arXiv 1503.009952015-03-03math-phmath.MP

解析函数的复幂与QFT中的亚纯重整化

Complex powers of analytic functions and meromorphic renormalization in QFT

Nguyen Viet Dang

备注：: Feedback welcome ! arXiv admin note: text overlap with arXiv:1305.3535 by other authors

摘要：

本文利用Hironaka的奇点分解研究了亚纯分布族$(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p) \in \mathbb{C}^p}$的泛函解析性质，然后利用最近关于具有线性极点的亚纯芽的分解的工作，在分布空间中对解析函数$\prod_{i=1}^p(f_j+i0)^{k_j}, k_j \in \mathbb{Z}$的幂积进行了重整化.我们还研究了$(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p)\in\mathbb{C}^p}$和$\prod_{i=1}^p (f_j+i0)^{k_j}, k_j \in \mathbb{Z}$的微局域性质。在第二部分中，我们证明了上述具有\emph{regular holonomic singularities}的分布族提供了一个描述Feynman振幅奇异性的普适模型，并作为第一部分思想的应用，给出了凸解析Lorentz时空上量子场论可重整化的新证明。

英文摘要：

In this article, we study functional analytic properties of the meromorphic families of distributions $(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p) \in \mathbb{C}^p}$ using Hironaka's resolution of singularities, then using recent works on the decomposition of meromorphic germs with linear poles, we renormalize products of powers of analytic functions $\prod_{i=1}^p(f_j+i0)^{k_j}, k_j \in \mathbb{Z}$ in the space of distributions. We also study microlocal properties of $(\prod_{i=1}^p (f_j+i0)^{λ_j})_{(λ_1,\dots,λ_p)\in\mathbb{C}^p}$ and $\prod_{i=1}^p (f_j+i0)^{k_j}, k_j \in \mathbb{Z}$. In the second part, we argue that the above families of distributions with \emph{regular holonomic singularities} provide a universal model describing singularities of Feynman amplitudes and give a new proof of renormalizability of quantum field theory on convex analytic Lorentzian spacetimes as applications of ideas from the first part.

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摘要：

英文摘要：

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