東京大学 大学院 理学系研究科
物理学専攻
2019年度 数学 第1問

1.

(i)

\begin{align} \left( i [A, B] \right)^\dagger &= \left( i \left( AB - BA \right) \right)^\dagger \\ &= -i \left( B^\dagger A^\dagger - A^\dagger B^\dagger \right) \\ &= -i \left( BA - AB \right) \\ &= i \left( AB - BA \right) \\ &= i [A, B] \\ \\ \text{tr} \left( i [A, B] \right) &= i \left( \text{tr} (AB) - \text{tr} (BA) \right) \\ &= i \left( \text{tr} (AB) - \text{tr} (AB) \right) \\ &= 0 \end{align}

(ii)

\begin{align} U^\dagger U &= \left( e^{iA} \right)^\dagger e^{iA} \\ &= e^{-i A^\dagger} e^{iA} \\ &= e^{-iA} e^{iA} \\ &= I \end{align}

(iii)

\begin{align} \det U = \det e^{iA} = e^{i \text{tr} A} = e^0 = 1 \end{align}

2.

\begin{align} T_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} , T_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} , T_7 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \end{align}

3.

\begin{align} [T_7, T_1] &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \\ &= \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ &= \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = i \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = i T_2 \\ \therefore A_{71,2} &= 1 \\ [T_5, T_6] &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} - \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \\ &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & i & 0 \\ 0 & 0 & -i \end{pmatrix} - \begin{pmatrix} 0 & 0 & 0 \\ 0 & -i & 0 \\ 0 & 0 & i \end{pmatrix} \\ &= \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2i & 0 \\ 0 & 0 & -2i \end{pmatrix} = 2i \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} = 2i T_8 \\ \therefore A_{56,8} &= 2 \end{align}

4.

(i)

\begin{align} G'_a &= x'^\dagger T_a y' = (Ux)^\dagger T_a (Uy) = x^\dagger U^\dagger T_a U y \\ &\simeq x^\dagger (I - i \theta T) T_a (I + i \theta T) y \\ &\simeq x^\dagger (T_a + i \theta T_a T - i \theta T T_a ) y \\ &= G_a + i \theta x^\dagger \left( T_a \sum_{c=1}^8 \alpha_c T_c - \sum_{c=1}^8 \alpha_c T_c T_a \right) y \\ &= G_a + i \theta x^\dagger \sum_{c=1}^8 \alpha_c [T_a, T_c] y \\ &= G_a + i \theta x^\dagger \sum_{c=1}^8 \alpha_c \left( i \sum_{b=1}^8 A_{ac,b} T_b \right) y \\ &= G_a + i \theta \sum_{b=1}^8 \left( \sum_{c=1}^8 i \alpha_c A_{ac,b} \right) G_b \\ \therefore H_{ab} &= \sum_{c=1}^8 i \alpha_c A_{ac,b} \end{align}