Spinning particles in Taub-NUT space

Abstract

The geodesic motion of pseudo-classical spinning particles in Euclidean Taub-NUT space is analysed. The constants of motion are expressed in terms of Killing-Yano tensors. Some previous results from the literature are corrected. PACS number(s):04.20.Jb, 02.40.-K The configuration space of spinning particles (spinning space) is an supersymmetric extension of an ordinary Riemannian manifold, parametrized by local coordinates {x µ }, to a graded manifold parametrized by local coordinates {x µ , ψ µ }, with the first set of variables being Grassmann even (commuting) and the second set, Grassmann odd (anticommuting). The equation of motion of a spinning particle on a geodesic is derived from the action:The corresponding world-line Hamiltonian is given by:where Π µ = g µνẋ ν is the covariant momentum.

Hidden symmetries of dynamics in classical and quantum physics

“…A non-trivial canonical transformation that is of higher order in the momenta transforms the system in a new Hamiltonian system where the two conserved quantities are of second order in the new momenta (Verhoeven et al, 2002). With respect to the new variables now the system is also separable.…”

“…From eqs. (162) and (165) it is possible to generate Killing-Yano and, respectively, closed conformal KillingYano tensors onM when f is Killing-Yano and, respectively closed conformal Killing-Yano on M. In particular this gives new examples of Lorentzian metrics with conformal Killing-Yano tensors by lifting known conformal Killing-Yano tensors in Riemannian signature, for example when M is the Kerr-NUT-(A)dS metric or the Taub-NUT metric (Baleanu and Codoban, 1999;Cordani et al, 1988;Gibbons and Ruback, 1987;Vaman and Visinescu, 1998;Visinescu, 2000).…”

Symmetries of the Dirac operators associated with covariantly constant Killing–Yano tensors

“…In general, in such a case, the Stäckel-Killing tensor K µν in Eq. (1) is not covariantly constant and the corresponding non-standard supercharges Q a of hidden supersymmetries do not close on the Hamiltonian [16] (or on D 2 , as in standard Dirac theories), but {Q a , Q b } ∝ Jδ ab , where J is an invariant different from the Hamiltonian (or D 2 ).…”

“…In the pseudo-classical spinning particle models in curved spaces from covariantly constant K-Y tensors f µν can be constructed conserved quantities of the type f µν θ µ θ ν depending on the Grassmann variables {θ µ } [16]. The Grassmann variables {θ µ } transform as a tangent space vector and describe the spin of the particle.…”

f-symbols in Robertson-Walker space-time

“…An illustration of existing extra conserved quantities is provided by Kerr-Newmann [1] and Taub-NUT geometry. For the geodesic motion in the Taub-NUT space,the conserved vector which is analogous to the Runge-Lenz vector of the Kepler type problem is quadratic in 4-velocities; its components are Stackel-Killing tensors and they can be expressed as symmetrized products of Killing-Yano tensors [2,3,4,5,6]. The configuration space of spinning particles (spinning space) is an extension of an ordinary Riemannian manifold, parameterized by local coordinates {x µ }, to a graded manifold parameterized by local coordinates {x µ , ψ µ }, with the first set of variables being Grassmann-even (commuting) and the second set Grassmann-odd (anti-commuting).…”