An existence and uniqueness theorem for the Cauchy problem for the evolution component of the coupled Yang-Mills and Dirac equations in the Minkowski space is proved in a Sobolev space for the temporal gauge condition. The constraint set C is shown to be a smooth submanifold of P preserved by the evolution. The Lie algebra gs(P) of infinitesimal gauge symmetries of P is identified. Its topology is of Beppo Levi type. The group GS(P) of gauge symmetries is of Lie type; its topology is induced by the topology of its Lie algebra. The constraint equations define a closed ideal gs(P)o of gs(P). It generates a closed connected subgroup GS(P)o of GS(P), which is showh to act properly in P. The reduced phase space is the space of GS(P)o orbits in the constraint set C. It is a smooth quotient manifold of C endowed with an exact symplectic form. The quotient group GS(P)/GS(P)o is isomorphic to the structure group G of the theory. Its action in the reduced phase space is Hamiltonian. The associated conserved quantities are colour charges. Only the charges corresponding to the centre of the Lie algebra gs(P)/ gs(P)o admit well defined local charge densities.
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