Massive and massless particles in Mielke--Baekler geometries

Abstract

The Mielke--Baekler geometries are three-dimensional reductive homogeneous spacetimes together with a choice of invariant connection which is compatible with a lorentzian metric. The spacetimes generalise Minkowski and (anti)de~Sitter spacetimes in that the invariant metric connection can have torsion, a peculiarity of three dimensions. Using coadjoint orbits and the techniques of nonlinear realisations, we construct worldline actions for massive and massless spinning particles moving in these spacetimes. We pay particular attention to the so-called teleparallel branch, in which the curvature of the invariant connection vanishes. Apart from the trivial Minkowski case, this singles out anti-de~Sitter spacetime, as the only of these lorentzian manifolds admitting an invariant Weitzenb\"ock connection; that is, a flat connection with torsion. The introduction of a Wess--Zumino term describing spin has, as a main consequence, the appearance of dynamical sectors (denoted ``regular'' and ``critical''), with a different number of physical degrees of freedom. In particular, in the massive case, we discuss the formulation of the dynamics in terms of either the Weitzenb\"ock or the Levi-Civita connection, and the emergence of a Papapetrou-type forcing term in the critical sector. For the massless particle we study the Noether symmetries of the action, which, for the spinless case, include the conformal transformations. For nonzero spin, only the Killing subset survives as genuine Noether transformations in the regular sector, while in the critical sector any conformal Killing contribution can be set to zero by a gauge transformation.