Two-Component Spinorial Formalism Using Quaternions for Six-Dimensional Spacetimes

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, 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, 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, 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).

Automated summary

This article discusses various aspects of the two-component spinorial formalism for six-dimensional spacetimes, including the representation of chiral spinors by objects with two quaternionic components and the identification of the spin group as SL(2, H). It explores the fundamental representations of this group, the representation of vectors, bivectors, and 3-vectors in this spinorial formalism, as well as the complexification of spacetime to handle other signatures. The article also addresses the lack of group representation in objects built from tensor products of the fundamental representations of SL(2, H) due to the non-commutativity of quaternions. Additionally, it establishes a connection between quaternionic spinorial formalism for six-dimensional spacetimes and the four-component spinorial formalism over the complex field.