Why is the color invariant in equation 18.38 a linear combination?

[ndung200790]

Please teach me this:
In the book writing: ...consider the color invariant:
(t$^{a}$)$_{ij}$(t$^{a}$)$_{kl}$(18.38).The indices i,k transform according to to 3 representation of color; the indices j,l transform according to
3$^{-}$.Thus,(18.38) must be a linear combination of the two possible way to contract these indices,
Aδ$_{il}$δ$_{kj}$+Bδ$_{ij}$δ$_{kl}$(18.39).
The constant A and B can be determined by contracting (18.38) and (18.39) with δ$_{ij}$ and with δ$_{jk}$...
I do not understand why (18.38)must be a linear combination as (18.39)?
Thank you very much for your kind helping.

 

[ndung200790]

Here t$^{a}$ is generator of SU(3).

 

[Physics Monkey]

To get started with an argument, what would happen if you acted on all ijkl indices with an arbitrary matrix U in the fundamental of SU(3). In other words, what can you say about $(U t^a U^+)_{ij} (U t^a U^+)_{kl}$?

 

[ndung200790]

Please help me to consider Chapter &18.2 QFT book of Peskin & Schoeder.

 

[ndung200790]

The book writing:...and adjusting A and B so that the contractions of (18.39) obey the identities: tr[t$^{a}$](t$^{a}$)$_{kl}$=0;(t$^{a}$t$^{a}$)$_{il}$=(4/3)δ$_{il}$ (18.40).
This gives the identity:
(t$^{a}$)$_{ij}$(t$^{a}$)$_{kl}$=(1/2)(δ$_{il}$δ$_{kj}$-(1/3)δ$_{ij}$δ$_{kl}$) (18.41)

 

[ndung200790]

Now I think that (18.41) is correct because (18.40) are more loosely conditions than the conditions that t$^{a}$ make themself the Lie algebras.Is that correct?

 

[ndung200790]

If t$^{a}$ are satisfied (18.41)(in #5) then are t$^{a}$ still the generators of SU(3)?

 

[ndung200790]

I have heard that this can be solved by 't Hooft's double line formalism.Then what is this?