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