自然界中存在E8的实验证据和Gosset圆的半径
Experimental evidence for the occurrence of E8 in nature and the radii of the Gosset circles
摘要：
最近的一项实验发现涉及冷一维磁体中电子的自旋结构，该发现表明了涉及特殊李群$E_8$的扎莫洛奇科夫模型的有效性。该模型预测了8个粒子，并预测了它们的质量比。即，8维Gosset多面体的顶点与$E_8$的240个根相同。在多面体的2-D（Peter McMullen）投影下，顶点的图像被布置在8个同心圆中，这里被称为戈塞特圆。戈塞特圆被理解为对应于模型中的8个质量，并且被理解为它们的半径的比率与对应的几何质量的比率相同。两个最小的圆的比例（读作2个最小的质量）是黄金数。这些理论现在已经得到了实验的验证，至少在前五个质量上是如此。McMullen投影推广到任何秩大于1的复单李代数。戈塞特圆也推广，使用Coxeter元素的轨道。使用结果在1959年的论文我，我发现前一段时间一个非常容易定义的运营商$A$其频谱正是平方半径$r_i$这些广义戈塞特圈。作为确认，在$E_8$的情况下，仅使用$A$的适当倍数的特征值，Vogan计算了$r_i$的比率。令人高兴的是，它们与扎莫洛奇科夫质量的相应比值一致。运算符$A$被写为$\ell +1$秩为1的运算符之和，由扩展Dynkin图中的点参数化。这种展开涉及最高根的系数$n_i$。回想麦凯对应，在$E_8$的情况下，$n_i$和1是二元二十面体群的不可约表示的维数。
英文摘要：
A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a Zamolodchikov model involving the exceptional Lie group $E_8$. The model predicts 8 particles and predicts the ratio of their masses. I.e., the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of $E_8$. Under the 2-D (Peter McMullen) projection of the polytope, the image of the vertices are arranged in 8 concentric circles, here referred to as the Gosset circles. The Gosset circles are understood to correspond to the 8 masses in the model, and it is understood that the ratio of their radii is the same as the ratio of the corres-ponding conjectural masses. A ratio of the two smallest circles (read 2 smallest masses) is the golden number. The conjectures have been now validated experimentally, at least for the first five masses. The McMullen projection generalizes to any complex simple Lie algebra whose rank is greater than 1. The Gosset circles also generalize, using orbits of the Coxeter element. Using results in a 1959 paper of mine, I found some time ago a very easily defined operator $A$ whose spectrum is exactly the squares of the radii $r_i$ of these generalized Gosset circles. As a confirmation, in the $E_8$ case, using only the eigenvalues of a suitable multiple of $A$, Vogan computed the ratio of the $r_i$. Happily these agree with the corresponding ratio of the Zamolodchikov masses. The operator $A$ is written as a sum of $\ell +1$ rank 1 operators, parameterized by the points in the extended Dynkin diagram. Involved in this expansion are the coefficients $n_i$ of the highest root. Recalling the McKay correspondence, in the $E_8$ case, the $n_i$, together with 1, are the dimensions of the irreducible representations of the binary icosahedral group.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
