卡罗尔量子力学：类时、类空与混合扇区
Carrollian Quantum Mechanics: Time-like, Space-like and Hybrid Sectors
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中文总结 AI 辅助
本研究通过对克莱因-戈登方程做超相对论收缩，构建含类时、类空、混合扇区的卡罗尔量子力学框架，推导各扇区的连续性方程等，并研究了箱中粒子、隧穿等量子系统。
中文摘要 AI 辅助
我们通过在$c \to 0$的极限下对克莱因-戈登方程进行系统的超相对论收缩，构建了卡罗尔量子力学的完整理论框架。该极限过程揭示了三个不同的扇区——类时、类空和混合扇区，每个扇区均受卡罗尔不变波动方程支配，并伴随一致的概率诠释。类时扇区呈现出一种新颖的时间隧穿现象，其特征为$|\mathcal{T}|^2 = 1 + |\mathcal{R}|^2$，这反映了克莱因-戈登范数的不定性质。类空扇区中空间传播的存在要求存在快子色散关系，这使得能量本征态的密度消失，产生零范数（零）态，由于 underlying 的零几何结构，这些态在卡罗尔物理中获得了物理相关性。混合扇区结合了前两个扇区的特征，允许两种不同的表述，可为快子型或非快子型，可被诠释为非齐次克莱因-戈登方程。针对所有三个扇区，我们推导了对应的连续性方程、概率密度和流，并在卡罗尔 regime 内研究了正则量子系统——包括箱中粒子和隧穿现象。
英文摘要
We develop a comprehensive theoretical framework for Carrollian quantum mechanics by performing systematic ultra-relativistic contractions of the Klein--Gordon equation in the limit $c \to 0$. This limiting process uncovers three distinct sectors---time-like, space-like, and hybrid---each governed by a Carroll-invariant wave equation and accompanied by a consistent probabilistic interpretation. The time-like sector exhibits a novel temporal tunneling phenomenon, characterized by the relation $|\mathcal{T}|^2 = 1 + |\mathcal{R}|^2$, which reflects the indefinite character of the Klein--Gordon norm. In the space-like sector, the presence of spatial propagation necessitates a tachyonic dispersion relation, which forces the density to vanish for energy eigenstates, yielding zero-norm (null) states that gain physical relevance within Carrollian physics due to the underlying null geometric structure. The hybrid sector combines features of both sectors and admits two distinct formulations, which may be either tachyonic or non-tachyonic, and can be interpreted as an inhomogeneous Klein--Gordon equation. For all three sectors, we derive the corresponding continuity equations, probability densities, and currents, and investigate canonical quantum systems---including the particle in a box and tunneling phenomena---within the Carrollian regime.
