Construction of a quantum Yang-Mills theory over the Minkowski space

A quantization procedure for the Yang-Mills equations for the Minkowski space R 1,3 is carried out in such a way that fi eld maps satisfying Wightman axioms of Constructive Quantum Field Theory can be obtained. Moreover, by removing the infrared and ultraviolet cutoff s, the spectrum of the corresponding (non-local) QCD Hamilton operator is proven to be positive and bounded away from zero, except for the case of the vacuum state, which has vanishing energy level. The whole construction is invariant for all gauge transformations preserving the Coulomb gauge. As expected from QED, if the coupling constant converges to zero, then so does the mass gap. This is the case for the running coupling constant leading to asymptotic freedom.