Klein-Gordon equations for energy-momentum of the relativistic particle in rapidity space

Abstract

Two alternative ways of description an evolution constrained by mass-shell equation are given by the hyperbolic and the periodic angles. In the both cases the angles are proportional to the mass. The differential operators with respect to these coordinates are not commute, the commutation relations coincide with commutation relations of the fermi-like oscillator. The derivative of the periodic angle with respect to the hyperbolic angle is equal to the velocity. This relationship prompts us to conclude that the hyperbolic angle $\phi$ is the time-like parameter, whereas the periodic angle $\theta$ is the space-like parameter. The evolution equations admit an extension to the case of three (or more) dimensions. It is proved, the energy and momentum defined in the space of four-rapidity obey Klein-Gordon equations constrained by the classical trajectory of a relativistic particle. It is conjectured, for small values of a proper mass influence of the constraint is weakened and the classical motion gains features of a wave motion.