重温狭义相对论：闵可夫斯基时空的自然代数替代
Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime
摘要：
闵可夫斯基在1908年提出了著名的时空连续统概念，将三维空间与一个虚的时间维度$ i c t $合并，虚的单位产生正确的时空距离$ x^2 - c^2 t^2 $，以及爱因斯坦当时最近发展的狭义相对论的结果，从而为爱因斯坦的理论提供了一个解释。作为两个空间维和一个时间维的平面闵可夫斯基时空的替代方案，我们用Clifford双向量$ ι= e_1 e_2 $代替单位虚部$ i = \sqrt{-1} $，该双向量也平方为负1，但可以在不增加额外维数的情况下包含它，因为它是具有正交基$ e_1 $和$ e_2 $的实笛卡尔平面的一个组成部分。我们发现，在这个平面时空模型中，使用二维克利福德多向量，时空度规和洛伦兹变换立即作为代数的性质。这也导致动量和能量被表示为一个多向量的组成部分，我们给出了一个新的有效的推导康普顿散射公式，和一个简单的狄拉克和麦克斯韦方程。基于多重矢量的数学结构，我们提出了一个大质量粒子的半经典模型，它可以被看作是闵可夫斯基时空结构的起源，从而更深入地解释相对论效应。我们也发现了一个新的角度对时间的性质，现在是一个精确的数学定义为双向量的平面。
英文摘要：
Minkowski famously introduced the concept of a space-time continuum in 1908, merging the three dimensions of space with an imaginary time dimension $ i c t $, with the unit imaginary producing the correct spacetime distance $ x^2 - c^2 t^2 $, and the results of Einstein's then recently developed theory of special relativity, thus providing an explanation for Einstein's theory in terms of the structure of space and time. As an alternative to a planar Minkowski space-time of two space dimensions and one time dimension, we replace the unit imaginary $ i = \sqrt{-1} $, with the Clifford bivector $ ι= e_1 e_2 $ for the plane that also squares to minus one, but which can be included without the addition of an extra dimension, as it is an integral part of the real Cartesian plane with the orthonormal basis $ e_1 $ and $ e_2 $. We find that with this model of planar spacetime, using a two-dimensional Clifford multivector, the spacetime metric and the Lorentz transformations follow immediately as properties of the algebra. This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton's scattering formula, and a simple formulation of Dirac's and Maxwell's equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
