Modal Triplet Theory: From MTT to Quantum Field Theory on Curved Spacetime

We derive interacting quantum field theory (QFT) on curved spacetimes from Modal Triplet Theory (MTT). Extending the projection operator formalism of the MTT → quantum mechanics correspondence, we construct ΠQFT as a map from modal coherent sectors of the 10D MTT configuration space to the algebra of quantum fields on a globally hyperbolic 4D spacetime (Y4, g). The formulation employs the algebraic QFT framework, ensures the Hadamard property of physical states, and incorporates running couplings from the covariant functional renormalisation group (FRG). We derive the semiclassical Einstein equation with FRG-improved renormalised stress–energy ⟨Tμν⟩, analyse unitarity and implementability via the Shale–Stinespring criterion, and work through an example of scalar particle production in a spatially flat FRW universe with scale-dependent mass and curvature coupling. This work completes the triad MTT → (quantum mechanics, general relativity, QFT) and provides a rigorous bridge between modal dynamics and the standard framework of quantum fields in curved backgrounds.