Paper detail

Inclusive probability of particle creation on classical backgrounds

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.