Representation of the active Lorentz transformation for particle dynamics

The rank-2 tensor representation of a passive Lorentz transformation constructed by Krause solely from the four-velocities of two inertial observers is shown to lack sufficient generality to describe unambiguously the active Lorentz transformation completely specifying the dynamic change of a classical particle from an initial state to a later state. This lack of uniqueness is traced to an arbitrary three-parameter little group transformation. For states described both by their four-vector momentum and polarization, we construct a rank-2 tensor representation of the active Lorentz transformation, connecting initial and later states, which is constructed solely from initial and later momenta and polarizations. This representation now contains a single arbitrary parameter. A unique active Lorentz transformation is achieved by further specification of an additional four-vector associated with the initial and later states.