- #1

Ben Niehoff

Science Advisor

Gold Member

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## Main Question or Discussion Point

I've been playing around with the QCD Lagrangian to get a better understanding of how it works. I can derive some classical, Maxwell-like equations; the inhomogenous ones are

[tex]\nabla \cdot \vec E^a = -gf^{a}_{bc} \vec A^b \cdot \vec E^c + \rho^a[/tex]

[tex]\nabla \times \vec B^a - \frac{\partial}{\partial t} \vec E^a = \vec J^a + gf^a_{bc} (\Phi^b \vec E^c - \vec A^b \times \vec B^c)[/tex]

The problem is that I'm not quite sure how to interpret these equations. The (gluon) color indices {a, b, c} run from 1 to 8. But there are three kinds of color charge. So how do I interpret the sources [itex]\rho^a[/itex] and [itex]\vec J^a[/itex]?

One thing I attempted was to multiply both equations by the generators [itex]T^a_{ij}[/itex]. This eliminates the gluon color indices {a, b, c} in favor of the quark color indices ij (which then run from 1 to 3). But now there are two indices on everything! One for a color and one for an anti-color. Again, I can't quite figure out how to interpret what it means.

[tex]\nabla \cdot \vec E^a = -gf^{a}_{bc} \vec A^b \cdot \vec E^c + \rho^a[/tex]

[tex]\nabla \times \vec B^a - \frac{\partial}{\partial t} \vec E^a = \vec J^a + gf^a_{bc} (\Phi^b \vec E^c - \vec A^b \times \vec B^c)[/tex]

The problem is that I'm not quite sure how to interpret these equations. The (gluon) color indices {a, b, c} run from 1 to 8. But there are three kinds of color charge. So how do I interpret the sources [itex]\rho^a[/itex] and [itex]\vec J^a[/itex]?

One thing I attempted was to multiply both equations by the generators [itex]T^a_{ij}[/itex]. This eliminates the gluon color indices {a, b, c} in favor of the quark color indices ij (which then run from 1 to 3). But now there are two indices on everything! One for a color and one for an anti-color. Again, I can't quite figure out how to interpret what it means.