Twistor variables of relativistic mechanics

Abstract

This paper is concerned with the description of noninteracting dynamical systems in Penrose's twistor theory. We begin with an introduction to the formal treatment of massive systems relying on the angular momentum twistor ${A}^{\ensuremath{\alpha}\ensuremath{\beta}}$. Expressions for dynamical data in terms of ${A}^{\ensuremath{\alpha}\ensuremath{\beta}}$ are given. The internal-symmetry transformations involved in the parametrization of ${A}^{\ensuremath{\alpha}\ensuremath{\beta}}$ by one-index twistors are shown to be of the canonical type. The case of two supporting twistors is discussed in detail. Relative to the frame of these twistors (a rest frame of the dynamical system) the classical spin has the remarkable structure $\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}=\overline{\ensuremath{\psi}}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\sigma}}\ensuremath{\psi}$. When decomposing ${A}^{\ensuremath{\alpha}\ensuremath{\beta}}$ into three twistors, the minimal symmetry transformations define a 14-parameter group. We show that this group of symmetries is locally isomorphic to the inhomogeneous SU(3).