A question in QFT book of Peskin&Schoeder?

[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?