A first-order relativistic wave equation is constructed in five dimensions. Its solutions are eight-component spinors, which are interpreted as single-particle fermion wave functions in four-dimensional spacetime. Use of a ``cylinder condition'' (the removal of explicit dependence on the fifth coordinate) reduces each eight-component solution to a pair of degenerate four-component spinors obeying the Dirac equation. This five-dimensional method is used to obtain solutions for a free particle and for a particle moving in the Coulomb potential. It is shown that, under the cylinder condition, the results are the same as those from the Dirac equation. Without the cylinder condition, on the other hand, the equation predicts some interesting new phenomena. It implies the existence of a scalar potential, and for zero-mass particles it leads to a four-dimensional fermionic equation analogous to Maxwell's equation with sources.
Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background
Simple derivation of the Weyl and Dirac quantum cellular automata
我们考虑体心立方晶格上的量子元胞自动机，并给出了仅有的两个均匀、局部、各向同性和酉二维自动机的简单推导[G. M. D'Ariano和P.Perinotti，Physical Review A 90，062106（2014）]。我们的推导依赖于格拉姆矩阵的概念，并强调了转换矩阵之间的联系，转换矩阵的自动机和体心立方晶格：转换矩阵本质上是矩阵表示的顶点的晶格的原始细胞。正如预期的那样，这两个自动机的动力学在小波矢量和连续时间的限制下归结为Weyl方程。我们还简要地研究了四维的情况下，我们发现两个单参数的自动机，减少到狄拉克方程在一个适当的限制家庭。
