费米子入门指南
A fermion primer
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中文总结 AI 辅助
本文为一篇费米子领域的综述，梳理狄拉克、外尔、马约拉纳费米子的关键差异，探讨外尔费米子传播子缺失等争议问题，指出替换技巧的逻辑漏洞，明确正确处理方式及威克转动的非等价性。
中文摘要 AI 辅助
本文是一篇综述论文，旨在阐明关于狄拉克（Dirac）、外尔（Weyl）和马约拉纳（Majorana）费米子及其差异的若干关键问题。第一部分包含费米子的基础介绍，更详细地介绍了四维（4d）费米子，重点关注区分三种费米子的要素：手征性、螺旋性、质量、场方程、在离散对称变换下的性质、作用量（Actions）、可观测量（Observables）。在此基础上，我们探讨一系列具有挑战性且有时存在争议的问题：外尔费米子传播子不存在的问题及规避方法；外尔费米子振幅的正则化，特别是使用PV正则化的适用性；外尔费米子的泛函积分定义；外尔费米与无质量马约拉纳费米子的差异，以及狄拉克质量项与马约拉纳质量项的差异。最值得注意的是，我们得出结论：用不存在的外尔费米子传播子替换无质量狄拉克传播子的技巧，尽管在某些情况下可能产生正确结果，但从根源上存在逻辑漏洞。随后我们展示了这种情况下的正确处理方式。我们还考虑了在外尔费米子经典作用量（classical Action）层面应用威克转动（Wick rotation）的问题，得出该过程会导致非等价理论的结论。最后，虽然反常（anomalies）不是本文的核心焦点，但我们认为简要回顾外尔费米子传播子缺失（外尔-狄拉克算子不可逆性）与理论中出现危险反常之间的关系及其上同调（cohomological）基础是有用的。
英文摘要
This is a review paper intended to illustrate a few critical issues concerning Dirac, Weyl and Majorana fermions and their differences. The first part consists in basic introductions to fermions, and more in detail to fermions in 4d, focusing in particular in what differentiate the three type of fermions: chirality, helicity, mass, field equations, properties under discrete symmetry transformations, Actions, Observables. On this basis we tackle a series of challenging and sometime controversial problems: the non-existence of Weyl fermion propagators, and the ways to circumvent it; the regularizations for Weyl fermion amplitudes, in particular the appropriateness of using the PV regularization; the definition of a functional integral for Weyl fermion, the difference between Weyl and massless Majorana fermions and the difference between the Dirac and Majorana mass terms. Most notably we come to the conclusion that the trick of replacing the non-existing Weyl fermion propagator with a massless Dirac one, although it may yield in some cases correct results, is flawed at the very origin by a logical loophole. We then show which is the correct way to proceed in this case. We consider also the topic of applying the Wick rotation at the classical Action level for Weyl fermions and conclude that this procedure leads to a nonequivalent theory. Finally, although anomalies are not the central focus here, we have deemed it useful to summarily review the relation, and its cohomological basis, that exists between the lack of a Weyl fermion propagator (non-invertibility of the Weyl-Dirac operator) and the appearance of dangerous anomalies in the theory.
发表机构
- International School for Advanced Studies (SISSA)（国际高等研究院）
- Università di Bologna（博洛尼亚大学）
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