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This group has eight generators, which we can write as ''gi'', with ''i'' taking values from 1 to 8. They obey the commutation relations ''gj'' = ''i fijk gk'' where a sum over the index ''k'' is implied. The Structure Constant ''fijk'' is completely antisymmetric in the three indices and has values
: ''f''123 = 2, ''f''147 = ''f''165 = ''f''246 = ''f''257 = ''f''345 = ''f''376 = 1, ''f''458 = ''f''678 = √3.
Any set of Hermitian Matrices which obey these relations are allowed. A particular choice of matrices is called a Group Representation , because any element of SU(3) can be written in the form exp(''i θi gi''), where ''θi'' are real numbers and a sum over the index ''i'' is implied. Given one representation, another may be obtained by an arbitrary unitary transformation, since that leaves the commutator unchanged.

An important representation involves 3×3 matrices, because the group elements then act on complex vectors with 3 entries, i.e., on the Fundamental Representation of the group. A particular choice of this representation is
:\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}, \lambda_2 = \begin{pmatrix} 0 & -i & 0 \ i & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}, \lambda_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 0 \end{pmatrix}, \lambda_4 = \begin{pmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0 \end{pmatrix}, \lambda_5 = \begin{pmatrix} 0 & 0 & -i \ 0 & 0 & 0 \ i & 0 & 0 \end{pmatrix}, \lambda_6 = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}, \lambda_7 = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & -i \ 0 & i & 0 \end{pmatrix}, \lambda_8 = rac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -2 \end{pmatrix}.
They are Traceless , Hermitian, and obey the extra relation Tr (''λiλj'') = 2''δij''. These properties were chosen by Gell-Mann because they then generalize the Pauli Matrices .

In this representation it is clear that the Cartan Subalgebra is given by the set of two matrices ''λ''3 and ''λ''8, which commute with each other. There are 3 independent SU(2) subgroups: {''λ''1, ''λ''2, ''x''}, {''λ''4, ''λ''5, y}, and {''λ''6, ''λ''7, ''z''}, where the ''x'', ''y'', ''z'' must consist of linear combinations of ''λ''3 and ''λ''8.

These matrices form an useful representation for computations in the Quark Model , and, to a lesser extent, in Quantum Chromodynamics .


SEE ALSO



REFERENCES AND EXTERNAL LINKS

  • ''Lie algebras in particle physics'', by Howard Georgi (ISBN 0738202339)

  • '' The quark model '', by J. J. J. Kokkedee