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The generators for the isospin symmetry are given by

$$ T_{+}=\left|\uparrow\rangle\langle\downarrow\right|, \qquad T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, \qquad T_{3}=\frac{1}{2}\left(\left|\uparrow\rangle\langle\uparrow\right|-\left|\downarrow\rangle\langle\downarrow\right|\right),$$

where $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1/2 & 0\\ 0 & -1/2\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do get the correct commutation relations using the matrix representations of the generators?
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?

The generators for the isospin symmetry are given by

$$ T_{+}=\left|\uparrow\rangle\langle\downarrow\right|, \qquad T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, \qquad T_{3}=\frac{1}{2}\left(\left|\uparrow\rangle\langle\uparrow\right|-\left|\downarrow\rangle\langle\downarrow\right|\right),$$

where $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1/2 & 0\\ 0 & -1/2\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do not get the correct commutation relations using the matrix representations of the generators.
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?

The generators for the isospin symmetry are given by

$$T_{+}=|\uparrow\rangle\langle\downarrow|, \qquad T_{-}=|\downarrow\rangle\langle\uparrow|, \qquad T_{3}=\frac{1}{2}(|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|),$$

where $|\uparrow\rangle$ and $|\downarrow\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $|\uparrow\rangle$ and $|\downarrow\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1/2 & 0\\ 0 & -1/2\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do get the correct commutation relations using the matrix representations of the generators?
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?

The generators for the isospin symmetry are given by

$$ T_{+}=\left|\uparrow\rangle\langle\downarrow\right|, \qquad T_{-}=\left|\downarrow\rangle\langle\uparrow\right|, \qquad T_{3}=\frac{1}{2}\left(\left|\uparrow\rangle\langle\uparrow\right|-\left|\downarrow\rangle\langle\downarrow\right|\right),$$

where $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1/2 & 0\\ 0 & -1/2\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do get the correct commutation relations using the matrix representations of the generators?
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?

The generators for the isospin symmetry are given by

$$T_{+}=|\uparrow\rangle\langle\downarrow|, \qquad T_{-}=|\downarrow\rangle\langle\uparrow|, \qquad T_{3}=\frac{1}{2}(|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|),$$

where $|\uparrow\rangle$ and $|\downarrow\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $|\uparrow\rangle$ and $|\downarrow\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do get the correct commutation relations using the matrix representations of the generators?
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?

The generators for the isospin symmetry are given by

$$T_{+}=|\uparrow\rangle\langle\downarrow|, \qquad T_{-}=|\downarrow\rangle\langle\uparrow|, \qquad T_{3}=\frac{1}{2}(|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|),$$

where $|\uparrow\rangle$ and $|\downarrow\rangle$ form a $2$-dimensional basis of states.

In the $2$-dimensional basis of states $|\uparrow\rangle$ and $|\downarrow\rangle$, the generators $T_+$, $T_-$ and $T_3$ can be written as

$T_{+}=\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix},\qquad T_{+}=\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix},\qquad T_{3}=\begin{bmatrix} 1/2 & 0\\ 0 & -1/2\\ \end{bmatrix}.$

1. The generators $T_+$, $T_-$ and $T_3$ obey the $SU(2)$ algebra $[T_{i},T_{j}]=i\epsilon_{ijk}T_{k}$ (with $\epsilon_{+-3}=1$). However, I do get the correct commutation relations using the matrix representations of the generators?
2. Is it possible to rewrite the generators $T_+$, $T_-$ and $T_3$ in terms of an arbitrary number of states?