##SU(2, \mathbb C)## parametrization using Euler angles

martinbn said:

Off topic: Can you quote a textbook that uses the notation ##SU(2,\mathbb C)##?

cianfa72 said:

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.

pasmith said:

for example [itex](-\pi,\pi)[/itex] which covers everything except -1 and [itex](0, 2\pi)[/itex] which covers everything except 1.

martinbn said:

Off topic: Can you quote a textbook that uses the notation ##SU(2,\mathbb C)##?

fresh_42 said:

I try to use always the field. It is redundant in that case but a good habit in the general case, especially when groups with complex matrix entries are considered real manifolds. And it makes the mention of the characteristic unnecessary.

martinbn said:

But the point is the ##\mathbb C## is not the base field.

fresh_42 said:

Base field of what? ##SU(2)=SU(2,\mathbb{C})## has complex entries. If I consider it as a real manifold, I write ##SU_\mathbb{R}(2,\mathbb{C})## like I would indicate ##V_\mathbb{F}## as a vector space over ##\mathbb{F}## if there is a doubt about it. However, I try to avoid writing ##SU_\mathbb{R}(2,\mathbb{C}),## I prefer ##\mathfrak{su}_\mathbb{R}(2,\mathbb{C})## and let the group be what it is, complex. I generally do not like the fact that a) reals are most often automatically assumed to be the scalar field and b) that linear (in-)dependence almost never states the scalar field, which is sometimes a serious problem!

martinbn said:

It has real dimension three. It cannot be complex since three is not even.

fresh_42 said:

It is an algebraic group of matrices with complex entries like ##GL(n,\mathbb{C})## or ##SL(n,\mathbb{C})## are. I only use the same nomenclature that notes the field where the matrix entries are taken from.

Dimension kicks in if we consider them as Lie groups, i.e. consider the corresponding Lie algebra. And if we do so, it is even more important to distinguish the fields!

cianfa72 said:

You mean -1 and 1 as complex numbers (i.e. the pairs (-1,0) and (1,0) under the identification ##\mathbb C \cong \mathbb R^2##).

As far as I can understand what you said, If we define ##x =(\psi+\phi)/2## and ##y =(\psi - \phi)/2## then we obtain a two-chart atlas for ##SU(2)## starting from the parametrization in post#1. However to get that atlas we need a similar approach for ##\theta## that runs in the closed interval ##[0, \pi]##.

pasmith said:

You are only dealing with a quarter period in the [itex]\theta[/itex] dependence; the end points are distinct. You need only 4 charts whose domains are [itex](\theta,\psi,\phi) \in [0,\pi] \times A \times B[/itex] where [itex]A[/itex] and [itex]B[/itex] are independently either [itex](-\pi,\pi)[/itex] or [itex](0, 2\pi)[/itex].

cianfa72 said:

Take for instance the chart ##\varphi## with domain ##[0,\pi] \times (-\pi, \pi) \times (- \pi, \pi)##. Is this domain open in ##\mathbb R^3## ?

jbergman said:

No. You have a closed interval for the first part.