  In the lattice QCD formalism, we derive a gauge-invariant analytical relation
connecting the Polyakov loop and the Dirac modes on a temporally odd-number
lattice, where the temporal lattice size is odd, with the normal (nontwisted)
periodic boundary condition. This analytical relation indicates that low-lying
Dirac modes have little contribution to the Polyakov loop. Using lattice QCD
simulations, we numerically confirm the analytical relation and the negligible
contribution of low-lying Dirac modes to the Polyakov loop at the quenched
level, i.e., the Polyakov loop is almost unchanged by removing low-lying
Dirac-mode contribution from the QCD vacuum generated by lattice QCD in both
confinement and deconfinement phases. Thus, we conclude that there is no
one-to-one correspondence between confinement and chiral symmetry breaking in
QCD. As a new method, modifying the Kogut-Susskind formalism, we develop a
method for spin-diagonalizing the Dirac operator on the temporally odd-number
lattice.
