A Note on the Cosmological Dynamics in Finite-Range Gravity

In this note we consider the homogeneous and isotropic cosmology in the finite-range gravity theory recently proposed by Babak and Grishchuk. In this scenario the universe undergoes late time accelerated expansion if both the massive gravitons present in the model are tachyons. We carry out the phase space analysis of the system and show that the late-time acceleration is an attractor of the model.Comment: RevTex, 4 pages, two figures, New references added, To appear in IJMP

A symplectic realization of the Volterra lattice

We examine the multiple Hamiltonian structure and construct a symplectic realization of the Volterra model. We rediscover the hierarchy of invariants, Poisson brackets and master symmetries via the use of a recursion operator. The rational Volterra bracket is obtained using a negative recursion operator.Comment: 8 page

Efficient numerical diagonalization of hermitian 3x3 matrices

A very common problem in science is the numerical diagonalization of symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be too inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with an analytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. The analytical algorithm outperforms the others by more than a factor of 2, but becomes inaccurate or may even fail completely if the matrix entries differ greatly in magnitude. This can mostly be circumvented by using a hybrid method, which falls back to QL if conditions are such that the analytical calculation might become too inaccurate. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from http://www.mpi-hd.mpg.de/~globes/3x3/ .Comment: 13 pages, no figures, new hybrid algorithm added, matches published version, typo in Eq. (39) corrected; software library available at http://www.mpi-hd.mpg.de/~globes/3x3

Effective-Mass Dirac Equation for Woods-Saxon Potential: Scattering, Bound States and Resonances

Approximate scattering and bound state solutions of the one-dimensional effective-mass Dirac equation with the Woods-Saxon potential are obtained in terms of the hypergeometric-type functions. Transmission and reflection coefficients are calculated by using behavior of the wave functions at infinity. The same analysis is done for the constant mass case. It is also pointed out that our results are in agreement with those obtained in literature. Meanwhile, an analytic expression is obtained for the transmission resonance and observed that the expressions for bound states and resonances are equal for the energy values $E=\pm m$.Comment: 20 pages, 6 figure

Low Momentum Scattering in the Dirac Equation

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is -1 and hence the transmission coefficient T=0 in general. If however the potential supports a half-bound state at k=0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T=1.Comment: 12 pages; revised to include additional references; to be published in J Phys

Exotic Statistics for Ordinary Particles in Quantum Gravity

Objects exhibiting statistics other than the familiar Bose and Fermi ones are natural in theories with topologically nontrivial objects including geons, strings, and black holes. It is argued here from several viewpoints that the statistics of ordinary particles with which we are already familiar are likely to be modified due to quantum gravity effects. In particular, such modifications are argued to be present in loop quantum gravity and in any theory which represents spacetime in a fundamentally piecewise-linear fashion. The appearance of unusual statistics may be a generic feature (such as the deformed position-momentum uncertainty relations and the appearance of a fundamental length scale) which are to be expected in any theory of quantum gravity, and which could be testable.Comment: Awarded an honourable mention in the 2008 Gravity Research Foundation Essay Competitio

Asymptotic Infrared Fractal Structure of the Propagator for a Charged Fermion

It is well known that the long-range nature of the Coulomb interaction makes the definition of asymptotic ``in'' and ``out'' states of charged particles problematic in quantum field theory. In particular, the notion of a simple particle pole in the vacuum charged particle propagator is untenable and should be replaced by a more complicated branch cut structure describing an electron interacting with a possibly infinite number of soft photons. Previous work suggests a Dirac propagator raised to a fractional power dependent upon the fine structure constant, however the exponent has not been calculated in a unique gauge invariant manner. It has even been suggested that the fractal ``anomalous dimension'' can be removed by a gauge transformation. Here, a gauge invariant non-perturbative calculation will be discussed yielding an unambiguous fractional exponent. The closely analogous case of soft graviton exponents is also briefly explored.Comment: Updated with a corrected sign error, longer discussion of fractal dimension, and more reference