The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8, \(\mathbb {C})\) of complex \(8 \times 8\) matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti-)particle states are identified with basis elements of the vector space M(8, \(\mathbb {C})\). Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8, \(\mathbb {C})\) contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics.

中文翻译：

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统一标准模型

这项工作的目的是为我们已建立的粒子物理学描述找到一个简单的数学框架。我们证明了标准模型粒子的特定量规结构，群表示和电荷分配都由复杂的\（8 \ times 8 \）矩阵的代数M（8，\（\ mathbb {C}）\）捕获 。该代数与八进制的规范除法代数有着密切的联系，因此具有很好的动机。（反）粒子状态由向量空间M（8，\（\（mathbb {C}）\\）的基本元素来标识 。规范变换可以简单地由代数对其自身进行描述。我们的结果表明，标准模型的所有粒子和尺度结构都包含在所有四个范数除代数的张量积中，其中四元数提供洛伦兹表示。有趣的是，空间M（8，  \（\ mathbb {C}）\）包含两个独立于标准模型粒子的附加元素，这暗示着最少的新物理现象。