论文标题
量子场理论中大型N问题的随机PDE方法：一项调查
A stochastic PDE approach to large N problems in quantum field theory: a survey
论文作者
论文摘要
在这项调查中，我们回顾了量子场理论，随机量化和奇异随机PDE及其平均场限制问题的最新严格结果。特别是，我们讨论了两个和三维圆环上的O（n）线性Sigma模型。随机量化过程导致N相互作用$φ^4 $方程的耦合系统。在d = 2中，我们在n范围内显示了均匀的动力学和收敛到平均场奇异SPDE的均匀。对于足够大的质量或足够小的耦合，不变的度量（即O（n）线性Sigma模型）会在大量的高斯自由场中收敛，这是均值场距离的均值量度的独特不变测度。我们还获得了某些O（n）不变的可观测值的紧密度，作为合适的BESOV空间中的随机字段，如$ n \ to \ infty $，以及限制相关性的精确描述。在d = 3中，由于方程式更为单数，估计值变得更加参与。在这种情况下，我们讨论了如何证明与大型高斯自由领域的融合。这些结果的证明是基于奇异SPDE理论的最新进展，并结合了许多新技术，例如n估计值和动态平均场理论。这些是基于与斯科特·史密斯（Scott Smith），隆昌（Rongchan Zhu）和徐尚（Xiangchan Zhu）的联合论文。
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 $Φ^4$ 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.
