aus Wikisource, der freien Quellensammlung
(9,6)
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{\displaystyle {\begin{array}{r|c|c|c|c}&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{3}g^{4}&\sum \limits _{\textrm {Perm}}g^{1}g^{2}g^{3}g^{4}&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{1}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{2}\right)\\\hline \sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{2}g^{3}=\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{1}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)=&0&0&-{\frac {1}{3}}&0\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{1}\right)g^{1}g^{2}=\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{1}\right)\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)=&0&0&+{\frac {2}{3}}&-1\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{1}\right)^{2}=&0&0&-2&3\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{1}g^{3}=&0&-{\frac {1}{2}}&+{\frac {1}{6}}&0\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{1}g^{2}=&0&+1&-1&1\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)=&-{\frac {1}{2}}&0&+{\frac {1}{6}}&0\\\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)^{2}=&+1&0&-1&1\end{array}}}
Die linearen Beziehungen zwischen Gliedern wie
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{\displaystyle \sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{3}g^{4}}
,
welche den Positronenimpuls
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{\displaystyle -{\mathfrak {p}}}
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{\displaystyle {\mathfrak {p}}}
(9,7)
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{\displaystyle \left\{{\begin{aligned}{\overline {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{4}}{3\cdot 5}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {{p_{y}}^{2}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{6}}{3\cdot 5\cdot 7}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {{p_{y}}^{4}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{8}}{5\cdot 7\cdot 9}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{2}}{3}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\\{\overline {{p_{y}}^{2}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{4}}{3\cdot 5}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\\{\overline {{p_{y}}^{4}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{6}}{5\cdot 7}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right).\end{aligned}}\right.}
Die Behauptung der Eichinvarianz besagt nun, daß in der Tabelle (9,6) der linearen Beziehungen die 4. Spalte (in der Summe über alle Glieder des Dirac schen Matrixelements) nicht vorkommt, und die Behauptung der Lorentzinvarianz: daß die ersten drei Spalten (summiert über alle Glieder) im Verhältnis
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{\displaystyle 1:-2:1}
Wir verabreden nun (9,5 a), nur die Glieder der zweiten Spalte zu sammeln, d. h. (9‚6; 9,7) im Dirac schen Matrixelement nur die Glieder:
(9,8)
{
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{\displaystyle \left\{{\begin{aligned}&\sum \limits _{\textrm {Perm}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)g^{3}g^{4}=\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^{3}g^{4}=\sum \limits _{\textrm {Perm}}\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)g^{3}g^{4}\end{aligned}}\right.}
![{\displaystyle \left\{{\begin{aligned}{\overline {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{4}}{3\cdot 5}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {{p_{y}}^{2}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{6}}{3\cdot 5\cdot 7}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {{p_{y}}^{4}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)\left({\mathfrak {pg}}^{3}\right)\left({\mathfrak {pg}}^{4}\right)}}=&{\frac {p^{8}}{5\cdot 7\cdot 9}}\left[\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\left({\mathfrak {g}}^{3}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{3}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{4}\right)+\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{4}\right)\left({\mathfrak {g}}^{2}{\mathfrak {g}}^{3}\right)\right]\\{\overline {\left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{2}}{3}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\\{\overline {{p_{y}}^{2}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{4}}{3\cdot 5}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right)\\{\overline {{p_{y}}^{4}\cdot \left({\mathfrak {pg}}^{1}\right)\left({\mathfrak {pg}}^{2}\right)}}=&{\frac {p^{6}}{5\cdot 7}}\left({\mathfrak {g}}^{1}{\mathfrak {g}}^{2}\right).\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b1801000ce1b0ae1de44842f25aca02e00dde8)
