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Correct derivation of R1 ～ R5 in Tong's QHE page 130

物理

Last updated at 2023-07-14Posted at 2023-07-13

page 130 of http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf

$\Psi_1^\dagger =\frac{1}{2}\left(\gamma_1 - i\gamma_{2}\right) $, 　$\Psi_1 =\frac{1}{2}\left(\gamma_1 + i\gamma_{2}\right) $

$\Psi_1^\dagger \Psi_1 =\frac{1}{4}\left(\gamma_1 -i \gamma_{2} \right) \left( \gamma_1 + i\gamma_{2}\right) = \frac{1}{4}\left(\gamma_1^2 +i\gamma_1\gamma_2 -i\gamma_2\gamma_1 +\gamma_2^2\right)$
$=\frac{1}{4}\left(2-2i\gamma_2\gamma_1 \right ) = \frac{1}{2}\left(1-i\gamma_2\gamma_1 \right )$

$\frac{1}{\sqrt{2}} \left ( 1 + i -2i \Psi_1^\dagger \Psi_1 \right ) = \frac{1}{\sqrt{2}} \left ( 1 + i - i \left(1-i\gamma_2\gamma_1 \right )\right )$
$=\frac{1}{\sqrt{2}} \left ( 1 - \gamma_2\gamma_1 \right )$

This won't agree the above equation. So we have to have,

$\frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_1^\dagger \Psi_1 \right ) = \frac{1}{\sqrt{2}} \left ( 1 - i + i \left(1-i\gamma_2\gamma_1 \right )\right )$
$=\frac{1}{\sqrt{2}} \left ( 1 + \gamma_2\gamma_1 \right )$

So the correct expression should be,

$R_1 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_1^\dagger \Psi_1 \right )$
$R_3 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_2^\dagger \Psi_2 \right )$
$R_5 = \frac{1}{\sqrt{2}} \left ( 1 -i +2i \Psi_3^\dagger \Psi_3 \right )$

Now, we check whether the following is correct.

$\Psi_1^\dagger =\frac{1}{2}\left(\gamma_1 - i\gamma_{2}\right) $, 　$\Psi_1 =\frac{1}{2}\left(\gamma_1 + i\gamma_{2}\right) $

$\Psi_2 + \Psi_2^\dagger = \gamma_3$, 　$\Psi_1 - \Psi_1^\dagger = i\gamma_2$

So, this seems OK. Then,

$R_2 = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)$

$R_2|0\rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right) |0\rangle$
$=\frac{1}{\sqrt{2}} \left ( |0\rangle -i \left (\Psi_2\Psi_1 |0\rangle - \Psi_2\Psi_1^\dagger |0\rangle +\Psi_2^\dagger\Psi_1 |0\rangle - \Psi_2^\dagger\Psi_1^\dagger |0\rangle \right ) \right) $
$=\frac{1}{\sqrt{2}} \left ( |0\rangle -i \left ( - \Psi_2 |1\rangle - \Psi_2^\dagger|1\rangle \right ) \right) = \frac{1}{\sqrt{2}} \left ( |0\rangle +i \Psi_2^\dagger|1\rangle \right)$
$ = \frac{1}{\sqrt{2}} \left ( |0\rangle+i |3\rangle \right)$

$R_2|1 \rangle = R_2 \Psi_1^\dagger |0 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_1^\dagger |0 \rangle$
$= \frac{1}{\sqrt{2}} \left ( |1\rangle - i \Psi_2^\dagger |0\rangle \right ) = \frac{1}{\sqrt{2}} \left ( |1\rangle - i|2\rangle \right ) $

$R_2|2 \rangle = R_2 \Psi_2^\dagger |0 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_2^\dagger |0 \rangle$
$= \frac{1}{\sqrt{2}} \left ( |2\rangle - i \left ( -\Psi_2\Psi_1^\dagger\Psi_2^\dagger |0\rangle \right ) \right ) = \frac{1}{\sqrt{2}} \left ( |2\rangle + i \Psi_2\Psi_1^\dagger\Psi_2^\dagger |0\rangle \right )$
$= \frac{1}{\sqrt{2}} \left ( |2\rangle - i \Psi_2\Psi_2^\dagger\Psi_1^\dagger |0\rangle \right )= \frac{1}{\sqrt{2}} \left ( |2\rangle - i |1\rangle \right ) = \frac{1}{\sqrt{2}} \left ( - i |1\rangle + |2\rangle \right )$

$R_2|3 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_2^\dagger \Psi_1^\dagger |0 \rangle$
$ = \frac{1}{\sqrt{2}} \left ( |3\rangle -i \left (\Psi_2\Psi_1\Psi_2^\dagger \Psi_1^\dagger |0 \rangle\right ) \right) =
\frac{1}{\sqrt{2}} \left ( |3\rangle + i |0 \rangle \right) = \frac{1}{\sqrt{2}} \left ( i |0 \rangle + |3\rangle \right)$

$R_2|4 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_3^\dagger |0 \rangle$
$ = \frac{1}{\sqrt{2}} \left ( |4\rangle +i |7\rangle \right )$

$R_2|5 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_3^\dagger \Psi_1^\dagger |0 \rangle$
$= \frac{1}{\sqrt{2}} \left ( |5\rangle -i \Psi_2^\dagger\Psi_1\Psi_3^\dagger \Psi_1^\dagger |0 \rangle \right )= \frac{1}{\sqrt{2}} \left ( |5\rangle +i \Psi_2^\dagger\Psi_3^\dagger|0 \rangle \right )$
$= \frac{1}{\sqrt{2}} \left ( |5\rangle - i \Psi_3^\dagger\Psi_2^\dagger|0 \rangle \right )= \frac{1}{\sqrt{2}} \left ( |5\rangle - i |6 \rangle \right )$

$R_2|6 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_3^\dagger \Psi_2^\dagger |0 \rangle$
$= \frac{1}{\sqrt{2}} \left ( |6\rangle +i \Psi_2 \Psi_1^\dagger \Psi_3^\dagger \Psi_2^\dagger |0 \rangle \right )= \frac{1}{\sqrt{2}} \left ( |6\rangle +i \Psi_1^\dagger\Psi_3^\dagger|0 \rangle \right )$
$= \frac{1}{\sqrt{2}} \left ( |6\rangle - i \Psi_3^\dagger\Psi_1^\dagger|0 \rangle \right )= \frac{1}{\sqrt{2}} \left ( |6 \rangle - i |5 \rangle \right )= \frac{1}{\sqrt{2}} \left ( - i |5 \rangle + |6 \rangle \right )$

$R_2|7 \rangle = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_2\Psi_1 - \Psi_2\Psi_1^\dagger +\Psi_2^\dagger\Psi_1 - \Psi_2^\dagger\Psi_1^\dagger\right ) \right)\Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger|0 \rangle$
$= \frac{1}{\sqrt{2}} \left ( |7\rangle - i \Psi_2 \Psi_1 \Psi_3^\dagger \Psi_2^\dagger \Psi_1^\dagger|0 \rangle \right )= \frac{1}{\sqrt{2}} \left ( |7\rangle + i \Psi_3^\dagger |0 \rangle \right ) = \frac{1}{\sqrt{2}} \left ( |7\rangle + i |4 \rangle \right ) $
$= \frac{1}{\sqrt{2}} \left ( i |4 \rangle + |7\rangle \right ) $

Finally,

$R_2 = \frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & i & 0 & 0 & 0 &0 \\
0 & 1 & -i & 0 & 0 & 0 & 0 &0 \\
0 & -i & 1 & 0 & 0 & 0 & 0 &0 \\
i & 0 & 0 & 1 & 0 & 0 & 0 &0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 &i \\
0 & 0 & 0 & 0 & 0 & 1 & -i &0 \\
0 & 0 & 0 & 0 & 0 & -i & 1 & 0 \\
0 & 0 & 0 & 0 & i & 0 & 0 &1\end{array} \right)$

$\Psi_1^\dagger =\frac{1}{2}\left(\gamma_1 - i\gamma_{2}\right) $, 　$\Psi_1 =\frac{1}{2}\left(\gamma_1 + i\gamma_{2}\right) $
$\Psi_2^\dagger =\frac{1}{2}\left(\gamma_3 - i\gamma_{4}\right) $, 　$\Psi_2 =\frac{1}{2}\left(\gamma_3 + i\gamma_{4}\right) $
$\Psi_3^\dagger =\frac{1}{2}\left(\gamma_5 - i\gamma_{6}\right) $, 　$\Psi_3 =\frac{1}{2}\left(\gamma_5 + i\gamma_{6}\right) $

$\Psi_3 + \Psi_3^\dagger = \gamma_5$, 　$\Psi_2 - \Psi_2^\dagger = i\gamma_4$

$R_4 = \frac{1}{\sqrt{2}} \left ( 1-i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right ) \right)$

$R_4 | 0 \rangle = \frac{1}{\sqrt{2}} \left ( |0\rangle + i \Psi_3^\dagger\Psi_2^\dagger | 0 \rangle \right)= \frac{1}{\sqrt{2}} \left ( |0\rangle + i | 6 \rangle \right)$

$R_4 |1 \rangle = \frac{1}{\sqrt{2}} \left ( |1\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right ) \Psi_1^\dagger |0\rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |1\rangle +i|7\rangle \right)$

$R_4 | 2 \rangle = \frac{1}{\sqrt{2}} \left ( |2\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right ) \Psi_2^\dagger |0\rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |2\rangle -i|4\rangle \right)$

$R_4 | 3 \rangle = \frac{1}{\sqrt{2}} \left ( |3\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right )\Psi_2^\dagger \Psi_1^\dagger |0\rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |3\rangle -i|5\rangle \right)$

$R_4 | 4 \rangle = \frac{1}{\sqrt{2}} \left ( |4\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right )\Psi_3^\dagger | 0 \rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |4\rangle -i|2\rangle \right)=\frac{1}{\sqrt{2}} \left ( -i|2\rangle + |4\rangle \right) $

$R_4 |5 \rangle = \frac{1}{\sqrt{2}} \left ( |5\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right )\Psi_3^\dagger \Psi_1^\dagger |0\rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |5\rangle -i|3\rangle \right)=\frac{1}{\sqrt{2}} \left ( -i|3\rangle + |5\rangle \right) $

$R_4 | 6 \rangle = \frac{1}{\sqrt{2}} \left ( |6\rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right ) \Psi_3^\dagger \Psi_2^\dagger | 0 \rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |6\rangle +i|0\rangle \right)=\frac{1}{\sqrt{2}} \left ( i|0\rangle + |6\rangle \right) $

$R_4 | 7 \rangle = \frac{1}{\sqrt{2}} \left ( | 7 \rangle -i \left (\Psi_3\Psi_2 - \Psi_3\Psi_2^\dagger +\Psi_3^\dagger\Psi_2 - \Psi_3^\dagger\Psi_2^\dagger\right ) \Psi_3^\dagger\Psi_2^\dagger\Psi_1^\dagger | 0 \rangle \right)$
$=\frac{1}{\sqrt{2}} \left ( |7\rangle +i|1\rangle \right)=\frac{1}{\sqrt{2}} \left ( i|1\rangle + |7\rangle \right) $

Finally,

$R_4 = \frac{1}{\sqrt{2}}\left( \begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & i &0 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 &i \\
0 & 0 & 1 & 0 & -i & 0 & 0 &0 \\
0 & 0 & 0 & 1 & 0 & -i & 0 &0 \\
0 & 0 & -i & 0 & 1 & 0 & 0 &0 \\
0 & 0 & 0 & -i & 0 & 1 & 0 &0 \\
i & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & i & 0 & 0 & 0 & 0 & 0 &1\end{array} \right)$

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