Coarse-Graining and the Classification of Long-Range Correlations in Quantum Field Theory
A conservation-law-guided classification framework is developed for the execution of coarse-graining operations in quantum field theory. Coarse-graining produces a selection at the level of Feynman diagrams. Diagrams with nonzero momentum transfer are suppressed by oscillatory factors, zero-momentum-transfer ladder diagrams can accumulate through geometric series resummation to produce a spectral pole, and single-bubble topologies contribute only to the continuum. Conservation laws govern this classification. For operators protected by a Ward identity, the matrix element at zero momentum is guaranteed to be nonvanishing. For operators protected by BRST symmetry, the Slavnov-Taylor identities provide no mandatory suppression. The two cases differ in the strength of the algebraic guarantee. For unprotected operators the injection term vanishes and the spectral function remains continuous. The sign of the single-bubble contribution is determined by spin statistics. A positive sign leads to amplificative feedback in the ladder resummation, a negative sign leads to suppressive feedback. These two attributes, the nonvanishing of the injection term and the sign of the single-bubble contribution, are the defining criteria of the classification. As a direct application, a general classification of local operators is established within the emergence framework and verified on eight physical channels and three known solvable systems. The framework indicates which emergence paths are possible for each operator; whether the critical condition is reached is left for independent nonperturbative computation. The logical structure of this classification is parallel to the strategy used in deriving fluid equations from molecular kinetic theory in classical statistical physics.