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γ Ω=(0 16 1 16 1 16 0 16) \gamma_\Omega \;=\; \left( \array{ 0_{16} & 1_{16} \\ -1_{16} & 0_{16} } \right)
(γ Ω) 1=γ Ω (\gamma_\Omega)^{-1} \;=\; \gamma_\Omega
γ 1=diag(1 8, i 8, 1 8, i 8) \gamma_{1} \;=\; diag \big( \array{ 1_8, & i_{8}, & -1_{8}, & -i_{8} } \big)
(γ Ω) 1γ 1(γ Ω)=γ 1 (\gamma_{\Omega})^{-1} \gamma_1 (\gamma_{\Omega}) \;=\; \gamma_1
(γ Ω) 1(A) T(γ Ω)=A (\gamma_{\Omega})^{-1} (A)^{T} (\gamma_{\Omega}) \;=\; -A
(γ ΩA) T =(A) T(γ Ω) T =(A) T(γ Ω) =(γ Ω)(A) \begin{aligned} (\gamma_\Omega A)^T & = (A)^T (\gamma_\Omega)^T \\ & = - (A)^T (\gamma_\Omega) \\ & = (\gamma_\Omega) (A) \end{aligned}


e b 1b 2=F a[b 1δ b 2] 5dx a+13!ϵ b 1b 2b 3a 1a 2F a 1a 2dx b 3 e_{b_1 b_2} \;=\; F_{a [b_1} \delta_{b_2]}^5 \, d x^a + \tfrac{1}{3!} \epsilon_{b_1 b_2 b_3 a_1 a_2} F^{a_1 a_2} d x^{b_3}
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(0a i5e a 1 a 2e a 2 a 3e a 3 a 1)(F a 4b 4dx a 4dx b 4) =(30a i4e a 15e a 35e a 1a 3+0a i4e a 1a 2e a 2a 3e a 3 a 1)(F a 4b 4dx a 4dx b 4) =+12F a 1b 1dx a 1F a 2b 2dx a 2g a 3b 3dx a 3ϵ b 1b 2b 3c 1c 2F c 1c 2F a 4b 4dx a 4dx b 4 =+ϵ b 1 b 2c 1c 2c 3δ [b 2 b 1δ a 1 a 1δ a 2 a 2δ a 3] a 3F a 1a 2dx a 3F c 1c 2dx c 3F a 1a 2g a 4a 3dx a 4F a 4b 4dx a 4dx b 4 =+12F a 1b 1dx a 1F a 2c 2dx a 2g a 3b 3dx a 3ϵ b 1b 2b 3c 1c 2F c 1b 2F a 4b 4dx a 4dx b 4+ \begin{aligned} & \Big( \underset{ \mathclap{ 0 \leq a_i \leq 5 } }{\sum} e_{a_1}{}^{a_2} \wedge e_{a_2}{}^{a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \Big( 3 \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 5} \wedge e_{a_3 5} \wedge e^{a_1 a_3} + \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 a_2} \wedge e^{a_2 a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \phantom{+}\; \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 b_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 c_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & \phantom{=}\; + \epsilon^{b_1}{}_{b_2 c_1 c_2 c_3 } \delta^{b_1}_{[b_2} \delta^{a_1}_{a'_1} \delta^{a_2}_{a'_2} \delta^{a_3}_{a'_3]} F^{a'_1 a'_2} d x^{a'_3} F^{c_1 c_2} d x^{c_3} F_{a_1 a_2} g_{a_4 a_3} d x^{a_4} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & = \phantom{+}\; \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 c_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 b_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \;+\; \cdots \end{aligned}


(0a i5e a 1 a 2e a 2 a 3e a 3 a 1)(F a 4b 4dx a 4dx b 4) =(30a i4e a 15e a 35e a 1a 3+0a i4e a 1a 2e a 2a 3e a 3 a 1)(F a 4b 4dx a 4dx b 4) =12F a 1b 1dx a 1F a 2b 2dx a 2g a 3b 3dx a 3ϵ b 1b 2b 3c 1c 2F c 1c 2F a 4b 4dx a 4dx b 4 =12ϵ b 1b 2b 3c 1c 2F a 1b 1F a 2b 2g a 3b 3F c 1c 2F d 1d 2ϵ a 1a 2a 3d 1d 2dvol+ \begin{aligned} & \Big( \underset{ \mathclap{ 0 \leq a_i \leq 5 } }{\sum} e_{a_1}{}^{a_2} \wedge e_{a_2}{}^{a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \Big( 3 \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 5} \wedge e_{a_3 5} \wedge e^{a_1 a_3} + \underset{ \mathclap{ 0 \leq a_i \leq 4 } }{\sum} e_{a_1 a_2} \wedge e^{a_2 a_3} \wedge e_{a_3}{}^{a_1} \Big) \wedge \Big( F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \Big) \\ & = \tfrac{1}{2} F_{a_1 b_1} d x^{a_1} \wedge F_{a_2 b_2} d x^{a_2} \wedge g_{a_3 b_3} d x^{a_3} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{c_1 c_2} \wedge F_{a_4 b_4} d x^{a_4} \wedge d x^{b_4} \\ & = \tfrac{1}{2} \epsilon^{b_1 b_2 b_3 c_1 c_2} F_{a_1 b_1} F_{a_2 b_2} g_{a_3 b_3} F_{c_1 c_2} F_{d_1 d_2} \epsilon^{a_1 a_2 a_3 d_1 d_2} \cdot \mathrm{dvol} \;+\; \cdots \end{aligned}


=+ϵ a 1a 2a 3d 1d 2(F a 1b 1F a 2b 2g a 3b 3F c 1c 2F d 1d 2)ϵ b 1b 2b 3c 1c 2 =ϵ a 1a 2a 3d 1d 2(F a 1b 1F a 2c 2g a 3b 3F c 1b 2F d 1d 2)ϵ b 1b 2b 3c 1c 2 =+ϵ a 1a 2a 3d 1d 2(F a 1b 1F d 1c 2g a 3b 3F c 1b 2F a 2d 2)ϵ b 1b 2b 3c 1c 2 \begin{aligned} & \phantom{= +}\, \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{a_2 b_2} g_{a_3 b_3} F_{c_1 c_2} F_{d_1 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \\ & = - \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{a_2 c_2} g_{a_3 b_3} F_{c_1 b_2} F_{d_1 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \\ & = + \epsilon^{a_1 a_2 a_3 d_1 d_2} \big( F_{a_1 b_1} F_{d_1 c_2} g_{a_3 b_3} F_{c_1 b_2} F_{a_2 d_2} \big) \epsilon^{b_1 b_2 b_3 c_1 c_2} \end{aligned}

Last revised on September 2, 2019 at 15:04:46. See the history of this page for a list of all contributions to it.