In Wald Chapter 10, he discusses the initial value formalism of electromagnetism - how Maxwell's equations are actually a system of three equations plus an initial value constraint, and how we can resolve this issue by fixing a gauge. He took the Lorenz gauge $\partial_{\mu}A^{\mu}$, in which Maxwell's equations become $\partial^2A_{\nu} = 0$, and showed that as $\partial^2(\partial_{\mu}A^{\mu})=0$, if the Lorenz gauge holds initially, it will hold throughout. So, he showed that for any initial values of $A_{\mu}$ and $n^{\mu}A_{\mu}$ given on a spacelike hypersurface with normal $n^{\mu}$, Maxwell's equations have a unique solution (upto gauge transformation) which depends continuously on the initial parameters.

I was interested in whether we can do something like this for the Yang-Mills equations, and so tried to imitate the EM proof. However, I faced a problem with the gauge fixing. I was not able to show that the Lorenz gauge for the Yang-Mills equations holds throughout, and so was not able to proceed from there.

So, is it possible to have an initial value formulation of Yang-Mills equations just as for Maxwell's equations?



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