The exponential map for the unitary group SU(2,2)

In this article we extend our previous results for the orthogonal group, $SO(2,4)$, to its homomorphic group $SU(2,2)$. Here we present a closed, finite formula for the exponential of a $4\times 4$ traceless matrix, which can be viewed as the generator (Lie algebra elements) of the $SL(4,C)$ group. We apply this result to the $SU(2,2)$ group, which Lie algebra can be represented by the Dirac matrices, and discuss how the exponential map for $SU(2,2)$ can be written by means of the Dirac matrices.Comment: 10 page

Higher Derivative Fermionic Field Equation in the First Order Formalism

The generalized Dirac equation of the third order, describing particles with spin 1/2 and three mass states, is analyzed. We obtain the first order generalized Dirac equation in the 24-dimensional matrix form. The mass and spin projection operators are found which extract solutions of the wave equation corresponding to pure spin states of particles. The density of the electromagnetic current is obtained, and minimal and non-minimal (anomalous) electromagnetic interactions of fermions are considered by introducing three phenomenological parameters. The Hamiltonian form of the first order equation has been obtained.Comment: 16 pages, title changed, new section, appendixes, and references adde

Potential Scattering in Dirac Field Theory

We develop the potential scattering of a spinor within the context of perturbation field theory. As an application, we reproduce, up to second order in the potential, the diffusion results for a potential barrier of quantum mechanics. An immediate consequence is a simple generalization to arbitrary potential forms, a feature not possible in quantum mechanics.Comment: 7 page

Canonical and Lie-algebraic twist deformations of κ\kappa-Poincare and contractions to κ\kappa-Galilei algebras

We propose canonical and Lie-algebraic twist deformations of $\kappa$-deformed Poincare Hopf algebra which leads to the generalized $\kappa$-Minkowski space-time relations. The corresponding deformed $\kappa$-Poincare quantum groups are also calculated. Finally, we perform the nonrelativistic contraction limit to the corresponding twisted Galilean algebras and dual Galilean quantum groups.Comment: 16 pages, no figures, v3: few changes provided - version for journal, v2: submitted incidentally, v4: the page numbers for all references in preprint version are provide

Decomposition of time-covariant operations on quantum systems with continuous and/or discrete energy spectrum

Every completely positive map G that commutes which the Hamiltonian time evolution is an integral or sum over (densely defined) CP-maps G_\sigma where \sigma is the energy that is transferred to or taken from the environment. If the spectrum is non-degenerated each G_\sigma is a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on R with respect to a non-translation-invariant measure. As an example, I calculate this decomposition explicitly for the rotation invariant gaussian channel on a single mode. I address the question under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CP-map-valued measure (G_\sigma).Comment: latex, 33 pages, domains of unbounded operators are now explicitly specified. Presentation more detailed. Implementing the shift after the dephasing is sometimes more convenien

Conformally related massless fields in dS, AdS and Minkowski spaces

In this paper we write down the equation for a scalar conformally coupled field simultaneously for de Sitter (dS), anti-de Sitter (AdS) and Minkowski spacetime in d-dimensions. The curvature dependence appears in a very simple way through a conformal factor. As a consequence the process of curvature free limit, including wave functions limit and two-points functions, turns to be a straightforward issue. We determine a set of modes, that we call de Sitter plane waves, which become ordinary plane waves when the curvature vanishes.Comment: 7 pages, 1 figur

Superluminal X-shaped beams propagating without distortion along a coaxial guide

In a previous paper [Phys. Rev. E64 (2001) 066603; e-print physics/0001039], we showed that localized Superluminal solutions to the Maxwell equations exist, which propagate down (non-evanescence) regions of a metallic cylindrical waveguide. In this paper we construct analogous non-dispersive waves propagating along coaxial cables. Such new solutions, in general, consist in trains of (undistorted) Superluminal "X-shaped" pulses. Particular attention is paid to the construction of finite total energy solutions. Any results of this kind may find application in the other fields in which an essential role is played by a wave-equation (like acoustics, geophysics, etc.). [PACS nos.: 03.50.De; 41.20;Jb; 83.50.Vr; 62.30.+d; 43.60.+d; 91.30.Fn; 04.30.Nk; 42.25.Bs; 46.40.Cd; 52.35.Lv. Keywords: Wave equations; Wave propagation; Localized beams; Superluminal waves; Coaxial cables; Bidirectional decomposition; Bessel beams; X-shaped waves; Maxwell equations; Microwaves; Optics; Special relativity; Coaxial metallic waveguides; Acoustics; Seismology; Mechanical waves; Elastic waves; Guided gravitational waves.]Comment: plain LaTeX file (22 pages), plus 15 figures; in press in Phys. Rev.

Generating Complex Potentials with Real Eigenvalues in Supersymmetric Quantum Mechanics

In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being real and the other imaginary and PT symmetric. The latter turns out to be quasiexactly solvable with one known eigenvalue corresponding to a bound state. When the Weierstrass function degenerates to a hyperbolic one, the imaginary potential becomes PT non-symmetric and its known eigenvalue corresponds to an unbound state.Comment: 20 pages, Latex 2e + amssym + graphics, 2 figures, accepted in Int. J. Mod. Phys.

Klauder's coherent states for the radial Coulomb problem in a uniformly curved space and their flat-space limits

First a set of coherent states a la Klauder is formally constructed for the Coulomb problem in a curved space of constant curvature. Then the flat-space limit is taken to reduce the set for the radial Coulomb problem to a set of hydrogen atom coherent states corresponding to both the discrete and the continuous portions of the spectrum for a fixed \ell sector.Comment: 10 pages, no figure

Einstein-Podolsky-Rosen-Bohm experiment with relativistic massive particles

The EPRB experiment with massive partcles can be formulated if one defines spin in a relativistic way. Two versions are discussed: The one using the spin operator defined via the relativistic center-of-mass operator, and the one using the Pauli-Lubanski vector. Both are shown to lead to the SAME prediction for the EPRB experiment: The degree of violation of the Bell inequality DECREASES with growing velocity of the EPR pair of spin-1/2 particles. The phenomenon can be physically understood as a combined effect of the Lorentz contraction and the Moller shift of the relativistic center of mass. The effect is therefore stronger than standard relativistic phenomena such as the Lorentz contraction or time dilatation. The fact that the Bell inequality is in general less violated than in the nonrelativistic case will have to be taken into account in tests for eavesdropping if massive particles will be used for a key transfer.Comment: Figures added as appeared in PRA, two typos corrected (one important in the formula for eigenvector in Sec. IV); link to the unpublished 1984 paper containing the results (without typos!) of Sec. IV is adde