律则结构与约化

AI总结：

针对不规则律则结构的物理理论，通过辛约化将其转化为正则理论，并利用范畴论形式化分析状态空间同构与理论等价性。

摘要：

具有不规则律则结构的物理理论的正则表述是受约束的哈密顿理论，其中运动方程的不适定性与一种有害的过剩表征能力相关。通过辛约化过程，这类理论可以转化为具有规则律则结构和适定初值问题的理论。我们分析、综合并对比了不同方法对约束哈密顿理论的呈现与分析，借鉴了近期关于模型空间上律则结构形式化的工作（Gryb 和 Thébault 2024）以及通过范畴论对理论结构和表征能力的比较（Bradley 和 Weatherall 2020；Bradley 2025b）。我们认为，不规则律则结构的情况最适合采用范畴论呈现，其中状态空间是箭头，辛约化是箭头复合（Landsman 2005）。在这种方法下，我们得到自然结果：具有同构状态空间的理论是等价的，而约化状态空间同构的理论在其律则结构的正则表示层面是等价的。这一分析为具有不规则律则结构的理论的量子化提供了合适的基础，后者将在后续论文中讨论。

英文摘要：

The canonical formulation of physical theories with irregular nomic structure is as constrained Hamiltonian theories within which ill-posedness of the equations of motion is connected to a pernicious form of surplus representational capacity. Such theories can be converted into theories with regular nomic structure and a well-posed initial value problem via the process of symplectic reduction. We analyse, synthesise, and contrast different approaches to the presentation and analysis of constrained Hamiltonian theories, drawing upon recent work on formalisation of nomic structure on model spaces (Gryb and Thébault 2024) and comparisons of theoretical structure and representational capacity via category theory (Bradley and Weatherall 2020; Bradley 2025b). We suggest that the case of irregular nomic structure is most naturally suited to a category theoretic presentation in which state spaces are arrows and symplectic reduction is arrow composition (Landsman 2005). Under this approach one obtains the natural results that theories with isomorphic state spaces are equivalent and theories whose reduced state spaces are isomorphic are equivalent at the level of the regular representations of their nomic structure. This analysis provides a suitable foundation for the case of quantization of theories with irregular nomic structure, which will be in a companion paper.

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