Taub-NUT spinning space

The models of relativistic particles with spin have been proposed for a long time. The first published work concerning the Lagrangian description of the relativistic particle with spin was the paper by Frenkel which appeared in 1926. After that the literature on the particle with spin grew vast. The models involving only conventional coordinates are called the classical models while the models involving anticommuting coordinates are generally called pseudo-classical. In this paper we shall confine ourselves to discuss the relativistic spin one half particle models involving anticommuting vectorial degrees of freedom which are usually called the spinning particles. Spinning particles are in some sense the classical limit of the Dirac particles. After the first quantization these new anticommuting variables are mapped into the Dirac matrices and they disappear from the theory. The action of spin one half relativistic particle with spinning degrees of freedom described by Grassmannian (odd) variables was first proposed by Berezin and Marinov and soon after that was discussed and investigated in many papers. In the present paper we investigate the motion of pseudo-classical spinning point particles in curved spaces. The generalized Killing equations for the configuration space of spinning particles (spinning space) are analysed and the solutions of the homogeneous part of these equations are expressed in terms of Killing-Yano tensors. The general results are applied to the case of the four-dimensional Euclidean Taub-NUT spinning space. The motivation to carry out this example is two-fold. First of all, in the Taub-NUT geometry there are known to exist four Killing-Yano tensors. From this point of view the spinning Taub-NUT space is an exceedingly interesting space to exemplify the effective construction of all conserved quantities in terms of geometric ones, namely Killing-Yano tensors. On the other hand, the Taub-NUT geometry is involved in many modern studies in physics. For example the Kaluza-Klein monopole of Gross and Perry and of Sorkin was obtained by embedding the Taub-NUT gravitational instanton into five-dimensional Kaluza-Klein theory. Remarkably the same object has re-emerged in the study of monopole scattering. In the long distance limit, neglecting radiation, the relative motion of slow Bogomolny-Prasad-Sommerfield monopoles is described by the geodesics of this space. The problem of geodesic motion in this metric has therefore its own interest, independently of monopole scattering. (author)