  The quantum theories of boson and fermion fields with quadratic nonstationary
Hamiltoanians are rigorously constructed. The representation of the algebra of
observables is given by the Hamiltonian diagonalization procedure. The
sufficient conditions for the existence of unitary dynamics at finite times are
formulated and the explicit formula for the matrix elements of the evolution
operator is derived. In particular, this gives the well-defined expression for
the one-loop effective action. The ultraviolet and infrared divergencies are
regularized by the energy cutoff in the Hamiltonian of the theory. The possible
infinite particle production is regulated by the corresponding counterdiabatic
terms. The explicit formulas for the average number of particles $N_D$ recorded
by the detector and for the probability $w(D)$ to record a particle by the
detector are derived. It is proved that these quantities allow for the
regularization removal limit and, in this limit, $N_D$ is finite and
$w(D)\in[0,1)$. As an example, the theory of a neutral boson field with
stationary quadratic part of the Hamiltonian and nonstationary source is
considered. The average number of particles produced by this source from the
vacuum during a finite time evolution and the inclusive probability to record a
created particle are obtained. The infrared and ultraviolet asymptotics of the
average density of created particles are derived. As a particular case, quantum
electrodynamics with a classical current is considered. The ultraviolet and
infrared asymptotics of the average number of photons are derived. The
asymptotics of the average number of photons produced by the adiabatically
driven current is found.
