- #1

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- TL;DR Summary
- About the ##SU(2, \mathbb C)## parametrization using Euler angles.

Hi,

I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements

\begin{pmatrix}

e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\

ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}

\end{pmatrix}

where ##\theta, \psi## and ##\phi## are the three Euler angles. In particular ##\theta## runs in the closed interval ##[0, \pi]## whereas ##\psi## and ##\phi## in the range ##[0, 2\pi]## -- such a parametrization includes all and only ##SU(2 , \mathbb C)## group elements. That is actually a "closed box" in ##\mathbb R ^3## so to get a chart from it we need to exclude the box "boundary". This way we get a not global chart for ##SU(2)##. Since we know it is homeomorphic to ##\mathbb S^3## we can cover it with at least two charts.

Apart the above chart, which is one of the other charts for it ? Thanks.

I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements

\begin{pmatrix}

e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\

ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}

\end{pmatrix}

where ##\theta, \psi## and ##\phi## are the three Euler angles. In particular ##\theta## runs in the closed interval ##[0, \pi]## whereas ##\psi## and ##\phi## in the range ##[0, 2\pi]## -- such a parametrization includes all and only ##SU(2 , \mathbb C)## group elements. That is actually a "closed box" in ##\mathbb R ^3## so to get a chart from it we need to exclude the box "boundary". This way we get a not global chart for ##SU(2)##. Since we know it is homeomorphic to ##\mathbb S^3## we can cover it with at least two charts.

Apart the above chart, which is one of the other charts for it ? Thanks.

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