  Some many-body theories of optical transitions in solids were examined from
projection and population criteria. The results showed that state-independent
projection methods cannot be applied to electron systems with non-uniform
energy spectra. Moreover, neglecting some important terms in a second-order
approximation leads to invalidation of the population criterion. In addition, a
valid theory satisfying these two criteria can be obtained using a proper
state-dependent projection operator and Kang-Choi reduction identity, and the
result can be interpreted using a diagram, which can model the quantum dynamics
of electron in solids.
