电磁学的几何代数维脚手架
Dimensional scaffolding of electromagnetism using geometric algebra
摘要：
使用几何代数和微积分来表达电磁定律，我们能够以渐进的方式呈现大小和关系，逐步增加维度的数量。在一维情况下，电荷和电流密度、电场E以及标量势和矢量势在时空图中得到几何解释。应用于这些幅度的几何矢量导数产生简单的表达式，导致像位移电流，连续性和规范或延迟时间的概念，具有明确的几何意义。由于几何矢量导数是可逆的，我们引入简单的格林函数，并与此，它是可能的，以光速自然传播的延迟Liénard-Wiechert电位。在二维中，这些量值变得更加复杂，并且磁场B表现为在一维世界中不存在的赝标量。感应定律反映了E和B之间的关系，并有可能得出电容器、电路和坡印廷矢量的概念，解释能量的流动。在这个二维场景中，波动方程的解揭示了物理效应以光速传播的过程。这预示着在真实的三维世界中也会有同样的结果，但在这种情况下，它被赋予了一种在一维或三维中完全不存在的性质。因此，完全以光速传播的电磁波可以被看作是生活在一个具有奇数个空间维度的世界中的结果。最后，在真实的三维世界中，同一组简单的多向量微分表达式编码了电磁学的基本定律和概念。
英文摘要：
Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.
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