The Lorentz group and the Kronecker product of matrices

TL;DR

This paper presents a matrix algebra formulation of the spinor map relating SL(2,C) to Lorentz transformations using Kronecker products, facilitating algebraic manipulation in relativistic physics.

Contribution

It introduces a novel matrix algebra expression of the spinor map involving Kronecker products, enhancing the algebraic study of Lorentz transformations.

Findings

Derived a matrix algebra formula for the spinor map

Enabled manipulation of Lorentz transformations via matrix algebra

Facilitated applications in relativistic quantum mechanics

Abstract

The group $SL(2,\mathbb{C})$ of all complex $2\times 2$ matrices with determinant one is closely related to the group $\boldsymbol{\mathcal{L}}_{+}^\uparrow$ of real $4\times 4$ matrices representing the restricted Lorentz transformations. This relation, sometimes called the spinor map, is of fundamental importance in relativistic quantum mechanics and has applications also in general relativity. In this paper we show how the spinor map may be expressed in terms of pure matrix algebra by including the Kronecker product between matrices in the formalism. The so-obtained formula for the spinor map may be manipulated by matrix algebra and used in the study of Lorentz transformations.