A stochastic PDE approach to large N problems in quantum field theory: A survey
A stochastic PDE approach to large N problems in quantum field theory: A survey
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DOI：
10.1063/5.0089851
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发表时间：
2022-08
影响因子：
1.3
通讯作者：
Hao Shen
中科院分区：
文献类型：
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作者：
Hao Shen
在这篇综述中，我们回顾了量子场论中的大N问题，随机量子化和奇异随机偏微分方程（SPDE）及其平均场极限问题的一些最近的严格结果。特别是，我们讨论了O（N）线性sigma模型的二维和三维环面。随机量子化过程导致N个相互作用的Φ4方程的耦合系统。在d = 2，我们表现出均匀的N界的动力学和收敛到一个平均场奇异SPDE。对于足够大的质量或足够小的耦合，不变测度[即，O（N）线性sigma模型]收敛到大质量高斯自由场，平均场动力学的唯一不变测度，在Wasserstein距离。我们还得到了在适当的Besov空间中，当N → ∞时，某些O（N）不变观测量作为随机场的紧性，沿着，并给出了极限相关的精确描述。在d = 3时，由于方程更加奇异，估计变得更加复杂。在这种情况下，我们讨论如何证明收敛到大质量高斯自由场。这些结果的证明建立在奇异SPDE理论的最新进展之上，并结合了联合收割机许多新技术，如N估计的一致性和动力学平均场理论。这些是基于与Scott Smith、Rongchan Zhu和Xiangchan Zhu的联合论文。
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.