Faddov Ghosts and the non-Abelian Lagrangian

Elwin.Martin

So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as:
$G_{\mu \nu}\equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}-ig\left[ A_{\mu},A_{\nu}\right]$

$\mathcal{L} = -\frac{1}{4}Tr G_{\mu \nu} G^{\mu \nu}$

Dumb question:
Isn't $G_{\mu \nu} G^{\mu \nu}$ being summed over, and hence, scalar?
How is trace even defined on a scalar quantity? Is the trace only applying to the first G and is a scalar factor for the second?

I feel like I'm missing something obvious here.

Thanks for any and all advice.

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torquil

So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as:
$G_{\mu \nu}\equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}-ig\left[ A_{\mu},A_{\nu}\right]$

$\mathcal{L} = -\frac{1}{4}Tr G_{\mu \nu} G^{\mu \nu}$

Dumb question:
Isn't $G_{\mu \nu} G^{\mu \nu}$ being summed over, and hence, scalar?
Yes, a spacetime scalar. But not a scalar with respect to gauge group transformations. The field strength is an element of the Lie algebra, so it is a matrix when the gauge group is e.g. SU(n) for n>1. The rest of your questions should also be explained by this.

Matterwave

Gold Member
Remember that your object's indices indicate the space-time indices. They are groups of matrices in general. Just think about the Dirac matrices. You label $\gamma^\mu$. This is actually a set of 4 matrices. The trace means to take the trace of the resulting matrix.

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