The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge

We study the Cauchy problem for minimally coupled classical Yang-Mills and scalar fields inn+1 dimensional space-time in the temporal gauge. We prove the existence and uniqueness of solutions for small time intervals and for anyn. We then develop a general theory of solutions in local spaces and extend the previous local (in time) results to this more general setting. In space-time dimensions two and three, we prove the existence of global (in time) solutions by the method of a priori estimates, both in global and local spaces. In space-time dimension four, our estimates yield only partial results on the global existence problem.Laboratoire associé au Centre National de la Recherche Scientifique