离散相空间与连续时间相对论量子力学 II：皮亚诺圆、超环面相格与纤维丛
Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fibre bundles
AI总结：
本研究延续离散相空间与连续时间相对论量子力学框架，利用皮亚诺曲线构造皮亚诺圆与高维超环面相格，结合纤维丛给出几何解释，并通过克莱因–戈登方程分析自由玻色子与普朗克振子的共存，再与弦论维度比较。
摘要：
作为论文I[1]的延续，本文进一步研究相对论量子力学的离散相空间与连续时间表示。本文采用的核心数学构造是填充区域的皮亚诺曲线。我们证明，一类皮亚诺曲线序列的极限是一个记为$\bar{S}^{1}_{n}$的皮亚诺圆，即半径为$\sqrt{2n+1}$的圆，其中$n \in \{0,1,\cdots\}$。在我们的框架中，将这个二维皮亚诺圆解释为二维离散相平面内的一个相格。我们假设，质量极轻、尺寸小到超出当前实验探测能力的一次量子化普朗克振子占据该相格$\bar{S}^{1}_{n}$。这个皮亚诺圆的时间演化扫出一个二维竖直柱体，类似于弦论中的世界面。将这一结论推广到三维空间，我们引入$(2+2+2)$维相空间超环面$\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$作为物理维离散相空间中的对应相格。我们通过乘积纤维丛给出了该结构在态空间中的几何解释。我们还利用相对论偏差分-微分克莱因-戈登方程，研究了背景为$[(2+2+2)+1]$维离散相空间与连续时间态空间中的自由标量玻色子。该系统的二次量子化场量子可以与$\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$相格内的微小普朗克振子永久共存。最后，本文探讨了更高维$[(2+2+2)N+1]$离散态空间中的广义自由二次量子化克莱因-戈登方程，并将得到的离散相空间维度与部分主流弦论模型的关键空间维度进行了比较。
英文摘要：
The discrete phase space and continuous time representation of relativistic quantum mechanics is further investigated here as a continuation of paper I [1]. The main mathematical construct used here will be that of an area-filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as $\bar{S}^{1}_{n}$, a circle of radius $\sqrt{2n+1}$ where $n \in \{0,1,\cdots\}$. We interpret this two-dimensional Peano circle in our framework as a phase cell inside our two-dimensional discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell $\bar{S}^{1}_{n}$. The time evolution of this Peano circle sweeps out a two-dimensional vertical cylinder analogous to the world-sheet of string theory. Extending this to three dimensional space, we introduce a $(2+2+2)$-dimensional phase space hyper-tori $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fibre bundles. We also study free scalar Bosons in the background $[(2+2+2)+1]$-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein-Gordon equation. The second quantized field quantas of this system can cohabit with the tiny Planck oscillators inside the $\bar{S}^{1}_{n^1} \times \bar{S}^{1}_{n^2} \times \bar{S}^{1}_{n^3}$ phase cells for eternity. Finally, a generalized free second quantized Klein-Gordon equation in a higher $[(2+2+2)N+1]$-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.
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