To define the Yang-Mills Lagrangian, we need to define the ‘Trace’ of an End(E) valued form. Recall that the Trace of a matrix is the sum of its diagonal enTries. The Trace is independent of the choice of basis – an invariant notion that is independent of the choice of basis. A definition of the Trace that mkes this clear is as follows. Consider – an isomorphism that does not depend on any choice of basis – so the pairing between V and
defines a linear map
Now we can write down the Yang-Mills Lagrangian: If D is a connection on E, this is the n-form given by
where F is the curvature of D. Note that by the defintion of the hodge star operator (also in this collection of notes), we can write this in local co-ordinates as
If we integrate
This needs some elaboration. So let us explain these formulas better. We choose the physics convention
by another convention (there is a lot of ambiguity of signs in this subject). The first thing to note is that F has vector and Lie algebra indices. The Trace is over the Lie algebra, not over the vector indices. The vector indices are just those of the field sTrength in QED. In Yang-Mills the curvature form is Lie Algebra valued. \newline In this case
You may want to look at what I just posted on my Blog. It shows an exact solution for the Yang-Mills Largrangian which quantizes the theory and yields an exact propagator.
PS: My correct blog link is http://jayryablon.wordpress.com/2012/05/24/baryons-and-confinement-exact-quantum-yang-mills-propagators-mass-gap/