电磁场空间演化的Generator
Generator of spatial evolution of the electromagnetic field
摘要：
从麦克斯韦方程组开始，在傅立叶空间中定义正常变量，我们用哈密顿和拉格朗日形式写出电磁场的时间演化方程，明确了标准教科书中经常省略的所有中间步骤。然后，我们按照同样的步骤写这个场的演化方程沿空间维的哈密顿和拉格朗日形式。通过这种方式，我们得到了有源电磁场空间演化的生成器的显式形式，并表明它具有通过与传播方向正交的给定平面传递的动量模量的物理意义。在自由场的特定情况下，这个发生器与场的全部动量在传播方向上的投影一致，用负号表示。当考虑场的空间演化而不是时间演化时，空间演化的哈密顿和拉格朗日公式对于场的正确量子化是必不可少的，特别是对于等空间对易关系的正确定义。
英文摘要：
Starting with Maxwell's equations and defining normal variables in the Fourier space, we write the equations of temporal evolution of the electromagnetic field with sources in the Hamiltonian and Lagrangian forms, making explicit all intermediate steps often omitted in standard textbooks. Then, we follow the same steps to write the equations of evolution of this field along a spatial dimension in the Hamiltonian and Lagrangian forms. In this way, we arrive at the explicit form of the generator of spatial evolution of the electromagnetic field with sources and show that it has a physical meaning of the modulus of momentum transferred through a given plane orthogonal to the direction of propagation. In a particular case of free field this generator coincides with the projection of the full momentum of the field on the propagation direction, taken with a negative sign. The Hamiltonian and Lagrangian formulations of the spatial evolution are indispensable for a correct quantization of the field when considering its spatial rather than temporal evolution, in particular, for a correct definition of the equal-space commutation relations.
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