March and Weatherall prove a minimal coupling, in their precise category-theoretic sense, is unique up to isomorphism when one exists

philosophy-of-physicsgeneral-relativitysymmetric-hyperbolic-systemsnatural-bundlesminimal-couplingcategory-theoryarxiv

Eleanor March and James Owen Weatherall give a precise, category-theoretic definition of minimal coupling — a natural equation on a bundle X = B × G depending on the Lorentzian-metric quotient G to at most first order — and prove that a minimal coupling in this exact sense is unique up to a natural notion of isomorphism, whenever one exists (Theorem 2, "Uniqueness of minimal coupling," in "Minimally-coupled Quasilinear Symmetric Hyperbolic Systems, Naturally," arXiv:2610.39107). The result runs against a claim the authors attribute to the existing literature — that a given special-relativistic theory may admit many inequivalent minimal couplings to general relativity:

"We will now show that that is not true for minimal couplings in the present sense: that is, if a system of equations admits a minimal coupling, then it is unique, up to a natural notion of isomorphism."

The proof's load-bearing lemma is not new to this paper: it leans on Theorem 5 of Samuel C. Fletcher and Weatherall's 2023 paper, "The local validity of special relativity, part 1: Geometry" (Philosophy of Physics 1, 7). What is new here is the uniqueness result itself. Minimal coupling, informally the "comma goes to semicolon" recipe for turning a special-relativistic field equation into a general-relativistic one, had previously been treated as admitting multiple, inequivalent versions once general covariance enters the picture; the paper's contribution is to show that apparent plurality dissolves once "minimal coupling" is given a sufficiently precise definition.