Relativistically Invariant Classical Hamiltonian Mechanics with Extra Variables

Abstract

It is shown that one can construct a relativistically invariant classical mechanics in Hamiltonian form by utilizing as independent dynamical variables the position of each particle, together with its canonical conjugate variable, and the velocity of each particle, together with its canonical conjugate variable. The ten generators of the Poincaré group, obeying the correct Poisson bracket relations between themselves and the position variables, are constructed in their most general form. By restricting this form, it is possible to construct a Hamiltonian theory where the equations of motion for the particle's positions and velocities depend only upon the positions and velocities (and not upon their conjugate variables): these turn out to be the most general relativistically invariant classical equations of motion of this type. This is not useful as a starting point for constructing a physically interesting quantum theory, since the probability density in the quantized theory obeys the classical Liouville equation. This approach is also applied to the Galilean-invariant classical mechanics.