The Geometry of Reduction: Model Embedding, Compound Reduction, and Overlapping State Space Domains

Abstract

The relationship according to which one physical theory encompasses the domain of empirical validity of another is widely known as "reduction." Here it is argued that one popular methodology for showing that one theory reduces to another, associated with the so-called "Bronstein cube" of physical theories, rests on an over-simplified characterization of the type of mathematical relationship between theories that typically underpins reduction. An alternative methodology, based on a certain simple geometrical relationship between dis- tinct state space models of the same physical system, is then described and illustrated with examples. Within this approach, it is shown how and under what conditions inter-model reductions involving distinct model pairs can be composed or chained together to yield a direct reduction between theoretically remote descriptions of the same system. Building on this analysis, we consider cases in which a single reduction between two models may be effected via distinct composite reductions differing in their intermediate layer of description, and motivate a set of formal consistency requirements on the mappings between model state spaces and on the subsets of the model state spaces that characterize such reductions. These constraints are explicitly shown to hold in the reduction of a non-relativistic classical model to a model of relativistic quantum mechanics, which may be effected via distinct composite reductions in which the intermediate layer of description is either a model of non-relativistic quantum mechanics or of relativistic classical mechanics. Some brief speculations are offered as to whether and how this sort of consistency requirement between distinct composite reductions might serve to constrain the relationship that any unification of the Standard Model with general relativity must bear to these theories.