Inclusive probability of particle creation on classical backgrounds
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Regular Article - Theoretical Physics
Inclusive probability of particle creation on classical backgrounds Peter Kazinskia Physics Faculty, Tomsk State University, Tomsk 634050, Russia
Received: 15 March 2020 / Accepted: 1 August 2020 © The Author(s) 2020
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) ∈ [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.
1 Introduction The quantum field theories (QFTs) with quadratic Hamiltonians are the classical subject for investigation in theoretical physics. These models represent the base for perturbation a e-mail:
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theory and, per se, describe a wide range of phenomena. It is not surprising that there is a huge literature devoted to this subject (see, e.g., the books [1–12]). Although it appears that the problem of description of QFTs with quadratic Hamiltonians having linear equations of motion is somewhat trivial, the presence of infinite degrees of freedom considerably complicates the issue. There are different methods to tackle this problem and perhaps the most straightforward Hamiltonian approach is not the commonly used one. Our aim is to fill this gap. 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 paramet
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