  In view of the recent applications of chiral anomaly to various fields beyond
particle physics, we discuss some basic aspects of chiral anomaly which may
help deepen our understanding of chiral anomaly in particle physics also. It is
first shown that Berry's phase (and its generalization) for the Weyl model $H
=v_{F} \vec{\sigma}\cdot \vec{p}(t)$ assumes a monopole form at the exact
adiabatic limit but deviates from it off the adiabatic limit and vanishes in
the high frequency limit of the Fourier transform of $\vec{p}(t)$ for bounded
$|\vec{p}(t)|$. An effective action, which is consistent with the non-adiabatic
limit of Berry's phase, combined with the Bjorken-Johnson-Low prescription
gives normal equal-time space-time commutators and no chiral anomaly. In
contrast, an effective action with a monopole at the origin of the momentum
space, which describes Berry's phase in the precise adiabatic limit but fails
off the adiabatic limit, gives anomalous space-time commutators and a covariant
anomaly to the gauge current. We regard this anomaly as an artifact of the
postulated monopole and not a consequence of Berry's phase. As for the recent
application of the chiral anomaly to the description of effective Weyl fermions
in condensed matter and nuclear physics, which is closely related to the
formulation of lattice chiral fermions, we point out that the chiral anomaly
for each species doubler separately vanishes for a finite lattice spacing,
contrary to the common assumption. Instead a general form of pair creation
associated with the spectral flow for the Dirac sea with finite depth takes
place. This view is supported by the Ginsparg-Wilson fermion, which defines a
single Weyl fermion without doublers on the lattice and gives a well-defined
index (anomaly) even for a finite lattice spacing.
