Том 187, № 2 (2016)

We consider the peculiarities of the electron energy spectrum in the Coulomb field of a superheavy nucleus and discuss the long history of an incorrect interpretation of this problem in the case of a pointlike nucleus and its current correct solution. We consider the spectral problem in the case of a regularized Coulomb potential. For some special regularizations, we derive an exact equation for the point spectrum in the energy interval (-m,m) and find some of its solutions numerically. We also derive an exact equation for charges yielding bound states with the energy E = -m; some call them supercritical charges. We show the existence of an infinite number of such charges. Their existence does not mean that the oneparticle relativistic quantum mechanics based on the Dirac Hamiltonian with the Coulomb field of such charges is mathematically inconsistent, although it is physically unacceptable because the spectrum of the Hamiltonian is unbounded from below. The question of constructing a consistent nonperturbative second-quantized theory remains open, and the consequences of the existence of supercritical charges from the standpoint of the possibility of constructing such a theory also remain unclear.

We review the Reshetikhin–Turaev approach for constructing noncompact knot invariants involving Rmatrices associated with infinite-dimensional representations, primarily those constructed from the Faddeev quantum dilogarithm. The corresponding formulas can be obtained from modular transformations of conformal blocks as their Kontsevich–Soibelman monodromies and are presented in the form of transcendental integrals, where the main issue is working with the integration contours. We discuss possibilities for extracting more explicit and convenient expressions that can be compared with the ordinary (compact) knot polynomials coming from finite-dimensional representations of simple Lie algebras, with their limits and properties. In particular, the quantum A-polynomials and difference equations for colored Jones polynomials are the same as in the compact case, but the equations in the noncompact case are homogeneous and have a nontrivial right-hand side for ordinary Jones polynomials.

The associative algebra of symplectic reflections \(H: = {H_{1,{v_{1,}}{v_2}}}\left( {{I_2}\left( {2m} \right)} \right)\) based on the group generated by the root system I₂(2m) depends on two parameters, ν₁ and ν₂. For each value of these parameters, the algebra admits an m-dimensional space of traces. A trace tr is said to be degenerate if the corresponding symmetric bilinear form B_tr(x, y) = tr(xy) is degenerate. We find all values of the parameters ν₁ and ν₂ for which the space of traces contains degenerate traces and the algebra H consequently has a two-sided ideal. It turns out that a linear combination of degenerate traces is also a degenerate trace. For the ν₁ and ν₂ values corresponding to degenerate traces, we find the dimensions of the space of degenerate traces.

In the framework of the BRST-BV approach to the formulation of relativistic mechanics, we consider massless and massive fields of arbitrary spin propagating in a flat space and massless fields propagating in the AdS space. For such fields, we obtain BRST-BV Lagrangians invariant under gauge transformations. The Lagrangians and gauge transformations are constructed in terms of traceless gauge fields and traceless parameters of the gauge transformations. We consider the fields in the AdS space using the Poincaré parameterization of this space, which leads to a simple form of the BRST-BV Lagrangian. We show that in the Siegel gauge, the Lagrangian of the massless AdS fields leads to a decoupling of the equations of motion, and this substantially simplifies the study of the AdS/CFT correspondence. In a conformal algebra basis, we find a realization of the relativistic symmetries of fields and antifields in the AdS space.

We study the collective behavior of a system of Brownian agents each of which moves orienting itself to the group as a whole. This system is the simplest model of the motion of a “united drunk company.” For such a system, we use the functional integration technique to calculate the probability of transition from one point to another and to determine the time dependence of the probability density to find a member of the “drunk company” near a given point. It turns out that the system exhibits an interesting collective behavior at large times and this behavior cannot be described by the simplest mean-field-type approximation. We also obtain an exact solution in the case where one of the agents is “sober” and moves along a given trajectory. The obtained results are used to discuss whether such systems can be described by different theoretical approaches.

We construct a generalized Weyl correspondence for an electrically charged particle in the field of the Dirac magnetic monopole. Our starting points are a global Lagrangian description of this system as a constrained system with U(1) gauge symmetry given in terms of the fiber bundle theory and a reduction of the presymplectic structure arising on the constraint surface. In contrast to the recently proposed quantization scheme based on using a quaternionic Hilbert module, the quantum operators corresponding to classical observables in our construction act in the complex Hilbert space of U(1)-equivariant functions introduced by Greub and Petry. These functions are defined on the total space of a fiber bundle that is topologically equivalent to the Hopf fibration.