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We suggest a new (dynamical) Abelian projection of the lattice QCD. Itcontains no gauge condition imposed on gauge fields so that Gribov copying isavoided. Configurations of gauge fields that turn into monopoles in the Abelianprojection can be classified in a gauge invariant way. In the continuum limit,the theory respects the Lorentz invariance. A similar dynamical reduction ofthe gauge symmetry is proposed for studies of gauge-variant correlators (like agluon propagator) in lattice QCD. Though the procedure is harder for numericalsimulations, it is free of gauge fixing artifacts, like the Gribov horizon andcopies.Comment: plain Latex, 12 page

ABELIAN PROJECTION AND STUDIES OF GAUGE-VARIANT QUANTITIES IN THE LATTICE QCD WITHOUT GAUGE FIXING