I was watching some new lectures on QCD from Colorado and I have a few questions from what I heard -(adsbygoogle = window.adsbygoogle || []).push({});

--The ##\lambda^a_{ij}## are generators of SU(3) in the fundamental representation so are 3x3 matrices. That is because the ij indices are colour indices and they act on a 3x1 vector in colour space (the colour wavefunction of the quarks).There are 8 generators (labelled by the index 'a') so the lambda are vectors in what the prof in the Colorado video calls 'gluon space'. This gluon space is spanned by eight independent non trivially transforming colour octet states in a eight dimensional real hilbert space so each of these states can be mapped to a unit vector in the real hilbert space, ie each one is a 8x1 unit basis vector.

My understanding from the video was that the representation of the lambda in colour space are the 3x3 Gell-mann matrices acting on the colour component of the quark fields embedded in the fundamental representation. What is the representation of these lambda as vectors in the gluon space and what do they act on?

--In the equation ##A^{\mu} = A^{\mu}_a \lambda^a/2## are we saying that the gluons are exactly the generators of SU(3) in the fundamental representation which give rise to non trivial colour transformations in colour space which when act on quark colour wavefunctions mix around the colours? This equation also tells us the gluons live in the lie algebra of SU(3) since the gluon ##A^{\mu}## can be expanded in the basis of generators ##T^a = \lambda^a/2##. The lie algebra is 8 dimensional but why do we say they transform under the adjoint representation of SU(3)? I guess it makes contact with the above in that we can write down each possible gluon as a basis vector in a eight dim space but what is it that they are transformed by?

Thanks!

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# I A few questions about colour in QCD...

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