Classical mechanics with respect to an observer’s past light cone

Abstract

This work is motivated by the fact that it is impossible for an observer to know at time t0 all the initial data of a system, if that data is specified in the conventional manner on the spacelike surface t=t0. A Hamiltonian formulation for classical mechanics, first given by Dirac, is exploited, in which dynamical variables are specified by their values on an observer’s past light cones. Starting from initial data given on the past light cone of an observer at some initial space-time point, the values of the variables on the observer’s current past light cone are given by a canonical transform of the initial data. The method is illustrated for a spinless particle of mass m, which is either free or interacts with an external electromagnetic field. The remarkable result is obtained that the dynamics of this classical one-particle spin-0 system can be formulated in terms of a Dirac-like spinor, whose four components are formed from the generalized coordinates and momenta.