An explicit description of $SL(2,\mathbb{C})$ in terms of $SO^+(3,1)$ and vice versa

TL;DR

This paper provides explicit formulas connecting the Lorentz group $SO^+(3,1)$ with its spin group $SL(2, C)$ using complex matrices, motivated by Clifford algebra concepts, to clarify their relationship.

Contribution

It offers elementary, explicit formulas for the correspondence between $SO^+(3,1)$ and $SL(2, C)$, making the relationship more accessible without relying explicitly on Clifford algebra terminology.

Findings

Explicit formulas for the $SO^+(3,1)$ and $SL(2, C)$ correspondence

Elementary calculations using complex 2x2 matrices

Motivated by Clifford algebra structures

Abstract

In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation $SO^+(3,1)$, on the one hand, and its spin group $SL(2,\mathbb{C})$, on the other hand. Although we will not mention Clifford algebra terminology explicitly, it is hidden in our calculations by using complex $2\times2$-matrices. Nevertheless, our calculations are strongly motivated by the Clifford algebra $\mathfrak{gl}(4,\mathbb{C})$ of four-dimensional space-time.