Principal Bundles and Gauge Theories
Principal Bundles and Gauge Theories
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发表时间：
2011-06
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通讯作者：
Matthijs Vákár
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作者：
Matthijs Vákár
这套讲义填补了作者认为现有（前量子或首次量子化）规范理论文献中存在的一个漏洞：理论物理教科书在坐标中进行了大量计算，但未能理解其方程的潜在几何形状；数学课本把纤维束的几何理论发展得十分抽象，但是，在他们试图保持事物抽象的过程中，完全错过了对数学对象进行迷人的物理解释的描述。在这些笔记中，纤维束的数学理论是在很大的一般性基础上建立起来的，慢慢地增加了更多的结构，并收敛到我们在现代物理学中看到的结构。在这一点上，我们正处于一个完美的位置，可以将一些现代物理理论融入到这个框架中。这种非传统的材料展示有望为数学家和物理学家在这个主题上提供新的亮点，作为对其他学科观点的简要介绍。
This set of lecture notes fills a hole that in the authors opinion currently exists in the available literature on (prequantum or first quantised) gauge theory: theoretical physics texts do lots of calculations in coordinates and fail to appreciate the underlying geometry of their equations; mathematics texts develop the geometric theory of fibre bundles in its full abtract glory, but, in their attempt to keep things abstract, completely miss out on describing fascinating physical interpretations of their mathematical objects. In these notes, the mathematical theory of fibre bundles is built up from the ground in great generality, slowly adding more structure and converging to the structures we see in modern physics. At this point, we are in the perfect position to fit some modern theories of physics in the framework. This unconventional exposition of the material can hopefully shed new light on the topic for both mathematicians and physicists, serving as a brief introduction to the other discipline's point of view.