Poincaré代数的Z3-graded扩张
The Z3-graded extension of the Poincaré algebra
摘要：
最近的一系列论文提出了Dirac方程的Z3对称推广，讨论了它的性质和解。广义狄拉克算符作用于由六个泡利旋量组成的“色旋量”，描述了三种颜色和表征单个夸克态的粒子-反粒子自由度，从而组合了12分量广义波函数的Z2 x Z_2 x Z_3对称性。引入了Z3-阶化广义Lorentz代数的旋量表示，导致了额外的Z2 x Z2 x Z3对称性的出现，可能包含了同位旋、味和族的对称性。本文给出了Poincaré代数的Z3-分次扩张的一个构造。事实证明，这样的推广需要引入扩展的12维闵可夫斯基时空，其中包含通常的4维时空作为子空间，以及另外两个具有复值向量和度量张量的相互共轭“副本”。介绍了微分算子和广义Casimir算子的表示，并简要讨论了它们的对称性。
英文摘要：
A Z3 symmetric generalization of the Dirac equation was proposed in recent series of papers, where its properties and solutions discussed. The generalized Dirac operator acts on "coloured spinors" composed out of six Pauli spinors, describing three colours and particle-antiparticle degrees of freedom characterizing a single quark state, thus combining Z2 x Z_2 x Z_3 symmetries of 12-component generalized wave functions. Spinorial representation of the Z3-graded generalized Lorentz algebra was introduced, leading to the appearance of extra Z2 x Z2 x Z3 symmetries, probably englobing the symmetries of isospin, flavors and families. The present article proposes a construction of Z3-graded extension of the Poincaré algebra. It turns out that such a generalization requires introduction of extended 12-dimensional Minkowskian space-time containing the usual 4-dimensional space-time as a subspace, and two other mutually conjugate "replicas" with complex-valued vectors and metric tensors. Representation in terms of differential operators and generalized Casimir operators are introduced and their symmetry properties are briefly discussed.
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