Wavelet Electrodynamics I

A new representation for solutions of Maxwell's equations is derived. Instead of being expanded in plane waves, the solutions are given as linear superpositions of spherical wavelets dynamically adapted to the Maxwell field and well-localized in space at the initial time. The wavelet representation of a solution is analogous to its Fourier representation, but has the advantage of being local. It is closely related to the relativistic coherent-state representations for the Klein-Gordon and Dirac fields developed in earlier work.Comment: 8 Pages in Plain Te

Quasi-particle interaction in nuclear matter from chiral pion-nucleon dynamics

Based on a recent chiral approach to nuclear matter we calculate the in-medium interaction of nucleons at the Fermi surface $|\vec p_{1,2}|=k_f$. The isotropic part of this quasi-particle interaction is characterized by four density dependent (dimensionful) Fermi-liquid parameters: $f_0(k_f), f_0'(k_f), g_0(k_f)$ and $g_0'(k_f)$. In the approximation to $1\pi$-exchange and iterated $1\pi$-exchange (which as such leads already to a good nuclear matter equation of state) we find a spin-isospin interaction strength of $g_0'(2m_\pi) = 1.14$fm$^2$, compatible with existing empirical values. In the next step we include systematically the contributions from $2\pi$-exchange with virtual $\Delta(1232)$-isobar excitation which have been found important for good single-particle properties and spin-stability of nuclear matter. Without any additional short distance terms the spin-dependent Fermi-liquid parameters $g_0(k_{f0})$ and $g'_0(k_{f0})$ come out far too large. Estimates of these short-distance parameters from realistic NN-potentials go in the right direction, but sizeable enhancement factors are still needed to reproduce the empirical values of $g_0(k_{f0})$ and $g_0'(k_{f0})$. This points towards the importance of higher order iterations subsumed in the induced interaction. We consider also the tensor part of the quasi-nucleon interaction at the Fermi surface. In comparison to the leading $1\pi$-exchange tensor interaction we find from the $2\pi$-exchange corrections almost a doubling of the isoscalar tensor strength $h_0(k_f)$, whereas the isovector tensor strength $h_0'(k_f)$ is much less affected. These features are not changed by the inclusion of the chiral $\pi N\Delta$-dynamics.Comment: 19 pages, 9 figures, to be published in Nuclear Physics A (2006

Deformation of string topology into homotopy skein modules

Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of quantum topology. Our results can be considered as weak extensions to all orientable 3-manifolds of classical results by Turaev and Goldman concerning intersection and skein theory on oriented surfaces.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-34.abs.htm

Chiral four-body interactions in nuclear matter

An exploratory study of chiral four-nucleon interactions in nuclear and neutron matter is performed. The leading-order terms arising from pion-exchange in combination with the chiral $4\pi$-vertex and the chiral NN$3\pi$-vertex are found to be very small. Their attractive contribution to the energy per particle stays below $0.6\,$MeV in magnitude for densities up to $\rho =0.4\,$fm$^{-3}$. We consider also the four-nucleon interaction induced by pion-exchange and twofold $\Delta$-isobar excitation of nucleons. For most of the closed four-loop diagrams the occurring integrals over four Fermi spheres can either be solved analytically or reduced to easily manageable one- or two-parameter integrals. After summing the individually large contributions from 3-ring, 2-ring and 1-ring diagrams of alternating signs, one obtains at nuclear matter saturation density $\rho_0=0.16\,$fm$^{-3}$ a moderate contribution of $2.35\,$MeV to the energy per particle. The curve $\bar E(\rho)$ rises rapidly with density, approximately with the third power of $\rho$. In pure neutron matter the analogous chiral four-body interactions lead, at the same density $\rho_n$, to a repulsive contribution that is about half as strong. The present calculation indicates that long-range multi-nucleon forces, in particular those provided by the strongly coupled $\pi N \Delta$-system with its small mass-gap of $293\,$MeV, can still play an appreciable role for the equation of state of nuclear and neutron matter.Comment: 19 pages, 15 figures, to be published in Eur. Phys. J.

Pion-photon exchange nucleon-nucleon potentials

We calculate in chiral perturbation theory the dominant next-to-leading order correction to the $\pi\gamma$-exchange NN-potential proportional to the large isovector magnetic moment $\kappa_v = 4.7$ of the nucleon. The corresponding spin-spin and tensor potentials $\widetilde V_{S,T}(r)$ in coordinate space have a very simple analytical form. At long distances $r \simeq 2$ fm these potentials are of similar size (but opposite in sign) as the leading order $\pi\gamma$-exchange potentials. We consider also effects from virtual $\Delta$-isobar excitation as well as other isospin-breaking contributions to the $2\pi$-exchange NN-potential induced by additional one-photon exchange.Comment: 7 pages, 2 figures, to be published in Phys. Rev. C (2006) Brief Report