A new method was developed to solve the Dirac equation and used to find analytical solutions for time-dependent electromagnetic fields that translate a given Dirac spinor along any desired trajectory in the $x\text{−}y$ plane without distortion. As is well known, negative energy solutions describing antiparticles play a prominent role in the Dirac theory of the electron, and a separation of the part of the wave function describing only electrons from the full solutions often needs to be done. Here, we clarify how the aforementioned separation is done, which is not clear in our recent publication [Campos and Cabrera, Phys. Rev. Research 2, 013051 (2020)].

中文翻译：

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开发了一种新的方法来求解Dirac方程，并用于找到时变电磁场的解析解，该电磁场将给定的Dirac旋子沿轨道中的任何所需轨迹平移。$X\text{-}ÿ$平面无变形。众所周知，描述反粒子的负能量溶液在电子的狄拉克理论中起着重要的作用，因此通常只需要从完整溶液中分离出仅描述电子的波函数部分即可。在这里，我们阐明了上述分离是如何完成的，这在我们最近的出版物[Campos and Cabrera，Phys。启研究 2，013051（2020）]。