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

TL;DR

This paper explores Klein-Gordon equations for relativistic particles in rapidity space, using hyperbolic and periodic angles to describe constrained evolution, revealing quantum-like properties and potential wave behavior in classical motion.

Contribution

It introduces a novel description of relativistic particle dynamics in rapidity space using hyperbolic and periodic angles, linking classical trajectories to Klein-Gordon equations.

Findings

Energy and momentum obey Klein-Gordon equations in rapidity space.

Angles are proportional to mass and relate to velocity and time-like parameters.

Classical motion may exhibit wave-like features at small mass influences.

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.