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SU(5) Grand Unified Theory, its Polytopes and 5-fold Symmetric Aperiodic Tiling

机译:sU(5)大统一理论,其多面体和5重对称非周期性   平铺

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摘要

We associate the lepton-quark families with the vertices of the 4D polytopes5-cell and the rectified 5-cell derived from the SU(5) Coxeter-Dynkin diagram.The off-diagonal gauge bosons are associated with the root poytope (1000)A4whose facets are tetrahedra and the triangular prisms. The edge-vertexrelations are interpreted as the SU(5) charge conservation. The Dynkin diagramsymmetry of the SU(5) diagram can be interpreted as a kind ofparticle-antiparticle symmetry. The Voronoi cell of the root lattice consistsof the union of the polytopes (1000)A4 + (0100)A4 + (0010)A4 + (0001)A4 whosefacets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the rootlattice as the alternating 5-cell and the rectified 5-cell, a kind of dual tothe Voronoi cell. The vertices of the Delone cells closest to the originconsists of the root vectors representing the gauge bosons. The faces of therhombohedra project onto the Coxeter plane as thick and thin rhombs leading toPenrose-like tiling of the plane which can be used for the description of the5-fold symmetric quasicrystallography. The model can be extended to SO(10) andeven to SO(11) by noting the Coxeter-Dynkin diagram embedding in A4 in D5 inB5. Another embedding can be made through the relation A4 in D5 in E6 for morepopular GUT's.
机译:我们将轻子夸克族与从SU(5)Coxeter-Dynkin图派生的4D polytopes5-cell和整流的5-cell的顶点相关联。对角规玻色子与根Poytope(1000)A4whoose相关刻面是四面体和三棱柱。边缘垂直关系被解释为SU(5)电荷守恒。 SU(5)图的Dynkin图对称性可以解释为一种粒子-反粒子对称性。根晶格的Voronoi细胞由多面体(20)的多面体(1000)A4 +(0100)A4 +(0010)A4 +(0001)A4的并集组成。我们将根晶格的Delone(Delaunay)细胞构建为交替的5细胞和整流的5细胞,这是Voronoi细胞的双重细胞。最接近原点的Delone细胞的顶点由代表量规玻色子的根向量组成。 Therhombohedra的面以粗细的菱形突出到Coxeter平面上,导致平面上出现Penrose状的平铺,可用于描述5倍对称准晶体学。通过注意将Coxeter-Dynkin图嵌入到B5的D5中的A4中,可以将该模型扩展到SO(10)甚至扩展到SO(11)。对于更受欢迎的GUT,可以通过E6中D5中的关系A4进行另一个嵌入。

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