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We propose the reformulations of the $SU(N)$ Yang-Mills theory toward quarkconfinement and mass gap. In fact, we have given a new framework forreformulating the $SU(N)$ Yang-Mills theory using new field variables. Thisincludes the preceding works given by Cho, Faddeev and Niemi, as a special casecalled the maximal option in our reformulations. The advantage of ourreformulations is that the original non-Abelian gauge field variables can bechanged into the new field variables such that one of them called therestricted field gives the dominant contribution to quark confinement in thegauge-independent way. Our reformulations can be combined with the $SU(N)$extension of the Diakonov-Petrov version of the non-Abelian Stokes theorem forthe Wilson loop operator to give a gauge-invariant definition for the magneticmonopole in the $SU(N)$ Yang-Mills theory without the scalar field. In theso-called minimal option, especially, the restricted field is non-Abelian andinvolves the non-Abelian magnetic monopole with the stability group $U(N-1)$.This suggests the non-Abelian dual superconductivity picture for quarkconfinement. This should be compared with the maximal option: the restrictedfield is Abelian and involves only the Abelian magnetic monopoles with thestability group $U(1)^{N-1}$, just like the Abelian projection. We give someapplications of this reformulation, e.g., the stability for the homogeneouschromomagnetic condensation of the Savvidy type, the large N treatment forderiving the dimensional transmutation and understanding the mass gap, and alsothe numerical simulations on a lattice which are given by Dr. Shibata in asubsequent talk.

Reformulations of the Yang-Mills theory toward quark confinement and mass gap