Nonadditivity in quantum field theory: Replica energies, scaling filters, and the renormalization group

Extensive systems have a simple thermodynamic signature; the dimensionless canonical free energy scales homogeneously with the size of the system. We show that the failure of this scaling, measured by the replica energy E , provides a useful bridge between statistical mechanics and quantum field theory. The associated differential operator ( 1 − 1 d L ∂ L ) removes the leading bulk contribution to F = β F can = − log Z and isolates the part that is sensitive to boundaries, topology, defects, long-range forces, or other sources of nonadditivity. In quantum field theory this thermodynamic idea has two closely related uses. For ordinary finite-volume or spherical partition functions, suitable higher-order versions of the same filter remove local counterterms and extract universal fixed-point data such as the central charge, the sphere free energy F , and the Euler anomaly coefficient a . For replica geometries with entangling defects, the same filtering principle gives the renormalized defect free energy. In 2 + 1 dimensions, its n → 1 limit gives the entropic F -function, with the sign fixed below by the standard free-energy convention. We use this perspective to distinguish ordinary finite-size corrections, topology-dependent constants in gapped phases, subextensive fracton degeneracies, and genuinely nonextensive systems with long-range interactions such as self-gravitating thermal matter. Replica energy therefore offers a common thermodynamic language for additivity, defect free energies, and renormalization-group irreversibility.