We identify an extended phase space P for minimally interacting Yang-Mills and Dirac fields in the Minkowski space. It is a Sobolev space of Cauchy data for which we prove the finite time existence and uniqueness of the evolution equations. We prove that the Lie algebra gs(P) of all infinitesimal gauge symmetries of P is a Hilbert-Lie algebra, carrying a Beppo Levi topology. The connected group GS(P) of the gauge symmetries generated by gs(P) is proved to be a Hilbert-Lie group acting properly in P. The Lie algebra gs(P) has a maximal ideal gs(P)0. We prove that the action in P of the connected group GS(P)0 generated by gs(P)0 is proper and free. The constraint set is shown to be the zero level of the equivariant momentum map corresponding to the action of GS(P)0 in P.
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