物理学中的有限对称群
Finite symmetry groups in physics
AI总结：
本文提出以有限对称群为基础、通过群代数构造李群的新思路，探究有限对称群与基本相互作用规范群的关系，证明其表示几何可预测标准模型中多个混合角的精确值。
摘要：
有限对称性在粒子物理学中广泛存在，从弱二重态、代三重态到重子八重态等诸多例子皆是如此。学界通常以李群为研究起点，通过选取特定的外尔群（Weyl group）副本实现对称性破缺。本文探讨了另一种可能性：将有限对称性视为基础，借助群代数构造由有限对称群构建出李群。\n有限群代数是将有限对称群（例如三代基本费米子的对称群）嵌入李群的天然代数结构，而李群是量子场论形式体系（包括基本相互作用的规范群）的必要组成部分。有限群代数也是描述群表示的天然代数结构，群表示则用于描述基本粒子及其量子性质。因此很自然会提出这样的问题：有限群代数能否为粒子物理学标准模型提供形式化基础？如果可以，这一基础能否解释模型中其他方式无法解释的特征，比如组合规范群的奇特结构，或是不同相互作用之间的混合角？\n本文分别研究了各基本相互作用的有限对称群与规范群之间的关系，也研究了二者组合后的关系，并证明有限群的表示几何可用于预测标准模型中多个混合角的精确值，包括电弱混合角、一个轻子混合角、一个夸克混合角以及其中一个CP破坏相。
英文摘要：
Finite symmetries abound in particle physics, from the weak doublets and generation triplets to the baryon octet and many others. These are usually studied by starting from a Lie group, and breaking the symmetry by choosing a particular copy of the Weyl group. I investigate the possibility of instead taking the finite symmetries as fundamental, and building the Lie groups from them by means of a group algebra construction. Finite group algebras are the natural algebraic structures in which finite symmetry groups, such as the symmetry group of three generations of elementary fermions, are embedded in Lie groups, that are necessary for the formalism of quantum field theory, including the gauge groups of the fundamental forces. They are also the natural algebraic structures for describing representations of groups, which are used for describing elementary particles and their quantum properties. It is natural therefore to ask the question whether finite group algebras can provide a formal underpinning for the standard model of particle physics, and if so, whether this foundation can explain any aspects of the model that are otherwise unexplained, such as the curious structure of the combined gauge group, or the mixing angles between the different forces. In this paper I investigate the relationships between finite symmetry groups and the gauge groups of each of the fundamental forces individually and in combination, and show that the geometry of representations of the finite groups can be used to predict accurate values for a number of the mixing angles in the standard model, including the electro-weak mixing angle, one lepton mixing angle, one quark mixing angle and one of the CP-violating phases.
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