D. M. Gitman

THE DIRAC HAMILTONIAN WITH A SUPERSTRONG COULOMB FIELD

AbstractWe consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge Ze. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with Z = α−1 = 137 because the standard expression for the lower bound-state energy yields complex values at overcritical charges. We show that from the mathematical standpoint, there is no problem in defining a self-adjoint Hamiltonian for any charge value. Furthermore, the transition through the critical charge does not lead to any qualitative changes in the mathematical description of the system. A specific feature of overcritical charges is a nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical (and larger than the subcritical charge with n

Quantization of a spinning particle with anomalous magnetic moment

Z = (sqrt 3 /2)alpha ^{ - 1} = 118

SYMMETRIES AND PHYSICAL FUNCTIONS IN GENERAL GAUGE THEORY

n). We present the spectra and (generalized) eigenfunctions for all self-adjoint Hamiltonians. We use the methods of the theory of self-adjoint extensions of symmetric operators and the Krein method of guiding functionals. The relation of the constructed one-particle quantum mechanics to the real physics of electrons in superstrong Coulomb fields where multiparticle effects may be crucially important is an open question.

Stability of a non-commutative Jackiw–Teitelboim gravity

The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0 + 1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.

Quantization of the damped harmonic oscillator revisited

A generalization of the pseudoclassical action of a spinning particle in the presence of an anomalous magnetic moment is given. The leading considerations, to write the action, are gotten from the path-integral representation for the causal Green function of the generalized (by Pauli) Dirac equation for the particle with anomalous magnetic moment in an external electromagnetic field. The action can be written in reparameterization and supergauge-invariant form. Both operator (Dirac) and path-integral quantization are discussed. The first one leads to the Dirac-Pauli equation whereas the second one gives the corresponding propagator. One of the non-trivial points in this case is that both quantization schemes demand, for consistency, that one takes into account an operator ordering problem.A generalization of the pseudoclassical action of a spinning particle in the presence of an anomalous magnetic moment is given. The leading considerations, to write the action, are gotten from the path integral representation for the causal Greens function of the generalized (by Pauli) Dirac equation for the particle with anomalous magnetic momentum in an external electromagnetic field. The action can be written in reparametrization and supergauge invariant form. Both operator (Dirac) and path-integral (BFV) quantization are discussed. The first one leads to the Dirac-Pauli equation, whereas the second one gives the corresponding propagator. One of the nontrivial points in this case is that both quantizations schemes demand for consistency to take into account an operators ordering problem.

Coherent states of SU(l, 1) groups

Reparametrization invariance treated as a gaugesymmetry shows some specific peculiarities. We studythese peculiarities both from a general point of viewand by concrete examples. We consider the canonical treatment of reparametrization-invariantsystems in which one fixes the gauge on the classicallevel by means of time-dependent gauge conditions. Insuch an approach one can interpret different gauges as different reference frames. We discuss therelation between different gauges and the problem ofgauge invariance in this case. Finally, we establish ageneral structure of reparametrizations and itsconnection with the zero-Hamiltonian phenomenon.

Coherent states of the SU(N) groups

The aim of the present paper is to describe the symmetry structure of a general gauge (singular) theory, and, in particular, to relate the structure of gauge transformations with the constraint structure of a theory in the Hamiltonian formulation. We demonstrate that the symmetry structure of a theory action can be completely revealed by solving the so-called symmetry equation. We develop a corresponding constructive procedure of solving the symmetry equation with the help of a special orthogonal basis for the constraints. Thus, we succeed in describing all the gauge transformations of a given action. We find the gauge charge as a decomposition in the orthogonal constraint basis. Thus, we establish a relation between the constraint structure of a theory and the structure of its gauge transformations. In particular, we demonstrate that, in the general case, the gauge charge cannot be constructed with the help of some complete set of first-class constraints alone, because the charge decomposition also contains second-class constraints. The above-mentioned procedure of solving the symmetry equation allows us to describe the structure of an arbitrary symmetry for a general singular action. Finally, using the revealed structure of an arbitrary gauge symmetry, we give a rigorous proof of the equivalence of two definitions of physicality condition in gauge theories: one of them states that physical functions are gauge-invariant on the extremals, and the other requires that physical functions commute with FCC (the Dirac conjecture).

Darboux transformation for two-level system

We start with a non-commutative version of the Jackiw–Teitelboim gravity in two dimensions which has a linear potential for the dilaton fields. We study whether it is possible to deform this model by adding quadratic terms to the potential but preserving the number of gauge symmetries. We find that no such deformation exists (provided one does not twist the gauge symmetries).