Newton-Cartan limit of Klein-Gordon AQFT and the collapse of Galilean modular structure

Abstract

We extend the established Galilean/relativistic structural divider in algebraic quantum field theory, namely, the absence of Reeh-Schlieder and of Tomita-Takesaki modular flow on local algebras of any Galilean Haag-Kastler net satisfying a natural axiom set augmented by the Bargmann-charge hypotheses (G7$^*$)(a) and (G7$^*$)(d) to curved backgrounds via the Newton-Cartan ($c\to\infty$) limit. We show, for the free Klein-Gordon field on Minkowski and on static globally hyperbolic spacetimes admitting a Post-Newtonian expansion, that a position-independent rest-energy rescaling produces in the limit a Galilean Haag-Kastler net satisfying the axioms of Ref. [1] in flat-space form (Minkowski) or in a curved-space modification (Killing-flow invariance and uniqueness of the vacuum replacing full translation invariance) appropriate to the static case. The Bargmann central charge equals the Klein--Gordon mass~$m$; the gravitational potential $V(x)$ enters the limiting Schr\"odinger Hamiltonian but not the algebraic structure obstructed by the Galilean Reeh-Schlieder no-go theorem. The strengthened obstruction theorem of Ref. [1] extends to the modified curved-space setting on Fock representations, and the limiting net carries no modular flow on local algebras. Schwarzschild is treated as a worked example: the Killing horizon shrinks to a point, the Hartle-Hawking thermal state has no $c\to\infty$ limit, and the Boulware vacuum limits to the gravitational hydrogenic ground state. The Reissner-Nordstr\"om metric is included as a sanity check confirming that leading Post-Newtonian misses the electromagnetic content of the background. We discuss how Newton's constant~$G$ enters the present (background-metric) framework only at the level of the limiting Hamiltonian, and indicate where dynamical-metric extensions would require $G$ to play a structural role.