标题： 相对论粒子在rapidity空间中的能量-动量的Klein-Gordon方程 显示英文标题

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

摘要： 描述由质量壳方程约束的演化有两种替代方法，分别是双曲角和周期角。 在两种情况下，角度都与质量成正比。 相对于这些坐标微分算子不交换，对易关系与费米型振子的对易关系一致。 周期角相对于双曲角的导数等于速度。 这种关系使我们得出结论，双曲角$\phi$是时间-like 参数，而周期角$\theta$是空间-like 参数。 演化方程可以扩展到三维（或更多）的情况。 证明了在四快速度空间中定义的能量和动量服从由相对论粒子经典轨迹约束的克莱因-戈登方程。 推测当固有质量较小时，约束的影响减弱，经典运动将表现出波运动的特征。 显示英文摘要

摘要： 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.