  The level crossing problem and associated geometric terms are neatly
formulated by using the second quantization technique both in the operator and
path integral formulations. The analysis of geometric phases is then reduced to
the familiar diagonalization of the Hamiltonian. If one diagonalizes the
Hamiltonian in one specific limit, one recovers the conventional formula for
geometric phases. On the other hand, 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 a large but finite ratio of two time scales is involved and $T$ is
identified with the period of the slower system.
