  In both research and textbook literature one often finds two ``different''
Kubo formulas for the zero-temperature conductance of a non-interacting Fermi
system. They contain a trace of the product of velocity operators and
single-particle (retarded and advanced) Green operators: $\text{Tr} (\hat{v}_x
\hat{G}^r \hat{v}_x \hat{G}^a)$ or $\text{Tr} (\hat{v}_x \text{Im} \hat{G}
\hat{v}_x \text{Im} \hat{G})$. The study investigates the relationship between
these expressions, as well as the requirements of current conservation, through
exact evaluation of such quantum-mechanical traces for a nanoscale (containing
1000 atoms) mesoscopic disordered conductor. The traces are computed in the
semiclassical regime (where disorder is weak) and, more importantly, in the
nonperturbative transport regime (including the region around
localization-delocalization transition) where concept of mean free path ceases
to exist. Since quantum interference effects for such strong disorder are not
amenable to diagrammatic or nonlinear $\sigma$-model techniques, the evolution
of different Green function terms with disorder strength provides novel insight
into the development of an Anderson localized phase.
