紧致闵可夫斯基空间：神话与事实
Compactified Minkowski Space: Myths and Facts
摘要：
Minkowski空间，共形群，紧化，共形无穷大，共形反演，无穷大光锥，SU（2，2），SO（4，2），Hodge星算子，Clifford代数，旋量，扭量，反线性算子，外代数，双向量，迷向子空间，零测地线，Lie球，Dupin圆线，引力} \abstract {麦克斯韦方程组不仅在洛伦兹群下不变，而且在共形群下不变。欧文·E西格尔证明了，虽然伽利略群是庞加莱群的变形，庞加莱群是共形群的变形，但共形群结束了道路，它是{ \em rigid\}。因此，有令人信服的数学和物理理由促进共形群的作用的基本对称性的空间-时间，更重要的庞加莱群，形成群论基础的狭义和广义相对论。虽然共形群在闵可夫斯基空间上的作用是奇异的，但它自然地扩展到紧化闵可夫斯基空间上的非奇异作用，在文献中通常称为“闵可夫斯基空间加上无穷远处的光锥”。不幸的是，在某些教科书中，紧致闵可夫斯基空间的真实结构有时会被歪曲，包括错误的证明和错误的陈述。在本文中，我们提出了，一个简单的方式，两种不同的结构紧Minkowski空间，都源于原来的想法罗杰彭罗斯，但把重点放在数学上微妙的点和有关的结构的Clifford代数工具。特别是鲜为人知的反线性霍奇星运营商的介绍，以连接真正的和复杂的代数结构。也指出了与Waldyr Rodrigues关于引力是闵可夫斯基真空的塑性形变的观点的可能关系。
英文摘要：
Minkowski space, conformal group, compactification, conformal infinity, conformal inversion, light cone at infinity, SU(2,2), SO(4,2), Hodge star operator, Clifford algebra, spinors, twistors, antilinear operators, exterior algebra, bivectors, isotropic subspaces, null geodesics, Lie spheres, Dupin cyclides, gravitation} \abstract{Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a deformation of the Poincaré group, and the Poincaré group is a deformation of the conformal group, the conformal group ends the road, it is {\em rigid\}. There are thus compelling mathematical and physical reasons for promoting the conformal group to the role of the fundamental symmetry of space--time, more important than the Poincaré group that formed the group-theoretical basis of special and general theories of relativity. While the action of the conformal group on Minkowski space is singular, it naturally extends to a nonsingular action on the compactified Minkowski space, often referred to in the literature as "Minkowski space plus light-cone at infinity". Unfortunately in some textbooks the true structure of the compactified Minkowski space is sometimes misrepresented, including false proofs and statements that are simply wrong. In this paper we present in, a simple way, two different constructions of the compactified Minkowski space, both stemming from the original idea of Roger Penrose, but putting stress on the mathematically subtle points and relating the constructions to the Clifford algebra tools. In particular the little-known antilinear Hodge star operator is introduced in order to connect real and complex structures of the algebra. A possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation of Minkowski's vacuum is also indicated.
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