Solving the matrix exponential function for the groups SU(3), SU(4) and Sp(2)

TL;DR

This paper extends the analytical formulas for matrix exponentials from SU(2) to higher groups like SU(3), SU(4), and Sp(2), involving solving cubic and quartic equations, and discusses potential formulas for even larger groups.

Contribution

The paper derives explicit analytical formulas for matrix exponentials in SU(3), SU(4), and Sp(2), generalizing the known SU(2) case and exploring higher groups.

Findings

Analytical formula for SU(3) involves roots of a cubic equation.

For SU(4), the formula involves roots of a quartic equation.

Sp(2) matrices have simplified formulas due to root structure.

Abstract

The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau \!\cdot\! \vec \varphi\,) = \cos|\vec \varphi\,| + i\vec \tau \!\cdot\! \hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with eight real parameters. The resulting analytical formula involves the sum over three real roots of a cubic equation, corresponding to the so-called irreducible case, where one has to employ the trisection of an angle. When going to the special unitary group $SU(4)$ with 15 real prameters, the analytical formula involves the sum over four real roots of a quartic equation. The associated cubic resolvent equation with three positive roots belongs again to the irreducible case. Furthermore, by imposing the pertinent condition on $SU(4)$ matrices one can also treat the symplectic group $Sp(2)$ with ten real parameters. Since there the roots occur as two pairs of opposite sign, this simplifies the analytical formula for $Sp(2)$ matrices considerably. An outlook to the situation with analytical formulas for $SU(5)$, $SU(6)$ and $Sp(3)$ is also given.