探索超光速速度变换的群结构
Towards a group structure for superluminal velocity boosts
AI总结：
本文构建了超光速速度变换的几何映射，证明了在(1+1)维时空中亚光速与超光速变换的集合构成一个群，为避免与实验相悖的超光速群结构研究提供了新途径。
摘要：
标准的亚光速 Lorentz 变换具有明确的几何解释。它们可以简洁地表示为双曲旋转，使得光锥内部的双叶双曲面族和光锥外部的单叶双曲面族均保持不变。在本研究中，我们构建了这两类超曲面之间的映射，并将相应的算子解释为超光速速度变换。尽管物理观察者无法“跃过”光速壁垒从一个区域进入另一个区域（至少在经典情况下如此），但超光速运动本身并不会引发悖论。本文探讨了这一构造对 Dragan 和 Ekert 近期提出的“相对论量子原理”的意义。该几何图景在 (1+1) 维中重现了他们的“超变换”算子，但在 (1+3) 维中的推广方式截然不同。这使得一种现有模型似乎无法实现的重要可能性得以保留，即将超光速变换嵌入群结构而不产生与现有实验结果相悖的额外不良现象的可能性。我们证明了在 (1+1) 维时空中，包含亚光速和超光速变换的集合构成一个群，并概述了将这些结果推广至更高维几何的研究计划。
英文摘要：
Canonical subluminal Lorentz boosts have a clear geometric interpretation. They can be neatly expressed as hyperbolic rotations, that leave both the family of $2$-sheet hyperboloids within the light cone, and the family $1$-sheet hyperboloids exterior to it, invariant. In this work, we construct a map between the two families of hypersurfaces and interpret the corresponding operators as superluminal velocity boosts. Though a physical observer cannot `jump' the light speed barrier, to pass from one regime to the other (at least not classically), the existence of superluminal motion does not, by itself, generate paradoxes. The implications of this construction for recent work on the `quantum principle of relativity', proposed by Dragan and Ekert, are discussed. The geometric picture reproduces their `superboost' operator in $(1+1)$ dimensions but generalises to $(1+3)$ dimensions in a very different way. This leaves open an important possibility, which appears to be closed to existing models, namely, the possibility of embedding the superluminal boosts within a group structure, without generating additional unwanted phenomenology, that contradicts existing experimental results. We prove that the set containing both subluminal and superluminal boosts forms a group, in $(1+1)$-dimensional spacetimes, and outline a program to extend these results to higher-dimensional geometries.
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