ディラック作用素とリーマン曲率テンソル

$\def\ou#1#2#3{\overset{#2}{\underset{#3}{#1}}}\def\os#1#2{\overset{#2}{#1}}\def\diff{\mathrm{d}}\def\biff{\mathrm{b}}\def\D{\mathrm{D}}\def\p{\mathrm{p}}\def\pu#1{\underset{#1}{\mathrm{p}}}\def\G#1#2{\overset{#1}{\underset{#2}{\Gamma}}}\def\abs#1{|#1|}$ $A\wedge\D\wedge\D=\{\ou{A}{\sigma}{}\ou{\gamma}{}{\sigma}\}\wedge\{\ou{\p}{}{\mu}\ou{\gamma}{\mu}{}\}\wedge\{\ou{\p}{}{\nu}\ou{\gamma}{\nu}{}\}\\=\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\gamma}{}{\sigma}+\ou{A}{\sigma}{}\ou{\gamma}{}{\sigma}\ou{\p}{}{\mu}\}\wedge\ou{\gamma}{\mu}{}\wedge\{\ou{\p}{}{\nu}\ou{\gamma}{\nu}{}\}\\=\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\gamma}{}{\sigma}+\ou{A}{\sigma}{}\G{\alpha}{\sigma\mu}\ou{\gamma}{}{\alpha}\}\wedge\ou{\gamma}{\mu}{}\wedge\{\ou{\p}{}{\nu}\ou{\gamma}{\nu}{}\}\\=\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\gamma}{}{\sigma}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\gamma}{}{\sigma}\}\wedge\ou{\gamma}{\mu}{}\wedge\{\ou{\p}{}{\nu}\ou{\gamma}{\nu}{}\}\\=\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\{\ou{\p}{}{\nu}\ou{\gamma}{\nu}{}\}\\=\{\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\p}{}{\nu}+\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\ou{\p}{}{\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}+\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\ou{\gamma}{}{\sigma}\ou{\p}{}{\nu}\wedge\ou{\gamma}{\mu}{}+\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\ou{\p}{}{\nu}\}\wedge\ou{\gamma}{\nu}{}\\=\{\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\ou{\p}{}{\nu}\G{\sigma}{\alpha\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}+\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\G{\beta}{\sigma\nu}\ou{\gamma}{}{\beta}\wedge\ou{\gamma}{\mu}{}-\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\}\ou{\gamma}{}{\sigma}\wedge\G{\mu}{\nu\beta}\ou{\gamma}{\beta}{}\}\wedge\ou{\gamma}{\nu}{}\\=\{\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\ou{\p}{}{\nu}\G{\sigma}{\alpha\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}+\{\ou{A}{\beta}{}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\}\G{\sigma}{\beta\nu}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}-\{\ou{A}{\sigma}{}\ou{\p}{}{\beta}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\beta}\}\ou{\gamma}{}{\sigma}\wedge\G{\beta}{\nu\mu}\ou{\gamma}{\mu}{}\}\wedge\ou{\gamma}{\nu}{}\\=\{\ou{A}{\sigma}{}\ou{\p}{}{\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\ou{\p}{}{\nu}\G{\sigma}{\alpha\mu}+\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\beta}{}\ou{\p}{}{\mu}\G{\sigma}{\beta\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}-\ou{A}{\sigma}{}\ou{\p}{}{\beta}\G{\beta}{\nu\mu}-\ou{A}{\alpha}{}\G{\sigma}{\alpha\beta}\G{\beta}{\nu\mu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\\=\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\\=\frac12\{\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}+\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\}\\=\frac12\{\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}+\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\nu}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\nu}\G{\sigma}{\beta\mu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\nu}{}\wedge\ou{\gamma}{\mu}{}\}\\=\frac12\{\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}\}-\{\ou{A}{\alpha}{}\G{\sigma}{\alpha\nu}\ou{\p}{}{\mu}+\ou{A}{\alpha}{}\G{\beta}{\alpha\nu}\G{\sigma}{\beta\mu}\}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\\=\frac12\ou{A}{\alpha}{}\{\G{\sigma}{\alpha\mu}\ou{\p}{}{\nu}-\G{\sigma}{\alpha\nu}\ou{\p}{}{\mu}+\G{\beta}{\alpha\mu}\G{\sigma}{\beta\nu}-\G{\beta}{\alpha\nu}\G{\sigma}{\beta\mu}\}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\\=\frac12\ou{A}{\alpha}{}\ou{R}{\sigma}{\alpha\nu\mu}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\\=\frac12\ou{A}{\alpha}{}\ou{R}{}{\rho\alpha\nu\mu}\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}$
基底 $\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}\ (\rho\neq\mu\neq\nu)$ の成分は
$\frac12\ou{A}{\alpha}{}\{\ou{R}{}{\rho\alpha\nu\mu}-\ou{R}{}{\rho\alpha\mu\nu}+\ou{R}{}{\mu\alpha\rho\nu}-\ou{R}{}{\mu\alpha\nu\rho}+\ou{R}{}{\nu\alpha\mu\rho}-\ou{R}{}{\nu\alpha\rho\mu}\}\\=\ou{A}{\alpha}{}\{\ou{R}{}{\rho\alpha\nu\mu}+\ou{R}{}{\mu\alpha\rho\nu}+\ou{R}{}{\nu\alpha\mu\rho}\}\\=\ou{A}{\alpha}{}\{\ou{R}{}{\rho\alpha\nu\mu}+\ou{R}{}{\rho\nu\mu\alpha}+\ou{R}{}{\rho\mu\alpha\nu}\}\\=0$
ちなみに $A\wedge\{\D\wedge\D\}$ と計算しても同値である．また $A$ の偏微分を含む項がキャンセルされたので
$\ou{\gamma}{}{\alpha}\wedge\D\wedge\D=\frac12\ou{R}{\sigma}{\alpha\nu\mu}\ou{\gamma}{}{\sigma}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}=\frac12\ou{R}{}{\rho\alpha\nu\mu}\ou{\gamma}{\rho}{}\wedge\ou{\gamma}{\mu}{}\wedge\ou{\gamma}{\nu}{}=0$
である．これらの計算をベクトル場の成分が共変で表されるときで行えば自然にマクスウェル方程式の3次のほうが導かれる(幾何代数と重力場中の物理 - うべの時空代数)．