The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length l is investigated. Fundamental physical constants such as ℏ, c, and l are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles Sn1 of radii 2En = 2n + 1 within the discrete (1 + 1)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities Sn1 × ℝ ⊂ ℝ2 and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion S# matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein–Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group ℐ[O(3, 1)].

中文翻译：

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离散相空间和连续时间相对论量子力学 I：普朗克振荡器和闭合弦状圆形轨道

涉及特征长度的相对论量子力学的离散相空间连续时间表示l被调查。基本物理常数，例如ℏ,C， 和l保留在论文的大部分部分。普朗克振子的能量特征值问题正是在这个框架中解决的。恒定能量的离散同圆轨道显示为圆形小号n1半径2乙n = 2n + 1离散内(1 + 1)维相平面。此外，这些轨道的时间演化像几何实体一样扫除了世界表小号n1 × ℝ ⊂ ℝ2因此显示为闭合的弦状几何配置。相空间中这些离散轨道的物理解释为退化的弦状相单元，以数学上严格的方式显示。在离散相空间量子力学领域中这些闭合的同圆轨道的存在，以低阶扩展的非奇异性而闻名小号#矩阵术语，已知存在，但直到现在还没有被充分探索。最后，离散偏微分-微分 Klein-Gordon 方程被证明在连续非齐正交群下是不变的ℐ[○(3, 1)].