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

Abstract

In this Review, we review some recent rigorous results on large N problems in quantum field theory, stochastic quantization, and singular stochastic partial differential equations (SPDEs) and their mean field limit problems. In particular, we discuss the O( N) linear sigma model on a two- and three-dimensional torus. The stochastic quantization procedure leads to a coupled system of N interacting Φ4 equations. In d = 2, we show uniformity 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 → ∞, 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 uniformity in N estimates and dynamical mean field theory. These are based on joint papers with Scott Smith, Rongchan Zhu, and Xiangchan Zhu.

Solving Coupled Differential Equation Groups Using PINO-CDE

“…Thanks to the higher resolution brought by PINO-CDE, we found that the roughness and isolated probability hills are, in fact, caused by the insufficient resolution of the results solved from the probabilistic equation (PDEM), which has been observed in many studies. This may provide a new perspective for research in other disciplines since probabilistic and stochastic equations are widely used in physics [51,52], biology [53], and finance [54]. In conclusion, this study integrates mechanics and deep learning-based operator regression technologies and enables engineers to solve complex systems efficiently and accurately.…”

Large N limit of the O(N) linear sigma model via stochastic quantization

“…which were first developed in [SSZZ22] and [SZZ22]. We refer to [SSZZ22, Section 1] or [She22] for more discussion on the background, motivation, and previous results for large N problems in quantum field theory. With these new techniques [SSZZ22] proved that the large N limit of ν N is given by the Gaussian free field, i.e.…”