AI 中文总结
研究经典与量子力学在不同表示下的情况，借助维格纳-外尔对应关系，从态、运动学、动力学及测量等层面进行比较，区分人为因素与结构差异，明确不同表示下理论的变化与不变之处。
AI 中文摘要
理论的物理内容并非固有地与任何单一数学形式主义相关联。经典力学和量子力学都有等效表示，特别是在相空间和希尔伯特空间中，通过维格纳-外尔对应关系相联系。虽然这种对应关系在数学物理中已被长期研究，但其基础和操作意义常常未被明确阐述。在此，我们系统地说明了当经典和量子理论用彼此的原生语言表达时，哪些发生了变化，哪些没有变化。这种表示观点将人为因素（如在某些映射下非正性或负性的出现）与跨表示持续存在的稳健结构差异区分开来，特别是非对易性及其与普朗克常数相关的经典代数的星变形。我们在态、运动学和动力学层面进行比较，并通过在同一框架内制定结果统计和态更新规则将其扩展到测量。
英文摘要
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its $\hbar$-dependent $\star$-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.
