In 1956 Mark Kac proposed a process related to the telegrapher equation where the particle travels at constant speed (say the speed of light c) and randomly inverts its velocity. This process had important applications concerning the path-integral solution and the probabilistic interpretation of the 1\(+\)1 dimensions Dirac equation. The extension to 3\(+\)1 dimensions requires that the particle only moves at light-speed, which implies that velocity can be represented as a point on the surface of a sphere of radius c. The realizations of the process for the velocity only may connect these points, and, by strict analogy with the Kac model, it can be assumed that the velocity jumps from one value to another. In this paper we follow a new and different strategy assuming that the velocity performs continuous trajectories (velocity changes direction in a continuous way) which are the realization of a Wiener process on the surface. The processes which emerge transform one in the other by Lorentz boost. The associate Forward Kolmogorov Equation for the joint probability density of position and velocity, which is the (3\(+\)1) dimensional analogous of the telegrapher equation, is examined and a simplification is performed by means of variables separation.

1956年，马克·卡克（Mark Kac）提出了一个与电报机方程有关的过程，其中粒子以恒定速度（例如光速c）传播，并随机地反转其速度。该过程在路径积分解和1 \（+ \） 1维Dirac方程的概率解释方面具有重要的应用。扩展到3 \（+ \） 1维要求粒子仅以光速运动，这意味着速度可以表示为半径为c的球面上的一个点。仅针对速度的过程的实现可以将这些点联系起来，并且通过严格类似于Kac模型，可以假定速度从一个值跳到另一个值。在本文中，我们假设速度执行连续轨迹（速度以连续方式改变方向），这是表面上维纳过程的实现，这是一种不同的新策略。出现的过程通过洛伦兹的推动而彼此转化。研究了位置和速度联合概率密度的关联前向Kolmogorov方程，该方程类似于电报机方程的（3 \（+ \） 1）维，并通过变量分离进行了简化。