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

Abstract

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