标准模型对称性与\(\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}\)的嵌套嵌入
Standard Model Symmetries and the Nested Embeddings of $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}$
- Humboldt-Universität zu Berlin（柏林洪堡大学）
- Nicolaus Copernicus University（尼古拉·哥白尼大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究标准模型内部对称性的起源，通过将\(\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}\)视为乘法代数的模，利用特定方法得到相关对称性，借助嵌入揭示其与嵌套序列的联系，还定义了自同态模型并探讨相关联系。
AI中文摘要：
标准模型的内部对称性从何而来？将\(\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}\)视为其自身乘法代数的一个模，能为标准模型的前希格斯对称性\(\mathfrak{g}_{SM}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y}\)和后希格斯对称性\(\mathfrak{g}_{LE}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}\)提供一个特定的起源故事。通过将这些自同态及其模识别为\(\mathbb{Z}_2^n\)分次代数，消除某些最高阶（体积）元素，并对反厄米算子施加等迹条件，精确地得到了\(\mathfrak{g}_{SM}\)和\(\mathfrak{g}_{LE}\)。弱超荷和电荷算子\(Y\)和\(Q\)具有非常简单的形式：\(\sum \frac{1}{n}\mathbb{I}_{n\times n}\)。借助辅助虚单位，这个15维实的\(\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}\)自然地作为向量空间嵌入到几个经过充分研究的16维实代数中，我们一般将其称为\(\mathbb{V}\)。通过这种嵌入，标准模型的内部对称性部分地源于嵌套包含序列：\(\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset \mathbb{O}\subset\mathbb{V}\)。在\(\mathbb{V} = \mathbb{S}\)的十六元数情形下，完整序列成为一个凯莱 - 迪克森塔。我们定义了粒子物理的自同态模型的概念，并将\(End_\mathbb{R}(\mathbb{V})\simeq Cl(0,8)\)与早期的博特周期粒子物理思想联系起来。我们评论了多重复结构的存在与重子不对称问题之间可能的联系。
英文摘要：
Where does the Standard Model's internal structure come from? Treating $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ as a module for its own multiplication algebra enables a particular origin story for the Standard Model's pre-Higgs, $\mathfrak{g}_{SM}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y},$ and post-Higgs, $\mathfrak{g}_{LE}:=\mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q},$ symmetries. In this model, weak hypercharge and electric charge operators, $Y$ and $Q,$ take on a remarkably simple form: $\sum \frac{1}{n}\mathbb{I}_{n\times n}$. Upon the introduction of Cayley-Dickson imaginary units, this 15 $\mathbb{R}$ dimensional $\mathbb{O}\oplus\mathbb{H}\oplus\mathbb{C}\oplus\mathbb{R}$ embeds naturally as a vector space into several well-studied 16 $\mathbb{R}$ dimensional algebras, which we generically refer to as $\mathbb{V}.$ With this embedding, the Standard Model's internal symmetries may then be seen to arise in part from the sequence of nested inclusions: $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset \mathbb{O}\subset\mathbb{V}.$ In the sedenionic case of $\mathbb{V}= \mathbb{S}$, the full sequence becomes a Cayley-Dickson tower. We define the notion of endomorphic models of particle physics, and connect $End_\mathbb{R}(\mathbb{V})\simeq Cl(0,8)$ to the earlier ideas of Bott Periodic Particle Physics. We comment on a possible connection between the existence of multiple complex structures and the baryon asymmetry problem. In closing, we identify an appearance in this model of the fully connected tree of division algebraic Hopf fibrations that starts at $S^{15},$ and simultaneously involves the four parallelizable spheres $S^7, S^3, S^1, S^0.$
