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

Giacomo Santoni1,* and Francesco Scardino1,2,†

  • 1Physics Department, INFN Roma1, Piazzale A. Moro 2, Rome, I-00185, Italy
  • 2Physics Department, Sapienza University, Piazzale A. Moro 2, Roma, I-00185, Italy

  • *Contact author: giacomo.santoni@alumni.sns.it
  • †Contact author: francesco.scardino@uniroma1.it
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Phys. Rev. D 114, 045023 – Published 24 August, 2026

DOI: https://doi.org/10.1103/t75t-tmm5

Abstract

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 (1−1dL∂L) removes the leading bulk contribution to F=βFcan=−logZ 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.

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Phys. Rev. D 114, 045023– Published 24 August, 2026
  • Received 12 June 2026
  • Accepted 31 July 2026
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