11.29主要内容

上节课内容补充

1. 补充Higgs mechanism“吃掉”\(\phi_2\)的过程
2. Nonlinear/Interaction

   

   Nonlinear/Interaction

群论基本知识

1. 重排定理(rearrangement theorem)
   \(gG=G\), for any \(g\in G\)
2. 子群(subgroup)
   H is a group, and \(H\subset G\)
3. 陪集(coset): left&right coset
   \(g_{i}, g_{j} \in G \Rightarrow \left\{ \begin{aligned} g_{i} H=g_{j} H \\ g_{i} H \cap g_{j} H=\varnothing \end{aligned} \right. \)

   

   Coset decomposition
   
   Coset decomposition
4. 正规/不变子群(normal/invariant subgroup)
   \(H\lhd G\), 当对应左右陪集相等\(g_i H=H g_i\)
5. 商群(quotient group)
   不变子群及其所有陪集构成群，\(G/H=\left\{g_i H\ |\ g_i \in G\right\}\)
examples: \(Z_2, C_{3v}, \mathbb{Z}_4\)

群表示论(group representation theory)

1. 将本为operator的群元表示成matrix
2. 群的不等价不可约表示
   (1)Schur lemma
   (2)Jordan decomposition
   (3)表示的完备性：
   有限群不等价不可约表示维数平方和等于群的阶数\(\sum_j m_{j}^{2}\)
   特征标\(\chi^j(G)\)构成正交完备基(体现在特征标表中横纵形成矢量之间正交\(\left \langle v_i | v_j \right \rangle=0\) )
3. Application: molecular(raman or infrared spectrum), solid state physics(Bloch bands)
   (1) \(O\) point-group in Hydrogen atom
   (2) Degeneracy in the particle-in-a-box problem. 原文下载

同态基本定理

1. 同态基本定理

   

   Fundamental theorem on homomorphisms
   
   1.\(G / \operatorname{ker}(f) \simeq \operatorname{Im}(f) \neq G^{\prime}\)
   2.\(\operatorname{ker}(f) \equiv H \lhd G\)是\(G\)的正规子群
   3.\(\operatorname{Im}(f) \subset G^{\prime}\)是\(G^{\prime}\)的子群
   If \(G^{\prime} \simeq \operatorname{Im}(f) \Rightarrow G / \operatorname{ker}(f) \simeq G^{\prime}\)
   应用： rank-nullity theorem(秩零定理)
2. Homotopy group & (co)Homology group 概述
同态(Homomorphism)、同构(Isomorphism)、同伦(Homotopy)、同调(Homology)、上同调(Cohomology)、同胚(Homeomorphism=Topological isomorphism)

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