- #1

CAF123

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In 1972, a model was proposed by Georgi and Glashow as a candidate theory describing W bosons and photons with Lagrangian $$\mathcal L = -\frac{1}{2} \text{Tr} F^{\mu \nu}F_{\mu \nu} + (D_{\mu} \phi)^T (D^{\mu} \phi) - \mu^2 \phi^T \phi - \lambda(\phi^T \phi)^2$$ with ##F_{\mu \nu}## the field strength tensor and ##A_{\mu}^a## the gauge fields of the gauge group SO(3), ##D_{\mu} = \partial_{\mu} + ig A_{\mu}^a \tau^a##, and ##\phi## is a 3 component real scalar field.

It's clear that the generator basis is ##i(\tau)_{jk} = \frac{1}{2} (\delta_{jk} - \delta_{kj})## for ##1 \leq j < k \leq 3## which may be applied to the vacuum expectation value $$\phi_{min} = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ 0 \\ v \end{pmatrix}$$ to deduce the number of Goldstone bosons, in accordance with Goldstone's theorem....

My question is simply, why is that a generator basis? First of all there are three generators in SO(3) so I expected to see another index on the ##\tau## to label each generator. Secondly, for any ##1 \leq j < k \leq 3## the components ##\tau_{jk}## are all identically zero (!). So clearly I am misunderstanding something here. Can anyone help?

Thanks!