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A question in QFT book of Peskin&Schoeder?

  1. Feb 8, 2012 #1
    Please teach me this:
    In the book writing: ...consider the color invariant:
    (t[itex]^{a}[/itex])[itex]_{ij}[/itex](t[itex]^{a}[/itex])[itex]_{kl}[/itex](18.38).The indices i,k transform according to to 3 representation of color; the indices j,l transform according to
    3[itex]^{-}[/itex].Thus,(18.38) must be a linear combination of the two possible way to contract these indices,
    Aδ[itex]_{il}[/itex]δ[itex]_{kj}[/itex]+Bδ[itex]_{ij}[/itex]δ[itex]_{kl}[/itex](18.39).
    The constant A and B can be determined by contracting (18.38) and (18.39) with δ[itex]_{ij}[/itex] and with δ[itex]_{jk}[/itex].....
    I do not understand why (18.38)must be a linear combination as (18.39)?
    Thank you very much for your kind helping.
     
  2. jcsd
  3. Feb 8, 2012 #2
    Here t[itex]^{a}[/itex] is generator of SU(3).
     
  4. Feb 8, 2012 #3

    Physics Monkey

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    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 [itex] (U t^a U^+)_{ij} (U t^a U^+)_{kl} [/itex]?
     
  5. Feb 8, 2012 #4
    Please help me to consider Chapter &18.2 QFT book of Peskin & Schoeder.
     
  6. Feb 8, 2012 #5
    The book writing:...and adjusting A and B so that the contractions of (18.39) obey the identities: tr[t[itex]^{a}[/itex]](t[itex]^{a}[/itex])[itex]_{kl}[/itex]=0;(t[itex]^{a}[/itex]t[itex]^{a}[/itex])[itex]_{il}[/itex]=(4/3)δ[itex]_{il}[/itex] (18.40).
    This gives the identity:
    (t[itex]^{a}[/itex])[itex]_{ij}[/itex](t[itex]^{a}[/itex])[itex]_{kl}[/itex]=(1/2)(δ[itex]_{il}[/itex]δ[itex]_{kj}[/itex]-(1/3)δ[itex]_{ij}[/itex]δ[itex]_{kl}[/itex]) (18.41)
     
  7. Feb 9, 2012 #6
    Now I think that (18.41) is correct because (18.40) are more loosely conditions than the conditions that t[itex]^{a}[/itex] make themself the Lie algebras.Is that correct?
     
  8. Feb 10, 2012 #7
    If t[itex]^{a}[/itex] are satisfied (18.41)(in #5) then are t[itex]^{a}[/itex] still the generators of SU(3)?
     
  9. Feb 11, 2012 #8
    I have heard that this can be solved by 't Hooft's double line formalism.Then what is this?
     
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