解析重整化群
The Analytic Renormalization Group
摘要：
量子力学或量子场论中的有限温度欧几里德两点函数的特征在于与松原频率$ν_{k}=2πk/β$相关联的傅立叶系数$G_{k}$、$k\in\mathbb Z$的离散集合。我们表明，解析意味着系数$G_{k}$必须满足无限多的模型无关的线性方程，我们写下来明确。特别是，我们构建“解析重整化群“线性映射$\mathsf A_μ$，对于任何选择的截止$μ$，允许表达的低能量傅立叶系数$|ν_{k}|<μ$（与零模式$G_{0}$的可能例外），连同实时的分解器和光谱功能，在高能量傅立叶系数$|ν_{k}|\geqμ$。操作一个简单的数值算法，我们表明，$G_{k}$上的精确的通用线性约束可以用来系统地改善任何随机近似数据集，例如，从蒙特卡罗模拟。我们的结果说明了几个明确的例子。
英文摘要：
Finite temperature Euclidean two-point functions in quantum mechanics or quantum field theory are characterized by a discrete set of Fourier coefficients $G_{k}$, $k\in\mathbb Z$, associated with the Matsubara frequencies $ν_{k}=2πk/β$. We show that analyticity implies that the coefficients $G_{k}$ must satisfy an infinite number of model-independent linear equations that we write down explicitly. In particular, we construct "Analytic Renormalization Group" linear maps $\mathsf A_μ$ which, for any choice of cut-off $μ$, allow to express the low energy Fourier coefficients for $|ν_{k}|<μ$ (with the possible exception of the zero mode $G_{0}$), together with the real-time correlators and spectral functions, in terms of the high energy Fourier coefficients for $|ν_{k}|\geqμ$. Operating a simple numerical algorithm, we show that the exact universal linear constraints on $G_{k}$ can be used to systematically improve any random approximate data set obtained, for example, from Monte-Carlo simulations. Our results are illustrated on several explicit examples.
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