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The quantum theories of boson and fermion fields with quadratic nonstationaryHamiltoanians are rigorously constructed. The representation of the algebra ofobservables is given by the Hamiltonian diagonalization procedure. Thesufficient conditions for the existence of unitary dynamics at finite times areformulated and the explicit formula for the matrix elements of the evolutionoperator is derived. In particular, this gives the well-defined expression forthe one-loop effective action. The ultraviolet and infrared divergencies areregularized by the energy cutoff in the Hamiltonian of the theory. The possibleinfinite particle production is regulated by the corresponding counterdiabaticterms. The explicit formulas for the average number of particles $N_D$ recordedby the detector and for the probability $w(D)$ to record a particle by thedetector are derived. It is proved that these quantities allow for theregularization 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 withstationary quadratic part of the Hamiltonian and nonstationary source isconsidered. The average number of particles produced by this source from thevacuum during a finite time evolution and the inclusive probability to record acreated particle are obtained. The infrared and ultraviolet asymptotics of theaverage density of created particles are derived. As a particular case, quantumelectrodynamics with a classical current is considered. The ultraviolet andinfrared asymptotics of the average number of photons are derived. Theasymptotics of the average number of photons produced by the adiabaticallydriven current is found.

Inclusive probability of particle creation on classical backgrounds