Hypercomplex representation of the Lorentz's group

TL;DR

This paper explores a hypercomplex number system based on Dirac matrices to represent Lorentz's group, simplifying transformations and laws related to relativistic symmetries, with potential applications in spin and particle physics.

Contribution

It introduces a reducible hypercomplex representation of Lorentz's group that simplifies transformation rules and related physical laws.

Findings

Simplifies Lorentz transformation composition rules

Unifies transformation of vectors, tensors, and bispinors

Facilitates analysis of spin connection and Wigner little group

Abstract

Lorentz's group represented by the hypercomplex system of numbers, which is based on dirac matrices, is investigated. This representation is similar to the space rotation representation by quaternions. This representation has several advantages. Firstly, this is reducible representation. That is why transformation of different geometrical objects (vectors, antisymmetric tensors of the second order and bispinors) are implemented by the same operators. Secondly, the rule of composition of two arbitrary Lorentz's transformations has a simple form. These advantages strongly simplify finding a lot of the laws related to the Lorentz's group. In particular they simplify investigation of the spin connection with Pauli-Lubanski pseudovector and Wigner little group.