taub-NUT space, constructing Dirac operators linked to hidden symmetries, including a conserved Runge-Lenz type operator, revealing new algebraic structures.

Contribution

It explicitly constructs Dirac operators involving Killing-Yano tensors in Taub-NUT space and introduces a conserved Runge-Lenz type operator related to hidden symmetries.

Findings

Dirac operators anticommute with standard Dirac operator

Constructed conserved Runge-Lenz type operator

Revealed algebraic properties of the hidden symmetries

Abstract

Fermions in D=4 self-dual Euclidean Taub-NUT space are investigated. Dirac-type operators involving Killing-Yano tensors of the Taub-NUT geometry are explicitly given showing that they anticommute with the standard Dirac operator and commute with the Hamiltonian as it is expected. They are connected with the hidden symmetries of the space allowing the construction of a conserved vector operator analogous to the Runge-Lenz vector of the Kepler problem. This operator is written down pointing out its algebraic properties.