Electromagnetic Klein-Gordon and Dirac equations in scale relativity

We present a new step in the foundation of quantum field theory with the tools of scale relativity. Previously, quantum motion equations (Schr\"odinger, Klein-Gordon, Dirac, Pauli) have been derived as geodesic equations written with a quantum-covariant derivative operator. Then, the nature of gauge transformations, of gauge fields and of conserved charges have been given a geometric meaning in terms of a scale-covariant derivative tool. Finally, the electromagnetic Klein-Gordon equation has been recovered with a covariant derivative constructed by combining the quantum-covariant velocity operator and the scale-covariant derivative. We show here that if one tries to derive the electromagnetic Dirac equation from the Klein-Gordon one as for the free particle motion, i.e. as a square root of the time part of the Klein-Gordon operator, one obtains an additional term which is the relativistic analog of the spin-magnetic field coupling term of the Pauli equation. However, if one first applies the quantum covariance, then implements the scale covariance through the scale-covariant derivative, one obtains the electromagnetic Dirac equation in its usual form. This method can also be applied successfully to the derivation of the electromagnetic Klein-Gordon equation. This suggests it rests on more profound roots of the theory, since it encompasses naturally the spin-charge coupling.Comment: 14 pages, no figure

Dirac Equation in Scale Relativity

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows to recover quantum mechanics as mechanics on a non-differentiable (fractal) space-time. The Schr\"odinger and Klein-Gordon equations have already been demonstrated as geodesic equations in this framework. We propose here a new development of the intrinsic properties of this theory to obtain, using the mathematical tool of Hamilton's bi-quaternions, a derivation of the Dirac equation, which, in standard physics, is merely postulated. The bi-quaternionic nature of the Dirac spinor is obtained by adding to the differential (proper) time symmetry breaking, which yields the complex form of the wave-function in the Schr\"odinger and Klein-Gordon equations, the breaking of further symmetries, namely, the differential coordinate symmetry ($dx^{\mu} \leftrightarrow - dx^{\mu}$) and the parity and time reversal symmetries.Comment: 33 pages, 4 figures, latex. Submitted to Phys. Rev.

Analogy electromagnetism-acoustics: Validation and application to local impedance active control for sound absorption

An analogy between electromagnetism and acoustics is presented in 2D. The propagation of sound in presence of absorbing material is modeled using an open boundary microwave package. Validation is performed through analytical and experimental results. Application to local impedance active control for free field sound absorption is finally described

Electrophoretic silica-coating process on a nano-structured copper electrode

A method for silica-coating at the nanoscale by electrophoretic deposition is presented here, using raw or grafted silica dispersions

Building profile reconstruction using TerraSAR-X data time-series and tomographic techniques

This work aims to show the potentialities of SAR Tomography (TomoSAR) techniques for the 3-D characterization (height, reflectivity, time stability) of built-up areas using data acquired by the satellite sensor TerraSAR-X. For this purpose 19 TerraSAR-X single-polarimetric multibaseline images acquired over Paris urban area have been processed applying classical nonparametric (Beamforming and Capon) and parametric (MUSIC) spectral estimation techniques

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.Comment: 34 page

Anderson Localization of Matter Waves in 3D Anisotropic Disordered Potentials

Using a cutoff-free formulation of the coherent transport theory, we show that the interference terms at the origin of localization strongly affect the transport anisotropy. In contrast to the common hypothesis, we then find that the anisotropies of incoherent and coherent diffusion are significantly different, in particular at criticality. There, we show that the coherent transport anisotropy is mainly determined by the properties of the disorder-averaged effective scattering medium while the incoherent transport contributions become irrelevant

Boundary singularities of positive solutions of some nonlinear elliptic equations

We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and existence or non existence and uniqueness results for solutions of some nonlinear elliptic equations on the half unit sphere.Comment: 6 page