γ1～γ6　の並べ替え　(Tong's QHE ~p.130)　 Rearrangement of γ1 to γ6 (Tong's QHE ~p.130)

様々な演算子は、$\gamma^1\sim\gamma^6$で構成されている。一方、ヒルベルトスペースの基底は $|0\rangle \sim |6\rangle$ としている。つまり、任意の状態は、
$|\Psi\rangle \equiv \left(\begin{array}{c} a_0 \\\ a_1 \\\ a_2 \\\ a_3 \\\ a_4 \\\ a_5 \\\ a_6 \\\ a_7\end{array}\right) \begin{array}{c} |0\rangle \\\ |1\rangle \\\ |2\rangle \\\ |3\rangle \\\ |4\rangle \\\ |5\rangle \\\ |6\rangle  \\\ |7\rangle\end{array}$
と書けるが、$|0\rangle \sim |6\rangle$ の各状態が与える $\gamma^7=\sigma^3 \otimes \sigma^3 \otimes \sigma^3 $ の固有値は、$\gamma^7 |0> = 1 |0>$, 　$\gamma^7 |3> = 1 |3>$, 　$\gamma^7 |5> = 1 |5>$, 　$\gamma^7 |6> = 1 |6>$,
$\gamma^7 |1> = -1 |1>$, 　$\gamma^7 |2> = -1 |2>$, 　$\gamma^7 |4> = -1 |4>$, 　$\gamma^7 |7> = -1 |7>$, であるので、次の変換行列$U$を考える。
$U=\left(\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right) \begin{array}{c} |0\rangle \\\ |1\rangle \\\ |2\rangle \\\ |3\rangle \\\ |4\rangle \\\ |5\rangle \\\ |6\rangle  \\\ |7\rangle\end{array}$
$U^\dagger=\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right) \begin{array}{c} |0\rangle \\\ |1\rangle \\\ |2\rangle \\\ |3\rangle \\\ |4\rangle \\\ |5\rangle \\\ |6\rangle  \\\ |7\rangle\end{array}$
$U^\dagger U = \left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)=I$
$U U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
=I$

なので、$U$はユニタリ変換行列である。$U\gamma^iU^\dagger$を考える。

$U\gamma^1U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}
0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 
1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 
0 & 0 & 0 & -1 & 0 & 0 & 0 &0 \\\ 
0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 
0 & 0 & 0 & 0 & 0 & -1 & 0 &0 \\\ 
0 & 0 & 0 & 0 & -1 & 0 & 0 &0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 &1 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 & 1 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 & -1 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & -1 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &1 \\\ 
1 & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 & -1 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 & -1 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 1 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 & 1 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 & -1 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 & -1 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & 1  \\\ 
1 &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 & -1 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 & -1 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 1 & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^2U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}
0 & -i & 0 & 0 & 0 & 0 & 0 &0 \\\ 
i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 
0 & 0 & 0 & i & 0 & 0 & 0 &0 \\\ 
0 & 0 & -i & 0 & 0 & 0 & 0 &0 \\\ 
0 & 0 & 0 & 0 & 0 & i & 0 &0 \\\ 
0 & 0 & 0 & 0 & -i & 0 & 0 &0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\\ 
0 & 0 & 0 & 0 & 0 & 0 & i &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &-i &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 & -i &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & -i &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &-i \\\ 
i & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  i &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  i & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & i &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 &-i &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 & -i &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 & -i & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & -i  \\\ 
i &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  i &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  i & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & i & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^3U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}0 & 0 & 1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 & 0 &  1 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 1 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &-1 \\\ 
0 & 0 &  0 &  0 & -1 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  1 &  0 &  0 & 0 &0 \\\ 
1 & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 &-1 &0 \\\ 
0 & 0 &  0 &  0 &  0 & -1 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 & 0 &  1 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 1 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & -1  \\\ 
0 &  0 &  0 & 0 & 0 &  0 & -1 & 0  \\\ 
0 &  1 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
1 &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & -1 & 0 &  0 &  0 & 0 \\\ 
0 &  0 & -1 & 0 & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^4U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}0 & 0 & -i & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & -i & 0 & 0 & 0 &0 \\\ i & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & i & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & i & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & i \\\ 0 & 0 & 0 & 0 & -i & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & -i & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 & 0 & -i &  0 &  0 &  0 & 0 &0 \\\ 
0 & i &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &i \\\ 
0 & 0 &  0 &  0 & -i &  0 & 0 &0 \\\ 
0 & 0 &  0 & -i &  0 &  0 & 0 &0 \\\ 
i & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & i &0 \\\ 
0 & 0 &  0 &  0 &  0 & -i & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 & 0 & -i &  0 & 0 \\\ 
0 &  0 &  0 & 0 & i &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & i  \\\ 
0 &  0 &  0 & 0 & 0 &  0 & -i & 0  \\\ 
0 & -i &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
i &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & i & 0 &  0 &  0 & 0 \\\ 
0 &  0 & -i & 0 & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^5U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & 1 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 & 0 &  0 &  0 &  1 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &1 \\\ 
0 & 1 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  1 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  1 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 1 &0 \\\ 
1 & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  1 &  0 &  0 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 & 0 &  0 &  1 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & 1  \\\ 
0 &  0 &  0 & 0 & 1 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  1 &  0 & 0  \\\ 
0 &  0 &  1 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 & 1 & 0 &  0 &  0 & 0 \\\ 
1 &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  1 &  0 & 0 & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^6U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc}0 & 0 & 0 & 0 & -i & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & 0 & -i & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & -i & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -i \\\ i & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & i & 0 & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & i & 0 & 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & i & 0 & 0 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 & 0 &  0 &  0 & -i &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &-i \\\ 
0 & i &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  i &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 & -i & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 &-i &0 \\\ 
i & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  i &  0 &  0 & 0 &0\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
0 &  0 &  0 & 0 & 0 &  0 & -i & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 &-i  \\\ 
0 &  0 &  0 & 0 & i &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & 0 &  i &  0 & 0  \\\ 
0 &  0 & -i & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  0 &  0 &-i & 0 &  0 &  0 & 0 \\\ 
i &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  i &  0 & 0 & 0 &  0 &  0 & 0\end{array}\right)$

$U\gamma^7U^\dagger = \left(\begin{array}{cccccccc} 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)
\left(\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & -1 & 0 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 &0 \\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\end{array}\right)
\left(\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
1 & 0 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  1 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  1 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 1 &0 \\\ 
0 &-1 &  0 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 & -1 &  0 &  0 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 & -1 &  0 & 0 &0 \\\ 
0 & 0 &  0 &  0 &  0 &  0 & 0 &-1\end{array}\right)
\left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0  \\\ 
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\ 
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\end{array}\right)$
$=\left(\begin{array}{cccccccc}
1 &  0 &  0 & 0 & 0 &  0 &  0 & 0 \\\ 
0 &  1 &  0 & 0 & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  1 & 0 & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 1 & 0 &  0 &  0 & 0  \\\ 
0 &  0 &  0 & 0 & -1&  0 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 & -1 &  0 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 & -1 & 0 \\\ 
0 &  0 &  0 & 0 & 0 &  0 &  0 & -1\end{array}\right)$