Roots of SU(3): Basic Constructs & Generators

[arroy_0205]

I have some doubts regarding SU(3). These are at very basic level.

First, how does one construct adjoint representation of SU(3)? What will be the dimensionality of the matrices? The defining matrices in terms of Gell-Mann matrices are 3x3 but in the case of adjoint representation the matrices have to satisfy the condition:
$$[T_a]_{bc}=-if_{abc}$$
and we know f_{147} etc are nonzero so in this case, b=4, c=7. Is this right?

Second: In the book "Lie algebras in Particle Physics", H. Georgi gives (p101, equation no 7.12) the forms of $$E_{\pm1,0}$$ etc for SU(3). I do not understand how these generators are calculated. Can anybody please help?

 

[Bill_K]

In SU(2), you typically use a representation in which J_z is diagonal, with certain eigenvalues. You form linear combinations of the other two, J_x +- i J_y that raise and lower the eigenvalues.

The treatment of SU(3) is a generalization of this. Two of the generators can be diagonalized simultaneously, and form what we call the Cartan subalgebra. Their eigenvalues in a particular representation are called the weights. You form linear combinations of the six remaining generators to raise and lower the weights. (In the adjoint representation, the action of one generator on another is defined by taking an 8x8 matrix commutator.) These are what Georgi calls E. His Eq 7.12 shows they are complex combinations, like T₁ +- i T₂, and the subscripts on the E's shows how each one of them changes the weights.

 

[arroy_0205]

Thanks for the explanation. I'll check it and come back soon.