Vol 198, No 3 (2019)

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge Z > Z_cr,1 and radius R₀, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom,” we study the behavior of both the vacuum density ρ_VP(r⃗) and the total induced charge and also the vacuum-polarization energy EVP. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for ρ_VP(r⃗) and ɛ_VP, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρ_VP and ɛ_VP. The most important result is that for Z ≫ Z_cr,1 in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z, which reaches large negative values and whose behavior is estimated from below (in absolute value) as ~ −|η_effZ³|/R₀. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with m_j = ±1/2,±3/2. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge Q_VP in the overcritical region in detail.

We consider the quantum dynamics of charge propagation over a two-dimensional lattice with impurity sites at the lattice edges. These sites simulate boundary (Tamm) states. We solve the nonstationary problem of the evolution of a quantum excitation over impurity sites at the lattice perimeter in the tightbinding approximation. We obtain the solution as an expansion in eigenfunctions of the unperturbed system Hamiltonian. We obtain analytically accurate results for the propagation of the wave function over impurity sites.

We formulate principal positions of a non-Hermitian model with a γ₅-extension of the fermion mass, which are often neglected in investigating this subject. A consistent approach to this problem requires the constraint m ≤ M, where M bounds the entire fermion mass spectrum. An analogous approach was proposed in the geometric model, which can be regarded as the first PT-symmetric non-Hermitian fermion model with a γ₅-extension of mass. Exotic particles appear in both these theories. A detailed consideration of the properties of these particles allows conjecturing that they are possible candidates in the structure of dark matter. We also discuss a simple estimate for determining the maximum admissible value of the fermion mass M.

We study Hawking radiation of relativistic particles from uncharged and charged black strings in 3+1 dimensions in detail. We use the method of quantum tunneling in the framework of the Hamilton–Jacobi approach. We show that the radial function of the action allows calculating the tunneling rate of the emitted relativistic particles. Using the Boltzmann formula, we derive the Hawking temperatures of uncharged and charged black strings. We discuss the influence of the temporal contribution to the tunneling rate.