- #1
michael879
- 698
- 7
I've come across countless sources that gauge fix SU(N) Yang-Mills fields using the typical U(1) gauges (e.g. Lorenz gauge, coulomb gauge, temporal gauge, etc). However, I can't find a single one where they prove that this gauge fixing is valid for all field configurations... I've tried to derive it myself, but the non-abelian nature of the problem makes it very difficult. I'll sketch out the basics below, if anyone can prove this to me or point me to a proof I would really appreciate it. I'm especially interested in a proof of the existence of a Lorenz gauge transformation for any field configuration in either SU(2) or the more general SU(N).
SU(N) vector potential is written as: [itex]A_\mu \equiv A^a_\mu T_a[/itex] where [itex]T_a[/itex] are the SU(N) generators and [itex][T_a,T_b] = if_{abc}T^c[/itex] for structure constants [itex]f_{abc}[/itex]. For SU(2) the generators are the Pauli matrices (divided by 2).
The vector potential transforms as: [itex]A_\mu \rightarrow UA_\mu U^\dagger +\dfrac{i}{g}U\partial_\mu U^\dagger[/itex] where [itex]U\equiv e^{i\theta}[/itex] is an element of SU(2) and [itex]\theta\equiv \theta^a T_a[/itex].
What I am trying to prove is that for any [itex]A_\mu[/itex] there exists a U such that [itex]\partial^\mu A'_\mu = 0[/itex] (or another gauge condition). For the U(1) case its very straightforward and you end up with a wave equation for [itex]\theta[/itex] with a source term which is a function of A. However for the general SU(N) case I'm stuck.. Even for the specific case of SU(2) I can't prove its existence (I would be happy to just have a proof for this case)!
SU(N) vector potential is written as: [itex]A_\mu \equiv A^a_\mu T_a[/itex] where [itex]T_a[/itex] are the SU(N) generators and [itex][T_a,T_b] = if_{abc}T^c[/itex] for structure constants [itex]f_{abc}[/itex]. For SU(2) the generators are the Pauli matrices (divided by 2).
The vector potential transforms as: [itex]A_\mu \rightarrow UA_\mu U^\dagger +\dfrac{i}{g}U\partial_\mu U^\dagger[/itex] where [itex]U\equiv e^{i\theta}[/itex] is an element of SU(2) and [itex]\theta\equiv \theta^a T_a[/itex].
What I am trying to prove is that for any [itex]A_\mu[/itex] there exists a U such that [itex]\partial^\mu A'_\mu = 0[/itex] (or another gauge condition). For the U(1) case its very straightforward and you end up with a wave equation for [itex]\theta[/itex] with a source term which is a function of A. However for the general SU(N) case I'm stuck.. Even for the specific case of SU(2) I can't prove its existence (I would be happy to just have a proof for this case)!