Coarse-Graining and the Classification of Long-Range Correlations in Quantum Field Theory

How can we analytically determine which operators in a quantum field theory (QFT) will develop long-range correlations during the coarse-graining process? While numerical tools like the functional renormalization group (FRG) or lattice gauge theory can compute spectral functions, they often lack an a priori criterion to identify which specific interaction channels drive the emergence of collective behavior.

The author introduces a classification framework that treats coarse-graining as a dynamical process analogous to the derivation of fluid equations from molecular kinetics. By analyzing Feynman diagrams, the framework applies three distinct criteria to filter out irrelevant contributions:

1. Momentum Transfer: Diagrams with non-zero momentum transfer are suppressed by oscillatory factors, leaving only zero-momentum-transfer configurations (like ladder diagrams) to accumulate.
2. Topology: Among zero-momentum-transfer diagrams, only ladder topologies can be resummed into a geometric series that produces a spectral pole, whereas single-bubble topologies contribute only to the continuum.
3. Symmetry Protection: The framework uses conservation laws (Ward identities) to determine if the injection term—the source of spectral weight—is non-vanishing. Operators protected by Ward identities are guaranteed to have non-vanishing matrix elements, whereas those protected by BRST symmetry or unprotected operators behave differently.

The framework provides a systematic decision tree for operator emergence. It demonstrates that the non-vanishing of the injection term and the sign of the single-bubble contribution (determined by spin statistics) are the primary drivers of long-range correlations. The author verifies this classification against eight physical channels and three known solvable systems, showing that it successfully distinguishes between operators that can develop spectral poles and those that cannot. This approach serves as a pre-screening tool that complements nonperturbative numerical methods by identifying the physical pathways for emergence before computation begins.