The exponential map for the unitary group SU(2,2)

TL;DR

This paper derives a closed-form exponential map for the $SU(2,2)$ group using Dirac matrices, extending previous work on orthogonal groups and providing explicit formulas for Lie algebra elements.

Contribution

It provides a finite, explicit formula for the exponential of traceless matrices in $SU(2,2)$, linking Lie algebra elements to Dirac matrices.

Findings

Closed-form exponential formula for $SU(2,2)$ matrices.

Representation of the exponential map using Dirac matrices.

Extension of previous results from $SO(2,4)$ to $SU(2,2)$.

Abstract

In this article we extend our previous results for the orthogonal group, $SO(2,4)$, to its homomorphic group $SU(2,2)$. Here we present a closed, finite formula for the exponential of a $4\times 4$ traceless matrix, which can be viewed as the generator (Lie algebra elements) of the $SL(4,C)$ group. We apply this result to the $SU(2,2)$ group, which Lie algebra can be represented by the Dirac matrices, and discuss how the exponential map for $SU(2,2)$ can be written by means of the Dirac matrices.