  A new quantum algebraic description of relativistic electrons, built on a
conformal dynamical symmetry (SO(4,2)), has recently been proposed to treat
localization in space-time. It is shown here that localization of an electron
may be represented by components of a SO(4,2) vector which are quantum
generalizations of the hexaspherical coordinates of classical projective
geometry. The shift of this vector under transformations to uniformly
accelerated frames is described by SO(4,2) rotations. Hexaspherical observables
also allow one to represent the quantum law of free fall under a form
explicitly compatible with the same dynamical symmetry.
