Gell-Mann matrices

The Gell- Mann matrices, named after Murray Gell-Mann, are a possible representation of the infinitesimal generators of the special unitary group SU (3).

This group has eight hermitian generators, which can be written as with. They meet the Kommutatorrelation (see: Lie algebra )

(where the Einstein summation convention has been used). These are called structure constants and are fully - antisymmetric with respect to interchange of the indices. For the SU ( 3) they have the values ​​:

Every set of matrices that satisfy the Kommutatorrelation, can be used as generators of the group.

The Gell- Mann matrices are a standard set of such matrices. With the above generators are ( analogous to the Pauli matrices ) linked by:

They are selected as a 3x3 matrix and in the form of:

In the SU (2) one has the three Pauli matrices instead of the eight matrices.

The matrices have the following properties:

• You are Hermitian, so have only real eigenvalues
• " Marking", that is,
• And " orthogonal" in the following sense:

They are used, for example, for calculations in quantum chromodynamics, which is described by an SU (3 ) theory. From this we can understand the choice as 3x3 matrices, because the matrices are to act on Farbladungstriplets.