  The level crossing problem is neatly formulated by the second quantized
formulation, which exhibits a hidden local gauge symmetry. The analysis of
geometric phases is reduced to a simple diagonalization of the Hamiltonian. If
one diagonalizes the geometric terms in the infinitesimal neighborhood of level
crossing, the geometric phases become trivial (and thus no monopole
singularity) for arbitrarily large but finite time interval $T$. The
topological proof of the Longuet-Higgins' phase-change rule, for example, thus
fails in the practical Born-Oppenheimer approximation where $T$ is identified
with the period of the slower system. The crucial difference between the
Aharonov-Bohm phase and the geometric phase is explained. It is also noted that
the gauge symmetries involved in the adiabatic and non-adiabatic geometric
phases are quite different.
