Nomic Structure and Reduction

Abstract: 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.