In this talk, we explore adaptations of semidefinite programming relaxations for solving many-body physics problems. Our approach transforms a high-dimensional PDE problem into a convex optimization problem, setting it apart from traditional non-convex methods that rely on nonlinear re-parameterizations of the solution. In the context of statistical mechanics, we demonstrate how a mean-field type solution for an interacting particle Fokker-Planck equation can be provably recovered without resorting to non-convex optimization. For quantum mechanical systems, we present a similar technique to obtain the ground state of a quantum system and introduce a near-linear time algorithm for solving the convex program using hierarchical matrices.