  Schwinger pair creation in a purely time-dependent electric field can be
described through a quantum Vlasov equation describing the time evolution of
the single-particle momentum distribution function. This equation exists in two
versions, both of which can be derived by a Bogoliubov transformation, but
whose equivalence is not obvious. For the spinless case, we show here that the
difference between these two evolution equations corresponds to the one between
the "in-out" and "in-in" formalisms. We give a simple relation between the
asymptotic distribution functions generated by the two Vlasov equations. As
examples we discuss the Sauter and single-soliton field cases.
