自由半群非相对论相空间态量子化
Free semigroup non-relativistic phase space states quantisation
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中文总结 AI 辅助
该研究将量子化视为经典点算符在相空间普朗克尺度上的作用，引入经典力学的态语言/狄拉克记号，通过投影算符管理态转换，推导勒让德变换，并定义态间夹角度量量子性，探索自由半群时间贡献。
中文摘要 AI 辅助
量子化被视为经典点算符在相空间普朗克尺度上对场态的作用。重点放在物理变量的序列上。引入了经典力学的“态语言/狄拉克记号”；在许多情况下，它取代了保守系统的驻定作用原理。观测到的经典动力学和量子动力学被视为时间自由半群贡献的不同向量化。有一个梦想，即把经典动力学描述为状态向量在这样一个自由半群上的复变换，而不是求解微分方程。从字母表到态、从态到态函数、从态函数到可测量量的过渡由投影算符管理。尝试推导勒让德变换。在坐标表示中势能与波函数的平凡乘法及其在动量表示中的非平凡算符作用，无需傅里叶变换或普朗克尺度即可获得。量子化过程分为两个阶段：第一，点状经典算符对场函数的作用；第二，在经典相空间中可能无穷小的背景下，考虑普朗克长度的物理有限尺度。引入了态之间夹角的概念，定义了“量子性”的度量。在玻恩规则的背景下研究了自由半群时间贡献。
英文摘要
Quantisation is considered as an action of classical point operators on field states at the Planck scale of the phase space. Emphasis is placed on the sequence of physical variables. A "state language/Dirac notation" for classical mechanics is introduced; in many cases, it replaces the principle of stationary action for conservative systems. Observed classical and quantum dynamics are considered as different vectorisations of the free semigroup contribution of time. There is a dream of describing classical dynamics as complex transformations of state vectors over such a free semigroup, rather than solving differential equations. The transition from the alphabet to states, from states to state functions, and from state functions to measurable quantities is managed by projectors. An attempt is made to provide a derivation of the Legendre transformation. The trivial multiplication of the potential energy by the wave function in the coordinate representation and its non-trivial operator action in the momentum representation are obtained without a Fourier transform nor the Planck scale. The quantisation procedure is divided into two stages: first, the action of point-like classical operators on field functions; and second, accounting for the physically finite scale of the Planck length against the backdrop of the possible infinitesimality in the classical phase space. The concept of an angle between states is introduced, defining the measure of "quantumness". The free semigroup time contribution is studied in the context of the Born rule.
