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D. M. Gitman

Publications and source records attributed to D. M. Gitman.

At least 127 records · Page 7Linked to original sources
arXiv2005

Symmetries and physical functions in general gauge theory

The aim of the present article 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).

hep-th↗
arXiv2005

Non-Volkov solutions for a charge in a plane wave

We focus our attention, once again, on the Klein--Gordon and Dirac equations with a plane-wave field. We recall that for the first time a set of solutions of these equations was found by Volkov. The Volkov solutions are widely used in calculations of quantum effects with electrons and other elementary particles in laser beams. We demonstrate that one can construct sets of solutions which differ from the Volkov solutions and which may be useful in physical applications. For this purpose, we show that the transversal charge motion in a plane wave can be mapped by a special transformation to transversal free particle motion. This allows us to find new sets of solutions where the transversal motion is characterized by quantum numbers different from Volkov's (in the Volkov solutions this motion is characterized by the transversal momentum). In particular, we construct solutions with semiclassical transversal charge motion (transversal squeezed coherent states). In addition, we demonstrate how the plane-wave field can be eliminated from the transversal charge motion in a more complicated case of the so-called combined electromagnetic field (a combination of a plane-wave field and constant colinear electric and magnetic fields). Thus, we find new sets of solutions of the Klein--Gordon and Dirac equations with the combined electromagnetic field.

hep-th↗
arXiv2005

Symmetries of dynamically equivalent theories

A natural and very important development of constrained system theory is a detail study of the relation between the constraint structure in the Hamiltonian formulation with specific features of the theory in the Lagrangian formulation, especially the relation between the constraint structure with the symmetries of the Lagrangian action. An important preliminary step in this direction is a strict demonstration, and this is the aim of the present article, that the symmetry structures of the Hamiltonian action and of the Lagrangian action are the same. This proved, it is sufficient to consider the symmetry structure of the Hamiltonian action. The latter problem is, in some sense, simpler because the Hamiltonian action is a first-order action. At the same time, the study of the symmetry of the Hamiltonian action naturally involves Hamiltonian constraints as basic objects. One can see that the Lagrangian and Hamiltonian actions are dynamically equivalent. This is why, in the present article, we consider from the very beginning a more general problem: how the symmetry structures of dynamically equivalent actions are related. First, we present some necessary notions and relations concerning infinitesimal symmetries in general, as well as a strict definition of dynamically equivalent actions. Finally, we demonstrate that there exists an isomorphism between classes of equivalent symmetries of dynamically equivalent actions.

hep-th↗
arXiv2004

On Superfield Covariant Quantization in General Coordinates

We propose a natural extension of the BRST-antiBRST superfield covariant scheme in general coordinates. Thus, the coordinate dependence of the basic scalar and tensor fields of the formalism is extended from the base supermanifold to the complete set of superfield variables.

hep-th↗
arXiv2004

General quadratic gauge theory. Constraint structure, symmetries, and physical functions

How can we relate the constraint structure and constraint dynamics of the general gauge theory in the Hamiltonian formulation with specific features of the theory in the Lagrangian formulation, especially relate the constraint structure with the gauge transformation structure of the Lagrangian action? How can we construct the general expression for the gauge charge if the constraint structure in the Hamiltonian formulation is known? Whether can we identify the physical functions defined as commuting with first-class constraints in the Hamiltonian formulation and the physical functions defined as gauge invariant functions in the Lagrangian formulation? The aim of the present article is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions. To fulfill such a program, we demonstrate the existence of the so-called superspecial phase-space variables in terms of which the quadratic Hamiltonian action takes a simple canonical form. On the basis of such a representation, we analyze a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analyzing a symmetry equation. We then use these results to identify the two definitions of physical functions and thus prove the Dirac conjecture.

hep-th↗
arXiv2004

Darboux transformation for two-level systems

We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level system, transforming only one real potential into another real potential. We apply the obtained Darboux transformation to known exact solutions of the two-level system. Thus, we find three classes of new solutions for the two-level system and the corresponding new potentials that allow such solutions.

math-ph↗
arXiv2004

Symmetries in Constrained Systems

We describe symmetry structure of a general singular theory (theory with constraints in the Hamiltonian formulation), and, in particular, we relate the structure of gauge transformations with the constraint structure. We show that any symmetry transformation can be represented as a sum of three kinds of symmetries: global, gauge, and trivial symmetries. We construct explicitly all the corresponding conserved charges as decompositions in a special constraint basis. The global part of a symmetry does not vanish on the extremals, and the corresponding charge does not vanish on the extremals as well. The gauge part of a symmetry does not vanish on the extremals, but the gauge charge vanishes on them. We stress that the gauge charge necessarily contains a part that vanishes linearly in the first-class constraints and the remaining part of the gauge charge vanishes quadratically on the extremals. The trivial part of any symmetry vanishes on the extremals, and the corresponding charge vanishes quadratically on the extremals.

hep-th↗
arXiv2004

Dirac fermions in a magnetic-solenoid field

We consider the Dirac equation with a magnetic-solenoid field (the superposition of the Aharonov--Bohm solenoid field and a collinear uniform magnetic field). Using von Neumann's theory of the self-adjoint extensions of symmetric operators, we construct a one-parameter family and a two-parameter family of self-adjoint Dirac Hamiltonians in the respective 2+1 and 3+1 dimensions. Each Hamiltonian is specified by certain asymptotic boundary conditions at the solenoid. We find the spectrum and eigenfunctions for all values of the extension parameters. We also consider the case of a regularized magnetic-solenoid field (with a finite-radius solenoid field component) and study the dependence of the eigenfunctions on the behavior of the magnetic field inside the solenoid. The zero-radius limit yields a concrete self-adjoint Hamiltonian for the case of the magnetic-solenoid field. In addition, we consider the spinless particle in the regularized magnetic-solenoid field. By the example of the radial Dirac Hamiltonian with the magnetic-solenoid field, we present an alternative, more simple and efficient, method for constructing self-adjoint extensions applicable to a wide class of singular differential operators.

hep-th↗
arXiv2004

Green functions of the Dirac equation with magnetic-solenoid field

Various Green functions of the Dirac equation with a magnetic-solenoid field (the superposition of the Aharonov-Bohm field and a collinear uniform magnetic field) are constructed and studied. The problem is considered in 2+1 and 3+1 dimensions for the natural extension of the Dirac operator (the extension obtained from the solenoid regularization). Representations of the Green functions as proper time integrals are derived. The nonrelativistic limit is considered. For the sake of completeness the Green functions of the Klein-Gordon particles are constructed as well.

math-ph↗
arXiv2004

Quantization of (2+1)-spinning particles and bifermionic constraint problem

This work is a natural continuation of our recent study in quantizing relativistic particles. There it was demonstrated that, by applying a consistent quantization scheme to a classical model of a spinless relativistic particle as well as to the Berezin-Marinov model of 3+1 Dirac particle, it is possible to obtain a consistent relativistic quantum mechanics of such particles. In the present article we apply a similar approach to the problem of quantizing the massive 2+1 Dirac particle. However, we stress that such a problem differs in a nontrivial way from the one in 3+1 dimensions. The point is that in 2+1 dimensions each spin polarization describes different fermion species. Technically this fact manifests itself through the presence of a bifermionic constant and of a bifermionic first-class constraint. In particular, this constraint does not admit a conjugate gauge condition at the classical level. The quantization problem in 2+1 dimensions is also interesting from the physical viewpoint (e.g. anyons). In order to quantize the model, we first derive a classical formulation in an effective phase space, restricted by constraints and gauges. Then the condition of preservation of the classical symmetries allows us to realize the operator algebra in an unambiguous way and construct an appropriate Hilbert space. The physical sector of the constructed quantum mechanics contains spin-1/2 particles and antiparticles without an infinite number of negative-energy levels, and exactly reproduces the one-particle sector of the 2+1 quantum theory of a spinor field.

hep-th↗
arXiv2003

On Modifications of the Sp(2) Covariant Superfield Quantization

We propose a modification of the Sp(2) covariant superfield quantization to realize a superalgebra of generating operators isomorphic to the massless limit of the corresponding superalgebra of the osp(1,2) covariant formalism. The modified scheme ensures the compatibility of the superalgebra of generating operators with extended BRST symmetry without imposing restrictions eliminating superfield components from the quantum action. The formalism coincides with the Sp(2) covariant superfield scheme and with the massless limit of the osp(1,2) covariant quantization in particular cases of gauge-fixing and solutions of the quantum master equations.

hep-th↗
arXiv2003

Dirac equation in the magnetic-solenoid field

We consider the Dirac equation in the magnetic-solenoid field (the field of a solenoid and a collinear uniform magnetic field). For the case of Aharonov-Bohm solenoid, we construct self-adjoint extensions of the Dirac Hamiltonian using von Neumann's theory of deficiency indices. We find self-adjoint extensions of the Dirac Hamiltonian in both above dimensions and boundary conditions at the AB solenoid. Besides, for the first time, solutions of the Dirac equation in the magnetic-solenoid field with a finite radius solenoid were found. We study the structure of these solutions and their dependence on the behavior of the magnetic field inside the solenoid. Then we exploit the latter solutions to specify boundary conditions for the magnetic-solenoid field with Aharonov-Bohm solenoid.

hep-th↗
arXiv2003

On Problems of the Lagrangian Quantization of W3-gravity

We consider the two-dimensional model of W3-gravity within Lagrangian quantization methods for general gauge theories. We use the Batalin-Vilkovisky formalism to study the arbitrariness in the realization of the gauge algebra. We obtain a one-parametric non-analytic extension of the gauge algebra, and a corresponding solution of the classical master equation, related via an anticanonical transformation to a solution corresponding to an analytic realization. We investigate the possibility of closed solutions of the classical master equation in the Sp(2)-covariant formalism and show that such solutions do not exist in the approximation up to the third order in ghost and auxiliary fields.

hep-th↗
arXiv2002

Canonical form of Euler-Lagrange equations and gauge symmetries

The structure of the Euler-Lagrange equations for a general Lagrangian theory is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter in the right hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler-Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proven that for local theories all the gauge generators are local in time operators.

hep-th↗
arXiv2002

Angular dependence of synchrotron radiation intensity

The detailed analysis of angular dependence of the synchrotron radiation (SR) is presented. In particular, we analyze the angular dependence of the integral SR-intensity and peculiarities of the angular dependence of the first harmonics SR. Studying spectral SR-intensities, we have discovered their unexpected angular behavior, completely different from that of the integral SR-intensity. Namely, for any given synchrotron frequency, maxima of the spectral SR-intensities recede from the orbit plane with increasing particle energy. Thus, in contrast with the integral SR-intensity, the spectral ones have the tendency to deconcentrate themselves on the orbit plane.

physics.acc-ph↗
arXiv2002

Hamiltonization of theories with degenerate coordinates

We consider a class of Lagrangian theories where part of the coordinates does not have any time derivatives in the Lagrange function (we call such coordinates degenerate). We advocate that it is reasonable to reconsider the conventional definition of singularity based on the usual Hessian and, moreover, to simplify the conventional Hamiltonization procedure. In particular, in such a procedure, it is not necessary to complete the degenerate coordinates with the corresponding conjugate momenta.

hep-th↗
arXiv2002

Solutions of relativistic wave equations in superpositions of Aharonov-Bohm, magnetic, and electric fields

We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein-Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov-Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special non-uniform fields. The solutions obtained can be useful to study Aharonov-Bohm effect in the corresponding electromagnetic fields.

hep-th↗