arXiv 2105.127512021-05hep-thcond-mat.stat-mechhep-latmath-phmath.MP

量子场论的紫外结构。第三部分：规范理论

The Ultraviolet Structure of Quantum Field Theories. Part 3: Gauge Theories

Djordje Radicevic

备注：: 121 pages, 4 figures. v2: Minor improvements. Comments from readers are welcome!

摘要：

本文对阿贝尔规范理论的所有常见例子，无论有无物质，都给出了一个详细的格点-连续对应。这些从格点抽取连续统理论的规则代表了严格定义连续统规范理论的基本方法。重点是（2+1）D，但这里开发的技术适用于所有维度。本文的前半部分致力于纯麦克斯韦理论。它精确地表明连续麦克斯韦理论如何出现在低能量在适当的参数制度的$\mathbb{Z}_K$格点规范理论在大$K$。这个理论的熟悉的特征--它的OPE结构、紧致和非紧致自由度之间的区别、红外“粒子-涡旋”对偶性以及它的威尔逊圈的库仑定律行为--都是直接从晶格中导出的。论文的其余部分研究了耦合到玻色子或费米子物质的规范场。标量QED的分析在同一水平的纯麦克斯韦理论的精度，其相结构和拓扑BF理论的连接与新的评论。普通QED（狄拉克费米子耦合到麦克斯韦理论）的构建着眼于正确定义旋量和避免全局异常。传统的连续QED所示，产生于一个特定的限制的起始理论，以平滑变化的领域。这样的平滑也被证明是陈-西蒙斯理论在路径积分框架中积分出大质量费米子的必要条件。为了找到一个不依赖于路径积分光滑化的Chern-Simons理论的正则起源，我们证明了一个简单的通量附加格点规范理论在约束区产生了一个类似Chern-Simons的作用量.

英文摘要：

This paper develops a detailed lattice-continuum correspondence for all common examples of Abelian gauge theories, with and without matter. These rules for extracting a continuum theory out of a lattice one represent an elementary way to rigorously define continuum gauge theories. The focus is on (2+1)D but the techniques developed here work in all dimensions. The first half of this paper is devoted to pure Maxwell theory. It is precisely shown how continuum Maxwell theory emerges at low energies in an appropriate parameter regime of the $\mathbb{Z}_K$ lattice gauge theory at large $K$. The familiar features of this theory -- its OPE structure, the distinction between compact and noncompact degrees of freedom, infrared "particle-vortex" dualities, and the Coulomb law behavior of its Wilson loop -- are all derived directly from the lattice. The rest of the paper studies gauge fields coupled to either bosonic or fermionic matter. Scalar QED is analyzed at the same level of precision as pure Maxwell theory, with new comments on its phase structure and its connections to the topological BF theory. Ordinary QED (Dirac fermions coupled to Maxwell theory) is constructed with an eye toward properly defining spinors and avoiding global anomalies. The conventional continuum QED is shown to arise from a specific restriction of the starting theory to smoothly varying fields. Such smoothing also turns out to be a necessary condition for Chern-Simons theory to arise from integrating out massive fermions in a path integral framework. In an effort to find a canonical origin of Chern-Simons theory that does not rely on path integral smoothing, a simple flux-attached lattice gauge theory is shown to give rise to a Chern-Simons-like action in the confining regime.

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英文摘要：

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