Keller O
Institute of Physics and Nanotechnology, Aalborg University Skjernvej 4A, DK-9220 Aalborg Øst, Denmark.
J Microsc. 2008 Feb;229(Pt 2):331-6. doi: 10.1111/j.1365-2818.2008.01909.x.
After having briefly reviewed the Hamilton-Jacobi theory of classical point-particle mechanics, its extension to the quantum regime and the formal identity between the Hamilton-Jacobi equation for Hamilton's characteristic function and the eikonal equation of geometrical optics, an eikonal theory for free photons is established. The space-time dynamics of the photon is described on the basis of the six-component Riemann-Silberstein energy wave function. Form-identical eikonal equations are obtained for the positive and negative helicity dynamics. Microscopic response theory is used to describe the linear photon-matter interaction. In the presence of matter the free-photon concept is replaced by a quasi-photon concept, and there is a quasi-photon for each of the two helicity states. After having established integro-differential equations for the wave functions of the two quasi-photons, the eikonal conditions for the quasi-photons are determined. It appears that the eikonal condition contains complicated space integrals of the gradient of the eikonal over volumes of near-field domain size. In these space integrals the dynamics of the electrons (matter particles) appears via transverse transition current densities between pairs of many-body states. Generalized microscopic polarization and magnetization fields are introduced to establish the connection between the quasi-photon and macroscopic eikonal theories.
在简要回顾了经典点粒子力学的哈密顿 - 雅可比理论、其向量子领域的扩展以及哈密顿特征函数的哈密顿 - 雅可比方程与几何光学的程函方程之间的形式恒等式之后，建立了自由光子的程函理论。基于六分量的黎曼 - 西尔伯斯坦能量波函数描述了光子的时空动力学。对于正螺旋度和负螺旋度动力学，得到了形式相同的程函方程。微观响应理论用于描述线性光子 - 物质相互作用。在有物质存在的情况下，自由光子概念被准光子概念所取代，并且对于两个螺旋度态中的每一个都有一个准光子。在建立了两个准光子波函数的积分 - 微分方程之后，确定了准光子的程函条件。结果表明，程函条件包含在近场域大小的体积上对程函梯度的复杂空间积分。在这些空间积分中，电子（物质粒子）的动力学通过多体态对之间的横向跃迁电流密度出现。引入广义微观极化和磁化场以建立准光子理论与宏观程函理论之间的联系。
