パウリ行列によるガンマ行列のディラック表示
パウリ行列によるガンマ行列のディラック表示
パウリ行列\(\sigma_{1},\sigma_{2},\sigma_{3}\)によるガンマ行列のディラック表示は次で定められる。
\begin{align*} \gamma^{0} & =\sigma_{3}\otimes I_{2}\\ & =\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right)\otimes I_{2}\\ & =\left(\begin{array}{cc} I_{2} & 0\\ 0 & -I_{2} \end{array}\right) \end{align*} \(j\in\left\{ 1,2,3\right\} \)として、
\begin{align*} \gamma^{j} & =i\sigma_{2}\otimes\sigma_{j}\\ & =i\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right)\otimes\sigma_{j}\\ & =\left(\begin{array}{cc} 0 & \sigma_{j}\\ -\sigma_{j} & 0 \end{array}\right) \end{align*} とおくと、\(\mu\in\left\{ 0,1,2,3\right\} \)として反交換関係
\begin{align*} \left\{ \gamma^{\mu},\gamma^{\nu}\right\} & =\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}\\ & =2\eta^{\mu\nu}I \end{align*} が成り立つのでガンマ行列となる。
ここで\(\eta^{\mu\nu}\)はミンコフスキー計量\(\eta^{\mu\nu}=\diag\left(1,-1,-1,-1\right)\)であり、\(A\otimes B\)はクロネッカー積である。
パウリ行列\(\sigma_{1},\sigma_{2},\sigma_{3}\)によるガンマ行列のディラック表示は次で定められる。
\begin{align*} \gamma^{0} & =\sigma_{3}\otimes I_{2}\\ & =\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right)\otimes I_{2}\\ & =\left(\begin{array}{cc} I_{2} & 0\\ 0 & -I_{2} \end{array}\right) \end{align*} \(j\in\left\{ 1,2,3\right\} \)として、
\begin{align*} \gamma^{j} & =i\sigma_{2}\otimes\sigma_{j}\\ & =i\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right)\otimes\sigma_{j}\\ & =\left(\begin{array}{cc} 0 & \sigma_{j}\\ -\sigma_{j} & 0 \end{array}\right) \end{align*} とおくと、\(\mu\in\left\{ 0,1,2,3\right\} \)として反交換関係
\begin{align*} \left\{ \gamma^{\mu},\gamma^{\nu}\right\} & =\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}\\ & =2\eta^{\mu\nu}I \end{align*} が成り立つのでガンマ行列となる。
ここで\(\eta^{\mu\nu}\)はミンコフスキー計量\(\eta^{\mu\nu}=\diag\left(1,-1,-1,-1\right)\)であり、\(A\otimes B\)はクロネッカー積である。
パウリ行列は
\[ \sigma_{1}=\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right) \] \[ \sigma_{2}=\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right) \] \[ \sigma_{3}=\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right) \] \[ \sigma_{0}=I=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \] である。
\[ \sigma_{1}=\left(\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right) \] \[ \sigma_{2}=\left(\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right) \] \[ \sigma_{3}=\left(\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right) \] \[ \sigma_{0}=I=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \] である。
\(j,k\in\left\{ 1,2,3\right\} \)とする。
\begin{align*} \left\{ \gamma^{j},\gamma^{k}\right\} & =\gamma^{j}\gamma^{k}+\gamma^{k}\gamma^{j}\\ & =\left(i\sigma_{2}\otimes\sigma_{j}\right)\left(i\sigma_{2}\otimes\sigma_{k}\right)+\left(i\sigma_{2}\otimes\sigma_{k}\right)\left(i\sigma_{2}\otimes\sigma_{j}\right)\\ & =-\left(\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{2}\otimes\sigma_{k}\right)-\left(\sigma_{2}\otimes\sigma_{k}\right)\left(\sigma_{2}\otimes\sigma_{j}\right)\\ & =-\left(\sigma_{2}^{2}\otimes\sigma_{j}\sigma_{k}\right)-\left(\sigma_{2}^{2}\otimes\sigma_{k}\sigma_{j}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =-\left(I\otimes\sigma_{j}\sigma_{k}\right)-\left(I\otimes\sigma_{k}\sigma_{j}\right)\\ & =-\left(I\otimes\left(\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}\right)\right)\cmt{\because\left(A+B\right)\otimes C=A\otimes C+B\otimes C}\\ & =-\left(I\otimes\left\{ \sigma_{j},\sigma_{k}\right\} \right)\\ & =-\left(I_{2}\otimes2I_{2}\delta_{jk}\right)\\ & =-2\delta_{jk}\left(I_{2}\otimes I_{2}\right)\\ & =2\eta^{jk}I \end{align*} \begin{align*} \left\{ \gamma^{0},\gamma^{j}\right\} & =\gamma^{0}\gamma^{j}+\gamma^{j}\gamma^{0}\\ & =\left(\sigma_{3}\otimes I_{2}\right)\left(i\sigma_{2}\otimes\sigma_{j}\right)+\left(i\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{3}\otimes I_{2}\right)\\ & =i\left(\sigma_{3}\otimes I_{2}\right)\left(\sigma_{2}\otimes\sigma_{j}\right)+i\left(\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{3}\otimes I_{2}\right)\\ & =i\left(\sigma_{3}\sigma_{2}\otimes I_{2}\sigma_{j}\right)+i\left(\sigma_{2}\sigma_{3}\otimes\sigma_{j}I_{2}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =i\left(-i\sigma_{1}\otimes\sigma_{j}\right)+i\left(i\sigma_{1}\otimes\sigma_{j}\right)\\ & =\left(\sigma_{1}\otimes\sigma_{j}\right)-\left(\sigma_{1}\otimes\sigma_{j}\right)\\ & =O_{4}\\ & =2\eta^{0j}I \end{align*} \begin{align*} \left\{ \gamma^{j},\gamma^{0}\right\} & =\left\{ \gamma^{0},\gamma^{j}\right\} \\ & =O_{4}\\ & =2\eta^{j0}I \end{align*} \begin{align*} \left\{ \gamma^{0},\gamma^{0}\right\} & =2\left(\gamma^{0}\right)^{2}\\ & =2\left(\sigma_{3}\otimes I_{2}\right)^{2}\\ & =2\left(\sigma_{3}^{2}\otimes I_{2}^{2}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =2\left(I_{2}\otimes I_{2}\right)\\ & =2I_{4}\\ & =2\eta^{00}I_{4} \end{align*} これらより、
\begin{align*} \left\{ \gamma^{\mu},\gamma^{\nu}\right\} & =\begin{cases} 2\eta^{\mu\nu}I & \mu,\nu\in\left\{ 1,2,3\right\} \\ 2\eta^{0j}I & \mu=0\land\nu\in\left\{ 1,2,3\right\} \\ 2\eta^{j0}I & \nu\in\left\{ 1,2,3\right\} \land\mu=0\\ 2\eta^{00}I & \mu=0\land\nu=0 \end{cases}\\ & =2\eta^{\mu\nu}I \end{align*} となるので、\(\gamma^{\mu},\mu\in\left\{ 0,12,3\right\} \)はガンマ行列となる。
\begin{align*} \left\{ \gamma^{j},\gamma^{k}\right\} & =\gamma^{j}\gamma^{k}+\gamma^{k}\gamma^{j}\\ & =\left(i\sigma_{2}\otimes\sigma_{j}\right)\left(i\sigma_{2}\otimes\sigma_{k}\right)+\left(i\sigma_{2}\otimes\sigma_{k}\right)\left(i\sigma_{2}\otimes\sigma_{j}\right)\\ & =-\left(\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{2}\otimes\sigma_{k}\right)-\left(\sigma_{2}\otimes\sigma_{k}\right)\left(\sigma_{2}\otimes\sigma_{j}\right)\\ & =-\left(\sigma_{2}^{2}\otimes\sigma_{j}\sigma_{k}\right)-\left(\sigma_{2}^{2}\otimes\sigma_{k}\sigma_{j}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =-\left(I\otimes\sigma_{j}\sigma_{k}\right)-\left(I\otimes\sigma_{k}\sigma_{j}\right)\\ & =-\left(I\otimes\left(\sigma_{j}\sigma_{k}+\sigma_{k}\sigma_{j}\right)\right)\cmt{\because\left(A+B\right)\otimes C=A\otimes C+B\otimes C}\\ & =-\left(I\otimes\left\{ \sigma_{j},\sigma_{k}\right\} \right)\\ & =-\left(I_{2}\otimes2I_{2}\delta_{jk}\right)\\ & =-2\delta_{jk}\left(I_{2}\otimes I_{2}\right)\\ & =2\eta^{jk}I \end{align*} \begin{align*} \left\{ \gamma^{0},\gamma^{j}\right\} & =\gamma^{0}\gamma^{j}+\gamma^{j}\gamma^{0}\\ & =\left(\sigma_{3}\otimes I_{2}\right)\left(i\sigma_{2}\otimes\sigma_{j}\right)+\left(i\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{3}\otimes I_{2}\right)\\ & =i\left(\sigma_{3}\otimes I_{2}\right)\left(\sigma_{2}\otimes\sigma_{j}\right)+i\left(\sigma_{2}\otimes\sigma_{j}\right)\left(\sigma_{3}\otimes I_{2}\right)\\ & =i\left(\sigma_{3}\sigma_{2}\otimes I_{2}\sigma_{j}\right)+i\left(\sigma_{2}\sigma_{3}\otimes\sigma_{j}I_{2}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =i\left(-i\sigma_{1}\otimes\sigma_{j}\right)+i\left(i\sigma_{1}\otimes\sigma_{j}\right)\\ & =\left(\sigma_{1}\otimes\sigma_{j}\right)-\left(\sigma_{1}\otimes\sigma_{j}\right)\\ & =O_{4}\\ & =2\eta^{0j}I \end{align*} \begin{align*} \left\{ \gamma^{j},\gamma^{0}\right\} & =\left\{ \gamma^{0},\gamma^{j}\right\} \\ & =O_{4}\\ & =2\eta^{j0}I \end{align*} \begin{align*} \left\{ \gamma^{0},\gamma^{0}\right\} & =2\left(\gamma^{0}\right)^{2}\\ & =2\left(\sigma_{3}\otimes I_{2}\right)^{2}\\ & =2\left(\sigma_{3}^{2}\otimes I_{2}^{2}\right)\cmt{\because\left(A\otimes B\right)\left(C\otimes D\right)=\left(AC\right)\otimes\left(BD\right)}\\ & =2\left(I_{2}\otimes I_{2}\right)\\ & =2I_{4}\\ & =2\eta^{00}I_{4} \end{align*} これらより、
\begin{align*} \left\{ \gamma^{\mu},\gamma^{\nu}\right\} & =\begin{cases} 2\eta^{\mu\nu}I & \mu,\nu\in\left\{ 1,2,3\right\} \\ 2\eta^{0j}I & \mu=0\land\nu\in\left\{ 1,2,3\right\} \\ 2\eta^{j0}I & \nu\in\left\{ 1,2,3\right\} \land\mu=0\\ 2\eta^{00}I & \mu=0\land\nu=0 \end{cases}\\ & =2\eta^{\mu\nu}I \end{align*} となるので、\(\gamma^{\mu},\mu\in\left\{ 0,12,3\right\} \)はガンマ行列となる。
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パウリ行列の定義と性質
\[
\sigma_{j}\sigma_{k}=I\delta_{jk}+i\sum_{l=1}^{3}\epsilon_{jkl}\sigma_{l}
\]
エルミート形式・2次形式
\[
f\left(\boldsymbol{x}\right)=\boldsymbol{x}^{*}A\boldsymbol{x}
\]
恒等的に成り立つ行列
\[
A=B\Leftrightarrow\forall\boldsymbol{x}\in K^{n},A\boldsymbol{x}=B\boldsymbol{x}
\]
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\[
P=A\left(A^{*}A\right)^{-1}A^{*}
\]