AI 中文总结
本文针对时空非对易性相关物理意义存争议的问题，构建首个在非对易时空上两粒子的完全相对论性初量子化模型，刻画了变形对称代数及其极限，构建单粒子模型，展示两粒子系统描述，揭示对称生成元与相互作用定律变形的关联。
AI 中文摘要
在过去几十年里，时空非对易性及相关相对论对称性变形引发了诸多关注，因一些量子引力现象学窗口正接近真正的普朗克尺度灵敏度。然而，处理时空非对易性所引入数学结构的物理意义仍存争议，关键部分缺失或理解不足。有人提议通过分析精心设计的初量子化玩具模型来深入了解这些概念挑战。本文认真对待此提议，构建了首个在非对易时空上传播和相互作用的两个粒子的完全一致且完全相对论性的初量子化模型。我们模型中实现的特定时空非对易性，即‘时间对易κ平面’，此前已被提议用于初量子化分析，但先前研究多为启发式，未能完整描述变形的相对论对称性。我们在此超越了那些开创性尝试：全面刻画了适当变形的庞加莱对称代数及其伽利略极限；构建了携带变形伽利略代数不可约表示的单粒子量子模型；展示了通过变形谐振势相互作用的两个量子粒子系统的两种一致的伽利略相对论描述，发现两粒子对称生成元的结构与相互作用定律的变形紧密相连。
英文摘要
In the last decades, spacetime noncommutativity and the associated deformations of relativistic symmetries have attracted a lot of interest, as several phenomenological windows into quantum gravity are approaching genuine Planck-scale sensitivity. However, the physical significance of the mathematical structures introduced to deal with spacetime noncommutativity is still debated, and some crucial pieces are missing or poorly understood. From time to time it has been suggested that valuable insight into these conceptual challenges could come from the analysis of appropriately designed first-quantized toy models, in which major technical and interpretive issues can be effectively managed or sidestepped altogether. In this paper we take seriously this suggestion and develop the first fully consistent and fully relativistic first-quantized model of two particles propagating and interacting on a noncommutative spacetime. The specific spacetime noncommutativity implemented in our model, which we call 'time-commutative $κ$-plane', had already been proposed as a suitable arena for a first-quantized analysis, but previous studies were mostly heuristic and failed to provide a full description of the deformed relativistic symmetries. We here go much beyond those pioneering attempts: we find a full characterization of the appropriate deformed Poincaré symmetry algebra as well as its Galilean limit; we build a single-particle quantum model carrying an irreducible representation of the deformed Galilei algebra; we exhibit two consistent, Galilean-relativistic descriptions of a system of two quantum particles interacting via a deformed harmonic potential, finding that the structure of the two-particle symmetry generators is intimately connected with the deformation of the interaction law.
Comments33 pages
