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

Abstract

AbstractThe well known analytical formula forSU(2) matrices$$U = \exp (i \vec \tau \!\cdot \! \vec \varphi \,) = \cos |\vec \varphi \,|\mathbf{1} + i\vec \tau \!\cdot \! {\hat{\varphi }} \, \sin |\vec \varphi \,|$$U=exp(iτ→·φ→)=cos|φ→|1+iτ→·φ^sin|φ→|is extended to theSU(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 for solution the trisection of an angle. When going to the special unitary groupSU(4) with 15 real parameters, 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 onSU(4) matrices one can also treat the symplectic groupSp(2) with ten real parameters. Since there the roots occur as two pairs of opposite sign, this simplifies the analytical formula forSp(2) matrices considerably. An outlook to the situation with quasi-analytical formulas forSU(5),SU(6) andSp(3) is also given.