The exponentiation of Pauli vector $\vec \sigma=(\sigma_x,\sigma_y,\sigma_z)$ is trivial as we have the identity:$$e^{ia(\vec n\cdot \vec \sigma)}=I cos(a)+i(\vec n \cdot \vec \sigma)sin(a)$$
I have been studying the properties of Gell-Mann matrices and was wondering whether a similar exponentiation is possible or not. I used Mathematica to explicitly compute the exponentials of individual Gell-Mann matrices and I am getting exact solutions.
Though, the exponentials of Gell-Mann matrices can be obtained explicitely I was trying to obtain an identity similar to the above one for Pauli matrices. The major property of Pauli matrices which enable the derivation of the above identity is that $\sigma_i^2=I$. But, the Gell-Mann matrices in general does not have this property.
Is it possible to derive a similar identity? If yes, I would like to receive some hints on how to achieve the same.
Best Answer
This question is answered affirmatively and in detail in the following paper:
Thomas L. Curtright and Cosmas K. Zachos, Elementary results for the fundamental representation of SU(3), Rept. Math. Phys. 76 (2015), 401-404. e-Print: arXiv:1508.00868v2 [math.RT]