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

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. The connection to the quantum problem is then established by applying the variational approach and deriving a quantitative convergence of the quantum correlation functions to those of the Hartree classical field.