States and IR divergences in factorization algebras
Masashi Kawahira, Tomohiro Shigemura
TL;DR
The paper develops a rigorous, factorization-algebra framework for free real scalar fields to address infrared divergences, particularly in the massless theory. It introduces and analyzes three states—$\langle-\rangle_{\rm aug}$ (natural augmentation), $\langle-\rangle_{\rm cptf}$ (compactification), and $\langle-\rangle_{\rm Sch}$ (Schwartz)—and defines the compactification and Schwartz constructions in both massive and massless settings, using BV cohomology and gauge-theoretic data. The main result is the equivalence of these states in both the massive and massless cases, achieved by explicit quasi-isomorphisms, the removal of IR modes via the maps $j$, and a careful treatment of long-wavelength modes through harmonic-polynomial structures. The work clarifies IR behavior in quantum field theory within a mathematically precise framework, with implications for thermodynamic limits, translation invariance, and potential extensions to conformal and duality structures. It also outlines future directions, including Virasoro symmetry in low dimensions and Koszul duality, to deepen the connection between factorization algebras and fundamental QFT concepts.
Abstract
In field theory, one can consider a variety of states. Within the framework of factorization algebras, one typically works with the natural augmentation state $\langle-\rangle_{\rm aug}$. In physics, however, other states arise naturally, such as the compactification state $\langle-\rangle_{\rm cptf}$ or the Schwartz state $\langle-\rangle_{\rm Sch}$, defined by imposing Schwartz boundary conditions. At first sight, the relation among these three states is not obvious. This paper gives a definition of the compactification state in factorization algebras and provides a method for handling infrared divergences in the massless theory. We then prove that the three states are equivalent in both the massive and massless cases.