Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My apology to Yemon Choi, Will Jagi , Theo Johnson-Freyd and all other readers. My question was formulated extremely short without any context. It is the first time I am asking the MathOverflow community for help. And it is hard to judge how far to go in the description of the theory behind the question and the context of the question. So, let me try again. Cristi gave a good hint I think but the question is not answered yet.
The special unitary group SU(3) is a inherent component of the standard model of particle physics. It models the gauge field of the color charge property that is related the strong particle interaction.
Generally for SU(n), matrices are used as a representation of the generators of its Lie algebra.
Those matrices (L_i) are complex, traceless and antihermitian and their Lie bracket is the commutator.
The structure constants f_abc are defined by [L_a,L_b]=2i*f_abc*L_c
The matrices in this set that are diagonal, form a basis for the Cartan subalgebra.
Gell-Mann proposed a set of eight 3D-matrices (L_i, i=1 to 8) to be used for SU(3) in the standard model.
They are similar to the Pauli matrices in the SU(2)-case.
Additionally Gell-Mann requires that all eight matrices are trace orthogonal, or that
tr(L_a L_b)=2*delta(a,b) for a and b =1 to 8.
This means that the 9-dimensional vectors, corresponding to the matrices, are orthogonal.
The basis for the Cartan-algebra in this case is the pair L_3 and L_8.
All entries in L_3 and L_8 are zero except
L_3(1,1)=-L_3(2,2)=sqrt(3)*L_8(1,1)=sqrt(3)*L_8(2,2)=-.5*sqrt(3)*L_8(3,3)=1.
The completely antisymmetric structure constants are:
.5* f_123=f_147=f_165=f_246=f_257=f_345=f_376=.5, f_458=f_678=sqrt(3)*.5
In G-M’s particular choice of the matrices you see the appearance of the sqrt(3). I have the feeling that the expressions could be simpler (without sqrt(3)) if L_8 is replaced by (L_3+L_8),
so that the entries of L_8 are all zero except L_8(2,2)= -L_8(3,3)=1.
Of course the structure constants would change:
F_123=f_678=f_458=f_345=1,f_147=f_165=f_246=f_257=f_376=f_128=.5
However we would get tr(L_3 L_8) =.5 instead of 0.
So the question is:
Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
Best Answer
I'm no expert, and I haven't asked Professors Gell-Mann or Ne'eman, but with their choice of matrices, L_3 measures the familiar Heisenberg iso-spin quantum number, while L_8 measures the then-novel hypercharge. Mixing the operators would mix the quantum numbers.