  We develop a formalism to describe the particle production out of equilibrium
in terms of dynamical spectral functions, i.e. Wigner transformed
Pauli-Jordan's and Hadamard's functions. We take an explicit example of a
spatially homogeneous scalar theory under pulsed electric fields and
investigate the time evolution of the spectral functions. In the out-state we
find an oscillatory peak in Hadamard's function as a result of the mixing
between positive- and negative-energy waves. The strength of this peak is of
the linear order of the Bogoliubov mixing coefficient, whereas the peak
corresponding to the Schwinger mechanism is of the quadratic order. Between the
in- and the out-states we observe a continuous flow of the spectral peaks
together with two transient oscillatory peaks. We also discuss the medium
effect at finite temperature and density. We emphasise that the entire
structure of the spectral functions conveys rich information on real-time
dynamics including the particle production.
