Relativistic Motion in a Constant Electromagnetic Field
Relativistic Motion in a Constant Electromagnetic Field
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恒定电磁场中的相对论运动
DOI：
10.1063/1.3064796
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发表时间：
2008
期刊：
影响因子：
--
通讯作者：
S. Chin
中科院分区：
文献类型：
--
作者：
S. Chin
对于在恒定电磁场中运动的相对论性带电粒子，其速度4矢量已得到充分研究。然而，尽管电磁场和运动方程都是纯实数，但所得到的 4 速度似乎是由复杂的电磁场引起的。这项工作表明，这并不是由于使用了某些复杂的形式主义（例如 Clifford 代数），而是本质上由于洛伦兹群的 $o(3,1)$ 李代数等价于两个交换复数 $su(2)$ 代数这一事实。用升压和旋转算子来表达复杂的 $su(2)$ 发电机，然后自然地引入了复杂的电磁场。这项工作不是以矩阵方程的形式求解运动方程，而是以洛伦兹群的生成元的形式求解算子演化方程。将真实演化算子分解为两个可交换复数演化算子，然后直接给出速度 4 向量的时间演化，而不涉及任何中间场。
For a relativistic charged particle moving in a constant electromagnetic field, its velocity 4-vector has been well studied. However, despite the fact that both the electromagnetic field and the equations of motion are purely real, the resulting 4-velocity is seemingly due to a complex electromagnetic field. This work shows that this is not due to some complex formalism used (such as Clifford algebra) but is intrinsically due to the fact that the $o(3,1)$ Lie algebra of the Lorentz group is equivalent to two commuting complex $su(2)$ algebras. Expressing the complex $su(2)$ generators in terms of the boost and rotation operators then naturally introduces a complex electromagnetic field. This work solves the equation of motion not as a matrix equation, but as an operator evolution equation in terms of the generators of the Lorentz group. The factorization of the real evolution operator into two commuting complex evolution operators then directly gives the time evolution of the velocity 4-vector without any reference to an intermediate field.