A PDE Construction of the Euclidean $$\Phi ^4_3$$ Quantum Field Theory
A PDE Construction of the Euclidean $$\Phi ^4_3$$ Quantum Field Theory
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DOI：
10.1007/s00220-021-04022-0
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发表时间：
2018-10
影响因子：
2.4
通讯作者：
M. Gubinelli;M. Hofmanová
中科院分区：
文献类型：
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作者：
M. Gubinelli;M. Hofmanová
提出了一种基于偏微分方程论的欧几里得量子场论的新构造。更准确地说，我们考虑了定义在网格大小和边长为M的周期格子上的随机量子化方程的近似。我们在加权空间中引入了一种新的重整化能量方法，并证明了相应的Gibbs测度的紧性。每个极限点都是非高斯的，满足反射正性、平移不变性和伸展指数可积性。这些性质允许证明欧几里得QFT的Osterwalder-Schrader公理，而不是旋转不变性和聚类性。我们的论证适用于任意正耦合常数、具有O(N)对称性的多分量模型和一些长程变种。此外，我们还建立了欧氏关联函数的Dyson-Schwinger方程族的分部积分公式。为此，我们将重整化的三次项定义为欧几里得空间上的一种分布。
We present a new construction of the Euclideanquantum field theory onbased on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation ondefined on a periodic lattice of mesh sizeand side lengthM. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as,. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models withO(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as adistributionon the space of Euclidean fields.