So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as:(adsbygoogle = window.adsbygoogle || []).push({});

[itex] G_{\mu \nu}\equiv \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}-ig\left[ A_{\mu},A_{\nu}\right] [/itex]

[itex] \mathcal{L} = -\frac{1}{4}Tr G_{\mu \nu} G^{\mu \nu} [/itex]

Dumb question:

Isn't [itex]G_{\mu \nu} G^{\mu \nu} [/itex] 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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# Faddov Ghosts and the non-Abelian Lagrangian

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