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

Abstract

We derive the ${\Phi }_3^4$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the ${\Phi }_3^4$ measure as a semiclassical limit which captures the formation of Bose--Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the ${\Phi }_3^4$ measure through a Hartree measure with nonlocal interaction, achieved by developing new techniques in paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models.

118 pages, add results for the case that ${\lambda }^{\eta }\le \epsilon$ with small $\eta >0$