QSG 000-002: Why Quantum Field Theory Has Not Yet Become a Fully Spectral Theory

This note discusses why quantum field theory, despite the presence of deep spectral structures, has not yet been formulated as a fully spectral theory. It reviews existing spectral layers in QFT, including the Källén--Lehmann representation, the Wightman spectral condition, algebraic QFT, and the spectral action in noncommutative geometry. The main argument is that the obstacle is not the absence of spectral language, but the structural complexity of QFT: locality, causality, gauge redundancy, renormalization, operator-valued distributions, asymptotic scattering, and the dependence of amplitudes on preparation and registration procedures cannot be captured by the spectrum of a single Hamiltonian. As a possible direction, the note proposes a program of spectral geometry of quantum-field processes. In this perspective, a Feynman diagram or a QFT amplitude is regarded not only as a computational expression, but also as an induced spectral object depending on preparation, internal channels, registration, phase structure, and an induced spectral measure. The note does not claim to construct a complete spectral formulation of QFT. Its role is preparatory: it identifies a possible operator-spectral entry point into QFT within the broader Quantum Spectral Geometry (QSG) program, while leaving the construction of a full process carrier and its stress tests to subsequent work.