aus Wikisource, der freien Quellensammlung
(9,8)
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{\displaystyle \left\{{\begin{aligned}&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{1}g^{2}=\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{3}g^{4},\\&\sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{1}g^{2}=\sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{3}g^{4}\\&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{1}g^{3}=-{\frac {1}{2}}\cdot \sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{3}g^{4},\\&\sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{1}g^{3}=-{\frac {1}{2}}\cdot \sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{3}g^{4}\end{aligned}}\right.}
und die, die zu ihnen führen können, auszurechnen und alle anderen fortzulassen.
Unter dieser Vereinfachung erhält man für die Zähler des Dirac schen Matrixelements durch Entwicklung von (5,15) nach
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{\displaystyle g^{i}/mc}
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(9,9)
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{\displaystyle \left\{{\begin{aligned}{\begin{array}{|c|}Z_{1}\\Z_{2}=Z_{3}\\Z_{4}\\Z_{5}=Z_{6}\end{array}}=&-8{\begin{array}{|c|}0\\0\\0\\1\end{array}}+{\frac {8{p_{y}}^{2}}{{p_{0}}^{2}}}{\begin{array}{|c|}1\\1\\1\\2\end{array}}-8{\frac {{p_{y}}^{4}}{{p_{0}}^{4}}}{\begin{array}{|c|}1\\1\\1\\1\end{array}}\\&-{\frac {4{p_{y}}^{2}}{{p_{0}}^{4}}}\left[\left({\mathfrak {pg}}^{1}\right){\begin{array}{|c|}3\\2\\1\\4\end{array}}+\left({\mathfrak {pg}}^{2}\right){\begin{array}{|c|}2\\1\\1\\2\end{array}}+\left({\mathfrak {pg}}^{4}\right){\begin{array}{|c|}-1\\-1\\-2\\-2\end{array}}\right]-{\frac {8{p_{y}}^{4}}{{p_{0}}^{6}}}\left({\mathfrak {p}},-2{\mathfrak {g}}^{1}-{\mathfrak {g}}^{2}+{\mathfrak {g}}^{4}\right){\begin{array}{|c|}1\\1\\1\\1\end{array}}\\&-2\left[{\frac {\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}-1\\0\\0\\-1\end{array}}+2\left[{\frac {\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}0\\0\\1\\1\end{array}}+2\left[{\frac {\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}1\\1\\1\\1\end{array}}\\&-2{\frac {{p_{y}}^{2}}{{p_{0}}^{4}}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right){\begin{array}{|r|}5\\3\\3\\5\end{array}}+\left({\mathfrak {g}}^{4},{\mathfrak {g}}^{1}+{\mathfrak {g}}^{2}\right){\begin{array}{|r|}1\\1\\1\\1\end{array}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}{{p_{0}}^{2}}}{\begin{array}{|r|}15\\7\\7\\15\end{array}}+{\frac {\left({\mathfrak {pg}}^{4}\right)\left({\mathfrak {p}},{\mathfrak {g}}^{1}+{\mathfrak {g}}^{2}\right)}{{p_{0}}^{2}}}{\begin{array}{|r|}1\\1\\1\\1\end{array}}\right]\\&-8{\frac {{p_{y}}^{4}}{{p_{0}}^{6}}}\left[-\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)+{\frac {4\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}-2{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}\right]{\begin{array}{|r|}1\\1\\1\\1\end{array}}.\end{aligned}}\right.}
![{\displaystyle \left\{{\begin{aligned}{\begin{array}{|c|}Z_{1}\\Z_{2}=Z_{3}\\Z_{4}\\Z_{5}=Z_{6}\end{array}}=&-8{\begin{array}{|c|}0\\0\\0\\1\end{array}}+{\frac {8{p_{y}}^{2}}{{p_{0}}^{2}}}{\begin{array}{|c|}1\\1\\1\\2\end{array}}-8{\frac {{p_{y}}^{4}}{{p_{0}}^{4}}}{\begin{array}{|c|}1\\1\\1\\1\end{array}}\\&-{\frac {4{p_{y}}^{2}}{{p_{0}}^{4}}}\left[\left({\mathfrak {pg}}^{1}\right){\begin{array}{|c|}3\\2\\1\\4\end{array}}+\left({\mathfrak {pg}}^{2}\right){\begin{array}{|c|}2\\1\\1\\2\end{array}}+\left({\mathfrak {pg}}^{4}\right){\begin{array}{|c|}-1\\-1\\-2\\-2\end{array}}\right]-{\frac {8{p_{y}}^{4}}{{p_{0}}^{6}}}\left({\mathfrak {p}},-2{\mathfrak {g}}^{1}-{\mathfrak {g}}^{2}+{\mathfrak {g}}^{4}\right){\begin{array}{|c|}1\\1\\1\\1\end{array}}\\&-2\left[{\frac {\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}-1\\0\\0\\-1\end{array}}+2\left[{\frac {\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}0\\0\\1\\1\end{array}}+2\left[{\frac {\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{4}}}\right]{\begin{array}{|r|}1\\1\\1\\1\end{array}}\\&-2{\frac {{p_{y}}^{2}}{{p_{0}}^{4}}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right){\begin{array}{|r|}5\\3\\3\\5\end{array}}+\left({\mathfrak {g}}^{4},{\mathfrak {g}}^{1}+{\mathfrak {g}}^{2}\right){\begin{array}{|r|}1\\1\\1\\1\end{array}}-{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}{{p_{0}}^{2}}}{\begin{array}{|r|}15\\7\\7\\15\end{array}}+{\frac {\left({\mathfrak {pg}}^{4}\right)\left({\mathfrak {p}},{\mathfrak {g}}^{1}+{\mathfrak {g}}^{2}\right)}{{p_{0}}^{2}}}{\begin{array}{|r|}1\\1\\1\\1\end{array}}\right]\\&-8{\frac {{p_{y}}^{4}}{{p_{0}}^{6}}}\left[-\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)+{\frac {4\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}-2{\frac {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}-{\frac {\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{4}\right)}{{p_{0}}^{2}}}\right]{\begin{array}{|r|}1\\1\\1\\1\end{array}}.\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3248ffbc7505691a09f688f492ec7f49f08e5944)
