Faddov Ghosts and the non-Abelian Lagrangian

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So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as:
[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.
 
649
2
So my text (Ryder 2nd edition, page 252) is defining the "pure gauge-field Lagrangian" as:
[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?
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

Science Advisor
Gold Member
3,951
324
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 [itex]\gamma^\mu[/itex]. This is actually a set of 4 matrices. The trace means to take the trace of the resulting matrix.
 

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