论文标题
几何代数的电动力学
Electrodynamics in geometric algebra
论文作者
论文摘要
我们考虑真空中电荷和电流的电动力学，然后将结果推广到介电和磁性材料介质的描述：首先在空间代数（SA）中，然后在时空代数（STA）中。引入两极分化的多活动$ \ tilde {p} = \ boldsymbol {\ tilde {p}}} - \，\，\ frac {1} {c} {c} \，\ boldsymbol {\ boldsymbol {\ tilde {m} \ tilde {p} $，我们在sa中的材料介质中表达麦克斯韦方程。在时空中引入绑定的电流向量$ \ tilde {j} = j - \，c \，\ nabla \ cdot \ tilde {p} $，然后在sta中表示麦克斯韦方程。材料介质中的波方程是通过取麦克斯韦方程的梯度获得的。 For a uniform electromagnetic medium consisting of induced electric and magnetic dipoles, the stress-energy momentum vector is written as $\dot{T}\left(\dot{\nabla}\right) = \frac{1}{c}\,J \cdot F = f$ where $f$ is the electromagnetic force density vector in space-time.最后，材料介质中的麦克斯韦方程式可以用STA作为潜在向量$ a $的波方程式写入。
We consider the electrodynamics of electric charges and currents in vacuum and then generalise our results to the description of a dielectric and magnetic material medium : first in spatial algebra (SA) and then in space-time algebra (STA). Introducing a polarisation multivector $\tilde{P} = \boldsymbol{\tilde{p}} -\,\frac{1}{c}\,\boldsymbol{\tilde{M}}$ and an auxiliary electromagnetic field multivector $G = \varepsilon_0\,F + \tilde{P}$, we express the Maxwell equation in the material medium in SA. Introducing a bound current vector $\tilde{J} = J -\,c\,\nabla\cdot\tilde{P}$ in space-time, the Maxwell equation is then expressed in STA. The wave equation in the material medium is obtained by taking the gradient of the Maxwell equation. For a uniform electromagnetic medium consisting of induced electric and magnetic dipoles, the stress-energy momentum vector is written as $\dot{T}\left(\dot{\nabla}\right) = \frac{1}{c}\,J \cdot F = f$ where $f$ is the electromagnetic force density vector in space-time. Finally, the Maxwell equation in the material medium can be written in STA as a wave equation for the potential vector $A$.
