On the action principle for a system of differential equations

We consider the problem of constructing an action functional for physical systems whose classical equations of motion cannot be directly identified with Euler-Lagrange equations for an action principle. Two ways of action principle construction are presented. From simple consideration, we derive necessary and sufficient conditions for the existence of a multiplier matrix which can endow a prescribed set of second-order differential equations with the structure of Euler-Lagrange equations. An explicit form of the action is constructed in case if such a multiplier exists. If a given set of differential equations cannot be derived from an action principle, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. There exists an ambiguity (not reduced to a total time derivative) in associating a Lagrange function with a given set of equations. We present a complete description of this ambiguity. The general procedure is illustrated by several examples.Comment: 10 page

Perturbative Study of Bremsstrahlung and Pair-Production by Spin-1/2 Particles in the Aharonov-Bohm Potential

In the presence of an external Aharonov-Bohm potential, we investigate the two QED processes of the emission of a bremsstrahlung photon by an electron, and the production of an electron-positron pair by a single photon. Calculations are carried out using the Born approximation within the framework of covariant perturbation theory to lowest non-vanishing order in \alpha. The matrix element for each process is derived, and the corresponding differential cross-section is calculated. In the non-relativistic limit, the resulting angular and spectral distributions and some polarization properties are considered, and compared to results of previous works.Comment: 15 pages, RevTex 4, 2 figures, submitted for publicatio

Electron-positron pair production in the Aharonov-Bohm potential

In the framework of QED we evaluate the cross section for electron-positron pair production by a single photon in the presence of the external Aharonov-Bohm potential in first order of perturbation theory. We analyse energy, angular and polarization distributions at different energy regimes: near the threshold and at high photon energies.Comment: LaTeX file, 13 page

Aharonov-Bohm scattering of charged particles and neutral atoms: the role of absorption

The Aharonov-Bohm scattering of charged particles by the magnetic field of an infinitely long and infinitely thin solenoid (magnetic string) in an absorbing medium is studied. We discuss the partial-wave approach to this problem and show that standard partial-wave method can be adjusted to this case. The effect of absorption leads to oscillations of the AB cross section. Based on this we investigate the scattering of neutral atoms with induced electric dipole moments by a charge wire of finite radius which is placed in an uniform magnetic field. The physical realistic and practically important case that all atoms which collide with the wire are totally absorbed at its surface, is studied in detail. The dominating terms of the scattering amplitude are evaluated analytically for different physical constellations. The rest terms are written in a form suitable for a numerical computation. We show that if the magnetic field is absent, the absorbing charged wire causes oscillations of the cross section. In the presence of the magnetic field the cross section increases and the dominating Aharonov--Bohm peak appears in the forward direction, suppressing the oscillations.Comment: 15 pages, LaTeXfile, 2 figure

A pragmatic approach to the problem of the self-adjoint extension of Hamilton operators with the Aharonov-Bohm potential

We consider the problem of self-adjoint extension of Hamilton operators for charged quantum particles in the pure Aharonov-Bohm potential (infinitely thin solenoid). We present a pragmatic approach to the problem based on the orthogonalization of the radial solutions for different quantum numbers. Then we discuss a model of a scalar particle with a magnetic moment which allows to explain why the self-adjoint extension contains arbitrary parameters and give a physical interpretation.Comment: 8 pages, LaTeX, to appear in J. Phys.

Polarization of the vacuum of a quantized scalar field by an impenetrable magnetic vortex of finite thickness

We consider the effect of the magnetic field background in the form of a tube of the finite transverse size on the vacuum of the quantized charged massive scalar field which is subject to the Dirichlet boundary condition at the tube. It is shown that, if the Compton wavelength associated with the scalar field exceeds considerably the transverse size of the tube, then the vacuum energy which is finite and periodic in the value of the magnetic flux enclosed in the tube is induced on a plane transverse to the tube. Some consequences for generic features of the vacuum polarization in the cosmic-string background are discussed.Comment: 13 pages, 10 figures, journal version, minor changes, some references adde

Scattering of scalar and Dirac particles by a magnetic tube of finite radius

We consider the Dirac equation in cylindrically symmetric magnetic fields and find its normal modes as eigenfunctions of a complete set of commuting operators. This set consists of the Dirac operator itself, the $z$-components of the linear and the total angular momenta, and of one of the possible spin polarization operators. The spin structure of the solution is completely fixed independently of the radial distribution of the magnetic field which influences only the radial modes. We solve explicitly the radial equations for the uniform magnetic field inside a solenoid of a finite radius and consider in detail the scattering of scalar and Dirac particles in this field. For particles with low energy the scattering cross section coincides with the Aharonov-Bohm scattering cross section. We work out the first order corrections to this result caused by the fact that the solenoid radius is finite. At high energies we obtain the classical result for the scattering cross section.Comment: LaTeX file, 17 page