$Φ^4_3$ Theory from many-body quantum Gibbs states

Abstract: We derive the $\Phi^4_3$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas, where the limiting classical measure describes the critical behavior of the Bose gas just above the Bose--Einstein phase transition. Since the quantum problem is typically formulated using a nonlocal interaction potential, a key challenge is to approximate the local $\Phi^4_3$ theory by a Hartree measure with a nonlocal interaction. This requires uniform estimates on the Hartree measure, which are achieved using techniques from recent development on stochastic quantization and paracontrolled calculus from \cite{GIP15}. The connection to the quantum problem is then established by applying the variational approach in \cite{LewNamRou-21}, where using a recent correlation inequality from \cite{DeuNamNap-25} we refine the analysis and derive a quantitative convergence of the quantum correlation functions to those of the Hartree classical field.