Local Feynman Diagrammatics in Curved Spacetime: A Consistent LMC Framework

We develop a general framework for quantum field theory in curved spacetime based on Local Minkowski Coordinates (LMC), which incorporates curvature effects into local Feynman diagrammatics. Gravitational influence enters through a curvature-dependent normalization function $B(x)$, derived from covariant current conservation, and a gravitational phase $S(x)$, obtained via the WKB approximation. These quantities enter through local phase accumulation and observer-dependent normalization of external states, without affecting global observables. As a first application, we analyze the local redshift normalization and phase structure of quantum amplitudes in the vicinity of a Schwarzschild black hole. Within their range of validity, the curvature-dependent factors $B(x)$ and $S(x)$ reproduce the expected gravitational redshift of field amplitudes in general relativity. When amplitudes are propagated to asymptotic infinity and evaluated in a standard global quantum state (such as the Unruh state), the resulting flux is consistent with the standard Hawking result. The framework refines the local WKB structure and clarifies the separation between local normalization effects and globally conserved fluxes.