- #1
spaghetti3451
- 1,344
- 33
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?
$$T_{+}=|\uparrow\rangle\langle\downarrow|, \qquad T_{-}=|\downarrow\rangle\langle\uparrow|, \qquad T_{3}=\frac{1}{2}(|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|),$$
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?