  Aiming at the relation between QCD and the quark model, we consider
projections of gauge configurations generated in quenched lattice QCD
simulations in the Coulomb gauge on a 16$^{\rm 3}$ $\rm \times$ 32, $\rm \beta$
= 6.0 lattice. First, we focus on a fact that the static quark-antiquark
potential is independent of spatial gauge fields. We explicitly confirm this by
performing $\vec{A}$ = 0 projection, where spatial gauge fields are all set to
zero. We also apply the $\vec{A}$ = 0 projection to light hadron masses and
find that nucleon and delta baryon masses are almost degenerate, suggesting
vanishing of the color-magnetic interactions. After considering the physical
meaning of the $\vec{A}$ = 0 projection, we next propose a generalized
projection, where spatial gauge fields are expanded in terms of Faddeev-Popov
eigenmodes and only some eigenmodes are left. We apply the proposed projection
to light hadron and glueball masses and find that the N-$\rm \Delta$ and
0$^{\rm ++}$-2$^{\rm ++}$ mass splittings become evident when projected with
more than 33 (0.10 \%) low-lying eigenmodes, suggesting emergence of the
color-magnetic interactions. We also find that the original hadron masses are
approximately reproduced with just 328 (1.00 \%) low-lying eigenmodes. These
findings indicate an important role of low-lying eigenmodes on hadron masses
and would be useful in clarifying the relation between QCD and the quark model.
