The Dirac Field in Taub–nut Background

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 Schrödinger 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-Kähler Taub-NUT space. The energy eigenspinors of the central modes in spherical coordinates are completely evaluated in explicit, closed form.Pacs 04..62.+v *

Hierarchy of Dirac, Pauli, and Klein–Gordon conserved operators in Taub-NUT background

“…2,4,7,14 In this Kaluza-Klein geometry there is a pentad gauge fixing 15 where the massless Dirac field, , satisfies a simple gauge-covariant Dirac equation, D " ϭ0, where D " ϭi␥ 0 ‫ץ‬ t ϪD " s . 16,5,7,9 In the standard representation of the Dirac matrices ͓with diagonal ␥ 0 ͑Ref. 17͔͒ the Hamiltonian operator 7,9 Hϭ␥ 0 D " s ϭ ͩ 0 ␣* ␣ 0 ͪ ͑2͒ has manifest supersymmetry.…”

“…16,5,7,9 In the standard representation of the Dirac matrices ͓with diagonal ␥ 0 ͑Ref. 17͔͒ the Hamiltonian operator 7,9 Hϭ␥ 0 D " s ϭ ͩ 0 ␣* ␣ 0 ͪ ͑2͒ has manifest supersymmetry. It is expressed in terms of Pauli operators,…”

“…We start in the second section with a brief review of some previous results [7][8][9] we need. In the next section we study the algebra of the conserved Dirac operators and introduce a new type of even operators which help us to define the projection operator that separates the physical parts.…”

Fermions tunneling from Plebański-Demiański black holes

“…Recently, the new developments in relativistic quantum field theory has shed light on it. Cotȃescu and Visinescu [10] investigated the Dirac field in Taub-NUT background. Kerner and Mann [11] obtained the temperature of Taub-NUT-anti-de Sitter BHs by using null-geodesic method and the Hamilton-Jacobi ansatz.…”

Geodesic Motions in Euclidean Taub-NUT Spinning Spaces

“…Moreover, the motion of well-separated monopole-monopole interactions is described approximately by the geodesics of this space [30][31][32][33]. The Euclidean Taub-NUT background contains also interesting specific features of the quantum theory in the case of the scalar fields [34] as well as for Dirac fields of spin-1 2 fermions [35][36][37]. There exist large algebras of conserved observables in both cases [38].…”