Abstract
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 no-regularization 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.
Highlights
Theory and, per se, describe a wide range of phenomena
We shall obtain the solution of this problem using the Hamiltonian formalism, i.e., we shall find the matrix elements of the finite time evolution operator generated by the nonstationary Hamiltonian of a general form for both bosons and fermions imposing rather mild assumption on the parameters of the Hamiltonian
For comparison we present here the expression for the matrix element of the evolution operator Utout,tin that is obtained without the Hamiltonian diagonalization procedure, i.e., written in terms of the creation-annihilation operators diagonalizing the Hamiltonian (1) without the source K A
Summary
The fact that we consider the evolution of QFT with nonstationary Hamiltonian during a finite interval of time plays a crucial role. Apart from the construction of unitary evolution, we shall investigate a class of observables that allow for removal of regularization even in the case when the dynamics are not unitary in this limit. The particles are defined by the Hamiltonian diagonalization procedure This definition of the representation of the algebra of observables in the Fock space is local in time, i.e., the representation is determined by the configuration of background fields at the present instant of time. 2, the general formalism is developed for description of quadratic QFTs. The regularization is defined by the energy cutoff of the generator of evolution in the time dependent basis diagonalizing the Hamiltonian of the system. The main papers and books that are immediately related to the subject matter and are known to the author are cited
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