  The canonical commutation relations in quantum mechanics are not maintained
in the anomalous Hall effect described by Berry's phase in the presence of the
electromagnetic vector potential. To define quantum mechanical formulation, one
may avoid the electromagnetic vector potential but then the anomalous Nernst
effect induced by Berry's curvature is not described using a modified phase
space volume, although the anomalous Nernst effect by itself may be generated
by the adiabatic Berry's curvature. We also comment on the Born-Oppenheimer
approximation if it describes the anomalous Hall effect in the absence of the
electromagnetic vector potential in quantum mechanics. An alternative view of
the Bjorken-Johnson-Low prescription which is consistent with the principle of
quantum mechanics and the existing Berry's phase theory of the anomalous Hall
effect is also mentioned.
