うべの時空代数

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時空代数と物理量[書き途中]

1次と1次の積

ab=\left\{a_0\gamma_0+\overrightarrow{a}\right\}\left\{b_0\gamma_0+\overrightarrow{b}\right\}\\=-a_0b_0+a_0\gamma_0\overrightarrow{b}-\gamma_0\overrightarrow{a}b_0+\overrightarrow{a}\overrightarrow{b}\\=-a_0b_0+a_1b_1+a_2b_2+a_3b_3\\+\left\{a_0b_1-a_1b_0\right\}\gamma_{01}\\+\left\{a_0b_2-a_2b_0\right\}\gamma_{02}\\+\left\{a_0b_3-a_3b_0\right\}\gamma_{03}\\+\left\{a_2b_3-a_3b_2\right\}\gamma_{23}\\+\left\{a_3b_1-a_1b_3\right\}\gamma_{31}\\+\left\{a_1b_2-a_2b_1\right\}\gamma_{12}

3次と3次の積

ab=\gamma_{0123}\tilde{a}\gamma_{0123}\tilde{b}=\tilde{a}\tilde{b}\\=-a_0b_0+a_1b_1+a_2b_2+a_3b_3\\+\left\{a_0b_1-a_1b_0\right\}\gamma_{01}\\+\left\{a_0b_2-a_2b_0\right\}\gamma_{02}\\+\left\{a_0b_3-a_3b_0\right\}\gamma_{03}\\+\left\{a_2b_3-a_3b_2\right\}\gamma_{23}\\+\left\{a_3b_1-a_1b_3\right\}\gamma_{31}\\+\left\{a_1b_2-a_2b_1\right\}\gamma_{12}

1次と3次の積

ab=a\gamma_{0123}\tilde{b}=-\gamma_{0123}a\tilde{b} \\=\left\{a_0b_0-a_1b_1-a_2b_2-a_3b_3\right\}\gamma_{0123}\\+\left\{a_2b_3-a_3b_2\right\}\gamma_{01}\\+\left\{a_3b_1-a_1b_3\right\}\gamma_{02}\\+\left\{a_1b_2-a_2b_1\right\}\gamma_{03}\\-\left\{a_0b_1-a_1b_0\right\}\gamma_{23}\\-\left\{a_0b_2-a_2b_0\right\}\gamma_{31}\\-\left\{a_0b_3-a_3b_0\right\}\gamma_{12}

3次と1次の積

ab=\gamma_{0123}\tilde{a}b\\=\left\{-a_0b_0+a_1b_1+a_2b_2+a_3b_3\right\}\gamma_{0123}\\-\left\{a_2b_3-a_3b_2\right\}\gamma_{01}\\-\left\{a_3b_1-a_1b_3\right\}\gamma_{02}\\-\left\{a_1b_2-a_2b_1\right\}\gamma_{03}\\+\left\{a_0b_1-a_1b_0\right\}\gamma_{23}\\+\left\{a_0b_2-a_2b_0\right\}\gamma_{31}\\+\left\{a_0b_3-a_3b_0\right\}\gamma_{12}

2次と2次の積

ab=\left\{\gamma_0\overrightarrow{\alpha}+\gamma_{123}\overrightarrow{a}\right\}\left\{\gamma_0 \overrightarrow{\beta}+\gamma_{123}\overrightarrow{b}\right\}\\=\overrightarrow{\alpha}\overrightarrow{\beta}+\gamma_{0123}\overrightarrow{\alpha}\overrightarrow{b}+\gamma_{0123}\overrightarrow{a}\overrightarrow{\beta}-\overrightarrow{a}\overrightarrow{b}\\=\alpha_{1}\beta_{1}+\alpha_{2}\beta_{2}+\alpha_{3}\beta_{3}\\-a_{1}b_{1}-a_{2}b_{2}-a_{3}b_{3}\\-\left\{\alpha_2b_3-\alpha_3b_2+a_2\beta_3-a_3\beta_2\right\}\gamma_{01}\\-\left\{\alpha_3b_1-\alpha_1b_3+a_3\beta_1-a_1\beta_3\right\}\gamma_{02}\\-\left\{\alpha_1b_2-\alpha_2b_1+a_1\beta_2-a_2\beta_1\right\}\gamma_{03}\\+\left\{\alpha_2\beta_3-\alpha_3\beta_2-a_2b_3+b_3a_2\right\}\gamma_{23}\\+\left\{\alpha_3\beta_1-\alpha_1\beta_3-a_3b_1+b_1a_3\right\}\gamma_{31}\\+\left\{\alpha_1\beta_2-\alpha_2\beta_1-a_1b_2+b_2a_1\right\}\gamma_{12}\\+\left\{\alpha_1b_1+\alpha_2b_2+\alpha_3b_3\\+a_1\beta_1+a_2\beta_2+a_3\beta_3\right\}\gamma_{0123}

1次と2次の積

ab=\left\{a_0\gamma_0+\overrightarrow{a}\right\}\left\{\gamma_0\overrightarrow{\beta}+\gamma_{123}\overrightarrow{b}\right\}\\=-a_0\overrightarrow{\beta}+\gamma_{0123}a_0\overrightarrow{b}-\gamma_0\overrightarrow{a}\overrightarrow{\beta}+\gamma_{123}\overrightarrow{a}\overrightarrow{b}\\=\left\{-a_1\beta_1-a_2\beta_2-a_3\beta_3\right\}\gamma_0\\+\left\{-a_0\beta_1-a_2b_3+a_3b_2\right\}\gamma_1\\+\left\{-a_0\beta_2-a_3b_1+a_1b_3\right\}\gamma_2\\+\left\{-a_0\beta_3-a_1b_2+a_2b_1\right\}\gamma_3\\+\left\{a_1b_1+a_2b_2+a_3b_3\right\}\gamma_{123}\\+\left\{a_0b_1-a_2\beta_3+a_3\beta_2\right\}\gamma_{023}\\+\left\{a_0b_2-a_3\beta_1+a_1\beta_3\right\}\gamma_{031}\\+\left\{a_0b_3-a_1\beta_2+a_2\beta_1\right\}\gamma_{012}

2次と1次の積

ab=\left\{\gamma_0\overrightarrow{\alpha}+\gamma_{123}\overrightarrow{a}\right\}\left\{b_0\gamma_0+\overrightarrow{b}\right\}\\=\overrightarrow{\alpha}b_0+\gamma_0\overrightarrow{\alpha}\overrightarrow{b}+\gamma_{0123}\overrightarrow{a}b_0+\gamma_{123}\overrightarrow{a}\overrightarrow{b}\\=\left\{\alpha_1b_1+\alpha_2b_2+\alpha_3b_3\right\}\gamma_0\\+\left\{\alpha_1b_0-a_2b_3+a_3b_2\right\}\gamma_1\\+\left\{\alpha_2b_0-a_3b_1+a_1b_3\right\}\gamma_2\\+\left\{\alpha_3b_0-a_1b_2+a_2b_1\right\}\gamma_3\\+\left\{a_1b_1+a_2b_2+a_3b_3\right\}\gamma_{123}\\+\left\{a_1b_0+\alpha_2b_3-\alpha_3b_2\right\}\gamma_{023}\\+\left\{a_2b_0+\alpha_3b_1-\alpha_1b_3\right\}\gamma_{031}\\+\left\{a_3b_0+\alpha_1b_2-\alpha_2b_1\right\}\gamma_{012}

3次と2次の積

ab=\gamma_{0123}\tilde{a}b\\=\left\{-a_1b_1-a_2b_2-a_3b_3\right\}\gamma_0\\+\left\{-a_0b_1+a_2\beta_3-a_3\beta_2\right\}\gamma_1\\+\left\{-a_0b_2+a_3\beta_1-a_1\beta_3\right\}\gamma_2\\+\left\{-a_0b_3+a_1\beta_2-a_2\beta_1\right\}\gamma_3\\+\left\{-a_1\beta_1-a_2\beta_2-a_3\beta_3\right\}\gamma_{123}\\+\left\{-a_0\beta_1-a_2b_3+a_3b_2\right\}\gamma_{023}\\+\left\{-a_0\beta_2-a_3b_1+a_1b_3\right\}\gamma_{031}\\+\left\{-a_0\beta_3-a_1b_2+a_2b_1\right\}\gamma_{012}

2次と3次の積

ab=a\gamma_{0123}\tilde{b}=\gamma_{0123}a\tilde{b}\\=\left\{-a_1b_1-a_2b_2-a_3b_3\right\}\gamma_0\\+\left\{-a_1b_0-\alpha_2b_3+\alpha_3b_2\right\}\gamma_1\\+\left\{-a_2b_0-\alpha_3b_1+\alpha_1b_3\right\}\gamma_2\\+\left\{-a_3b_0-\alpha_1b_2+\alpha_2b_1\right\}\gamma_3\\+\left\{\alpha_1b_1+\alpha_2b_2+\alpha_3b_3\right\}\gamma_{123}\\+\left\{\alpha_1b_0-a_2b_3+a_3b_2\right\}\gamma_{023}\\+\left\{\alpha_2b_0-a_3b_1+a_1b_3\right\}\gamma_{031}\\+\left\{\alpha_3b_0-a_1b_2+a_2b_1\right\}\gamma_{012}