A stochastic PDE approach to large N problems in quantum field theory: A survey

In this survey we review some recent rigorous results on large N problems in quantum field theory, stochastic quantization and singular PDEs, their mean limit problems. particular discuss the O(N) linear sigma model two three dimensional torus. The procedure leads to a coupled system of interacting $\Phi^4$ equations. d = 2, show uniform bounds for dynamics convergence mean-field SPDE. For enough mass or small coupling, invariant measures (i.e. model) converge massive Gaussian free field, unique measure dynamics, Wasserstein distance. We also obtain tightness certain observables as random fields suitable Besov spaces $N\to \infty$, along with exact descriptions limiting correlations. 3, estimates become more involved since equation is singular. case how prove field. proofs these build progress SPDE theory combine many new techniques such dynamical theory. These are based joint papers Scott Smith, Rongchan Zhu Xiangchan Zhu.