We know that two SU(2) fundamentals have multiplication decompositions, such that $$ 2 \otimes 2= 1 \oplus 3.$$
In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial representation of SU(2).
I hope to see the precise SO(3) rotation from the two SU(2) fundamental rotations.

So let us first write two SU(2) fundamental objects in terms of an SO(3) object. In particular, we can consider the following three:
$$
1,1\rangle= \begin{pmatrix}
1\\
0
\end{pmatrix}\begin{pmatrix}
1\\
0
\end{pmatrix}=  \uparrow \uparrow \rangle,$$
$$1,0\rangle
={1 \over \sqrt{2} }
(\begin{pmatrix}
1\\
0
\end{pmatrix}
\begin{pmatrix}
0\\
1
\end{pmatrix}
+
\begin{pmatrix}
0\\
1
\end{pmatrix}
\begin{pmatrix}
1\\
0
\end{pmatrix})={1 \over \sqrt{2} }( \uparrow \downarrow \rangle+  \downarrow \uparrow \rangle)
,$$
$$1,1\rangle = \begin{pmatrix}
0\\
1
\end{pmatrix}\begin{pmatrix}
0\\
1
\end{pmatrix}=  \downarrow \downarrow \rangle.
$$
where the $ \uparrow \rangle$ and $ \downarrow \rangle$ are in SU(2) fundamentals. And we shothand
$ \uparrow \uparrow \rangle \equiv  \uparrow \rangle \uparrow \rangle $ and so on.
question: How do we rotate between $1,1\rangle$, $1,0\rangle$, $1,1\rangle$, via two SU(2) rotations acting on two SU(2) fundamentals? Namely, that is, construct an SO(3) rotation inside the two SU(2) fundamental rotations?
The SU(2) has three generators, parametrized by $m_x,m_y,m_z$; how do we write down the generic SO(3) rotations from two SU(2) rotations?
Let us consider an example, an SU(2) rotation $U$ acting on the SU(2) fundamental $\begin{pmatrix}
1\\
0
\end{pmatrix}$ give rise to
$$
U \begin{pmatrix}
1\\
0
\end{pmatrix}=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2}) & (i m_x m_y) \sin(\frac{\theta}{2}) \\
(i m_x +m_y) \sin(\frac{\theta}{2}) & \cos(\frac{\theta}{2}){i m_z} \sin(\frac{\theta}{2}) \\
\end{pmatrix}
\begin{pmatrix}
1\\
0
\end{pmatrix}=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
(i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}
\equiv\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})
\begin{pmatrix}
1\\
0
\end{pmatrix}
+
(i m_x +m_y) \sin(\frac{\theta}{2})
\begin{pmatrix}
0\\
1
\end{pmatrix}
$$
In other words, the SU(2) rotation $U$ (with the $\vec{m}^2=1$) rotates SU(2) fundamentals. Two SU(2) rotations act as
$$
UU1,1\rangle =
U \begin{pmatrix}
1\\
0
\end{pmatrix}U \begin{pmatrix}
1\\
0
\end{pmatrix}
=
\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
(i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}\begin{pmatrix}
\cos(\frac{\theta}{2})+{i m_z} \sin(\frac{\theta}{2})\\
(i m_x +m_y) \sin(\frac{\theta}{2})
\end{pmatrix}
$$
Hint: Naively, we like to construct
$$
L_{\pm} =L_{x} \pm i L_y,
$$ such that $L_{\pm}$ is an operator out of two SU(2) rotations acting on two SU(2) fundamentals, such that it raises/lowers between $1,1\rangle$, $1,0\rangle$, $1,1\rangle$.
question 2: Is it plausible that two SU(2) are impossible to perform such SO(3) rotations, but we need two U(2) rotations?
This post imported from StackExchange Physics at 20201106 18:48 (UTC), posted by SEuser annie marie heart