论文标题
使用四元组进行六维空间的两部分旋转形式主义
Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes
论文作者
论文摘要
在本文中，我们构建和讨论了六维空间的两组分旋转形式主义的几个方面，其中手性旋转器由具有两个Quaternionic成分的对象表示，并用$ SL（2; \ Mathbb {H}）识别旋转组，这是Lorentz组的双重覆盖组的双重覆盖。我们介绍了该小组的基本表示，并展示了这种旋转形式主义中如何代表向量，双分和3个向量。我们还使时空复杂化，以便可以解决其他签名。我们认为，通常，由于$ sl（2; \ mathbb {h}）$的基本表示的张量产物构建的对象，由于四季度的非交换性，因此不携带该组的代表。获得了旋转组的Lie代数，并提出了其与$（5,1）$的Lie代数的联系，为$ SL（2; \ Mathbb {H}）$的元素提供了物理解释。最后，我们介绍了这种四维旋转形式主义之间的桥梁，与复杂领域的四个成分旋转形式主义之间的桥梁是由六维欧几里得空间中的旋转组带来的。
In this article we construct and discuss several aspects of the two-component spinorial formalism for six-dimensional spacetimes, in which chiral spinors are represented by objects with two quaternionic components and the spin group is identified with $SL(2;\mathbb{H})$, which is a double covering for the Lorentz group in six dimensions. We present the fundamental representations of this group and show how vectors, bivectors, and 3-vectors are represented in such spinorial formalism. We also complexify the spacetime, so that other signatures can be tackled. We argue that, in general, objects built from the tensor products of the fundamental representations of $SL(2;\mathbb{H})$ do not carry a representation of the group, due to the non-commutativity of the quaternions. The Lie algebra of the spin group is obtained and its connection with the Lie algebra of $SO(5,1)$ is presented, providing a physical interpretation for the elements of $SL(2;\mathbb{H})$. Finally, we present a bridge between this quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field that comes from the fact that the spin group in six-dimensional Euclidean spaces is given by $SU(4)$.
