Explicit closed-form parametrization of SU(3) and SU(4) in terms of complex quaternions and elementary functions

TL;DR

This paper presents simple closed-form formulas for SU(3) and SU(4) group elements using biquaternions, offering a more efficient way to handle complex matrix calculations in quantum chromodynamics.

Contribution

It introduces explicit quaternion-based parametrizations for SU(3) and SU(4), simplifying their representation and potential applications in physics.

Findings

Closed-form expressions for SU(3) and SU(4) groups.

Quaternion methods applicable to quantum chromodynamics.

Potential for simplifying complex matrix calculations.

Abstract

Remarkably simple closed-form expressions for the elements of the groups SU(n), SL(n,R), and SL(n,C) with n=2, 3, and 4 are obtained using linear functions of biquaternions instead of n x n matrices. These representations do not directly generalize to SU(n>4). However, the quaternion methods used are sufficiently general to find applications in quantum chromodynamics and other problems which necessitate complicated 3 x 3 or 4 x 4 matrix calculations.