Для группы S U ( 2 ) {\displaystyle \mathrm {SU} (2)} генераторы известны как матрицы Паули :
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σ 1 = ( 0 1 1 0 ) {\displaystyle \sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}
σ 2 = ( 0 − i i 0 ) {\displaystyle \sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}}
σ 3 = ( 1 0 0 − 1 ) {\displaystyle \sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
Аналогом матриц Паули для S U ( 3 ) {\displaystyle \mathrm {SU} (3)} служат матрицы Гелл-Манна :
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λ 1 = ( 0 1 0 1 0 0 0 0 0 ) {\displaystyle \lambda _{1}={\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}}}
λ 2 = ( 0 − i 0 i 0 0 0 0 0 ) {\displaystyle \lambda _{2}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}}
λ 3 = ( 1 0 0 0 − 1 0 0 0 0 ) {\displaystyle \lambda _{3}={\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}}}
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λ 4 = ( 0 0 1 0 0 0 1 0 0 ) {\displaystyle \lambda _{4}={\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}}}
λ 5 = ( 0 0 − i 0 0 0 i 0 0 ) {\displaystyle \lambda _{5}={\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}}}
λ 6 = ( 0 0 0 0 0 1 0 1 0 ) {\displaystyle \lambda _{6}={\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}}}
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λ 7 = ( 0 0 0 0 0 − i 0 i 0 ) {\displaystyle \lambda _{7}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}}
λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 − 2 ) {\displaystyle \lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}}
Генераторы для S U ( 3 ) {\displaystyle \mathrm {SU} (3)} определяются как T {\displaystyle T} с использованием соотношения:
T a = λ a 2 {\displaystyle T_{a}={\frac {\lambda _{a}}{2}}} .Они подчиняются следующим соотношениям:
[ T a , T b ] = i ∑ c = 1 8 f a b c T c {\displaystyle \left[T_{a},T_{b}\right]=i\sum _{c=1}^{8}{f_{abc}T_{c}}} , где f {\displaystyle f} — структурная константа , значения которой равны:f 123 = 1 {\displaystyle f_{123}=1} ,
f 147 = f 165 = f 246 = f 257 = f 345 = f 376 = 1 2 {\displaystyle f_{147}=f_{165}=f_{246}=f_{257}=f_{345}=f_{376}={\frac {1}{2}}} ,
f 458 = f 678 = 3 2 {\displaystyle f_{458}=f_{678}={\frac {\sqrt {3}}{2}}} ;tr ( T a ) = 0 {\displaystyle \operatorname {tr} (T_{a})=0} .
Эрмитовы матрицы генераторы для S U ( 4 ) {\displaystyle \mathrm {SU} (4)} , аналогичные матрицам Паули и матрицам Гелл-Манна , имеют вид:
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λ 1 = ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ) {\displaystyle \lambda _{1}={\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}}
λ 2 = ( 0 − i 0 0 i 0 0 0 0 0 0 0 0 0 0 0 ) {\displaystyle \lambda _{2}={\begin{pmatrix}0&-i&0&0\\i&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}}
λ 3 = ( 1 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 ) {\displaystyle \lambda _{3}={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}}
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λ 4 = ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) {\displaystyle \lambda _{4}={\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\end{pmatrix}}}
λ 5 = ( 0 0 − i 0 0 0 0 0 i 0 0 0 0 0 0 0 ) {\displaystyle \lambda _{5}={\begin{pmatrix}0&0&-i&0\\0&0&0&0\\i&0&0&0\\0&0&0&0\end{pmatrix}}}
λ 6 = ( 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 ) {\displaystyle \lambda _{6}={\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&0\end{pmatrix}}}
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λ 7 = ( 0 0 0 0 0 0 − i 0 0 i 0 0 0 0 0 0 ) {\displaystyle \lambda _{7}={\begin{pmatrix}0&0&0&0\\0&0&-i&0\\0&i&0&0\\0&0&0&0\end{pmatrix}}}
λ 8 = 1 3 ( 1 0 0 0 0 1 0 0 0 0 − 2 0 0 0 0 0 ) {\displaystyle \lambda _{8}={\frac {1}{\sqrt {3}}}{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-2&0\\0&0&0&0\end{pmatrix}}}
λ 9 = ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ) {\displaystyle \lambda _{9}={\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{pmatrix}}}
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λ 10 = ( 0 0 0 − i 0 0 0 0 0 0 0 0 i 0 0 0 ) {\displaystyle \lambda _{10}={\begin{pmatrix}0&0&0&-i\\0&0&0&0\\0&0&0&0\\i&0&0&0\end{pmatrix}}}
λ 11 = ( 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 ) {\displaystyle \lambda _{11}={\begin{pmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&1&0&0\end{pmatrix}}}
λ 12 = ( 0 0 0 0 0 0 0 − i 0 0 0 0 0 i 0 0 ) {\displaystyle \lambda _{12}={\begin{pmatrix}0&0&0&0\\0&0&0&-i\\0&0&0&0\\0&i&0&0\end{pmatrix}}}
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λ 13 = ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 ) {\displaystyle \lambda _{13}={\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}}}
λ 14 = ( 0 0 0 0 0 0 0 0 0 0 0 − i 0 0 i 0 ) {\displaystyle \lambda _{14}={\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-i\\0&0&i&0\end{pmatrix}}}
λ 15 = 1 6 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − 3 ) {\displaystyle \lambda _{15}={\frac {1}{\sqrt {6}}}{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-3\end{pmatrix}}}
Эти матрицы удовлетворяют выражению для следа :
T r ( λ k 2 ) = 2 ; k = 1..15 {\displaystyle Tr{(\lambda _{k}^{2})}=2;k=1..15} и тождеству Якоби :
[ [ λ l , λ k ] , λ j ] + [ [ λ k , λ j ] , λ l ] + [ [ λ j , λ l ] , λ k ] = 0 ; j < k < l ; j , k , l = 1..15 {\displaystyle [[\lambda _{l},\lambda _{k}],\lambda _{j}]+[[\lambda _{k},\lambda _{j}],\lambda _{l}]+[[\lambda _{j},\lambda _{l}],\lambda _{k}]=0;j<k<l;j,k,l=1..15} При этом коммутатор вычисляется как:
[ λ j , λ k ] = 2 i ∑ m f j k l λ l {\displaystyle [\lambda _{j},\lambda _{k}]=2i\sum _{m}f_{jkl}\lambda _{l}} Таблица структурных констант f j k l {\displaystyle f_{jkl}}
f 1 , 2 , 3 = 1 {\displaystyle f_{1,2,3}=1}
f 1 , 4 , 7 = f 2 , 4 , 6 = f 2 , 5 , 7 = f 3 , 4 , 5 = f 1 , 9 , 12 = f 2 , 9 , 11 = f 2 , 10 , 12 = f 3 , 9 , 10 = f 4 , 9 , 14 = f 5 , 10 , 14 = f 6 , 11 , 14 = f 7 , 11 , 13 = f 7 , 12 , 14 = 1 2 {\displaystyle f_{1,4,7}=f_{2,4,6}=f_{2,5,7}=f_{3,4,5}=f_{1,9,12}=f_{2,9,11}=f_{2,10,12}=f_{3,9,10}=f_{4,9,14}=f_{5,10,14}=f_{6,11,14}=f_{7,11,13}=f_{7,12,14}={\frac {1}{2}}}
f 1 , 5 , 6 = f 3 , 6 , 7 = f 1 , 10 , 11 = f 3 , 11 , 12 = f 4 , 10 , 13 = f 6 , 12 , 13 = − 1 2 {\displaystyle f_{1,5,6}=f_{3,6,7}=f_{1,10,11}=f_{3,11,12}=f_{4,10,13}=f_{6,12,13}=-{\frac {1}{2}}}
f 4 , 5 , 8 = f 6 , 7 , 8 = f 8 , 9 , 10 = f 8 , 11 , 12 = f 9 , 10 , 15 = f 11 , 12 , 15 = f 13 , 14 , 15 = 3 2 {\displaystyle f_{4,5,8}=f_{6,7,8}=f_{8,9,10}=f_{8,11,12}=f_{9,10,15}=f_{11,12,15}=f_{13,14,15}={\frac {\sqrt {3}}{2}}}
f 8 , 13 , 14 = − 3 2 {\displaystyle f_{8,13,14}=-{\frac {\sqrt {3}}{2}}}