  The nonadiabatic quantum Vlasov equation in spinor QED is derived, and its
relation to the well-known adiabatic one is established by three methods. One
is by an explicitly analytical expression, the second is by the Dirac equation
in the V gauge, and the last is by introducing a turn-off electric field.
Wherein what the first two of them are given is an instantaneous relation.
Moreover, the time evolution of the distribution function for a specific
momentum and the momentum distribution of created particle pairs after turning
off the electric field are calculated and compared with those in scalar QED. It
is found that both the oscillation periods of the distribution functions in
spinor and scalar QED equal pi divided by the total energy of a particle after
the electric field is turned off. The momentum distributions in spinor and
scalar QED show a novel oscillation and out-of-phase behavior that cannot be
explained by the Stokes phenomenon. These findings will further deepen our
understanding of the quantum Vlasov equation and its application in vacuum pair
production.
