'Using the following normalization in the su(3) algebra ##[\lambda_i, \lambda_j] = 2if_{ijk}\lambda_k##, we see that ##g_{ij} = 4f_{ikl}f_{jkl} = 12 \delta_{ij}## and, by expanding the anticommutator in invariant tensors, we have further that $$\left\{\lambda_i, \lambda_j\right\} = \frac{4}{3}\delta_{ij} + 2d_{ijk}\lambda_k.$$(adsbygoogle = window.adsbygoogle || []).push({});

The first statement about ##g_{ij}## I understand but how did the one about the anticommutator come about?

I can reexpress ##\left\{\lambda_i, \lambda_j\right\} = [\lambda_i, \lambda_j] + 2 \lambda_j \lambda_i = 2if_{ijk}\lambda_k + 2 \lambda_j \lambda_i##. Now, ##\lambda_j \lambda_i## is a second rank tensor so can be written as ##a \delta_{ij}##, for some a. I was thinking I could then consider a single case to determine a (i.e i=j=1) but this didn't work.

Any tips would be great!

Thanks!

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# Invariant tensors of SU(3)

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