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The Special Unitary Group of Order 3 |
The abstract group SU(3) is represented by a set of eight 3×3 matrices of complex elements which have determinant of unity. These elements of the group can be generated by eight special matrices. These matrices must be Hermitian; i.e., the transpose of their complex conjugates is the same as the matrix. These matrices do not have determinants of unity; instead all have traces (sums of elements on the principal diagonal) of zero. For a proof these proposition see Group Generators.
In the following i represents the imaginary unit √-1. Here are the generators for the matrix representation of SU(3).
| 0 | 1 | 0 | |
½ | 1 | 0 | 0 | |
| 0 | 0 | 0 | |
| 0 | −i | 0 | |
½ | i | 0 | 0 | |
| 0 | 0 | 0 | |
| 1 | 0 | 0 | |
½ | 0 | −1 | 0 | |
| 0 | 0 | 0 | |
| 0 | 0 | 1 | |
½ | 0 | 0 | 0 | |
| 1 | 0 | 0 | |
| 0 | 0 | −i | |
½ | 0 | 0 | 0 | |
| i | 0 | 0 | |
| 0 | 0 | 0 | |
½ | 0 | 0 | 1| |
| 0 | 1 | 0 | |
| 0 | 0 | 0 | |
½ | 0 | 0 | −i | |
| 0 | i | 0 | |
| 1 | 0 | 0 | |
1/(2√3) | 0 | 1 | 0 | |
| 0 | 0 | −2 | |
Let these matrices be labeled λ_{1}, λ_{1}, …, λ_{8}. Now consider the computation of the commutations of them.
λ_{1}λ_{2} is equal to
| i | 0 | 0 | |
¼ | 0 | -i | 0 | |
| 0 | 0 | 0 | |
| 1 | 0 | 0 | |
¼i | 0 | -1 | 0 | |
| 0 | 0 | 0 | |
Likewise λ_{2}λ_{1} is equal to
| -i | 0 | 0 | |
¼ | 0 | i | 0 | |
| 0 | 0 | 0 | |
| 1 | 0 | 0 | |
-¼i | 0 | -1 | 0 | |
| 0 | 0 | 0 | |
Thus the commutation of λ_{1} and λ_{2} is equal to
Similarly the commutations for the other matrices can be found and the results can be represented as
where ε_{jkm} is +1 if jkm is an even permutation of values of j,k,m written in the order of their sizes and −1 if jkm is an odd permutation. If there are repeated indices then ε_{jkm} is equal to zero.
(To be continued.)
(To be continued.)
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