The classical point-electron in Colombeau's theory of nonlinear generalized functions

The electric and magnetic fields of a pole-dipole singularity attributed to a point-electron-singularity in the Maxwell field are expressed in a Colombeau algebra of generalized functions. This enables one to calculate dynamical quantities quadratic in the fields which are otherwise mathematically ill-defined: The self-energy (i.e., `mass'), the self-angular momentum (i.e., `spin'), the self-momentum (i.e., `hidden momentum'), and the self-force. While the total self-force and self-momentum are zero, therefore insuring that the electron-singularity is stable, the mass and the spin are diverging integrals of delta-squared-functions. Yet, after renormalization according to standard prescriptions, the expressions for mass and spin are consistent with quantum theory, including the requirement of a gyromagnetic ratio greater than one. The most striking result, however, is that the electric and magnetic fields differ from the classical monopolar and dipolar fields by delta-function terms which are usually considered as insignificant, while in a Colombeau algebra these terms are precisely the sources of the mechanical mass and spin of the electron-singularity.Comment: 30 pages. Final published version with a few minor correction

Nonlinear Generalized Functions: their origin, some developments and recent advances

We expose some simple facts at the interplay between mathematics and the real world, putting in evidence mathematical objects " nonlinear generalized functions" that are needed to model the real world, which appear to have been generally neglected up to now by mathematicians. Then we describe how a "nonlinear theory of generalized functions" was obtained inside the Leopoldo Nachbin group of infinite dimensional holomorphy between 1980 and 1985. This new theory permits to multiply arbitrary distributions and contains the above mathematical objects, which shows that the features of this theory are natural and unavoidable for a mathematical description of the real world. Finally we present direct applications of the theory such as existence-uniqueness for systems of PDEs without classical solutions and calculations of shock waves for systems in non-divergence form done between 1985 and 1995, for which we give three examples of different nature: elasticity, cosmology, multifluid flows.Comment: 42 pages, 4 figure

Kernel Theorems in Spaces of Tempered Generalized Functions

In analogy to the classical isomorphism between $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m}) )$ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m})$, we show that a large class of moderate linear mappings acting between the space $\mathcal{G}\_{\mathcal{S}}(\mathbb{R}^{n})$ of Colombeau rapidly decreasing generalized functions and the space $\mathcal{G}\_{\tau}(\mathbb{R}^{n})$ of temperate ones admits generalized integral representations, with kernels belonging to $\mathcal{G}\_{\tau}(\mathbb{R}^{n+m})$. Furthermore, this result contains the classical one in the sense of the generalized distribution equality.Comment: 15 page

Border guards as an alien police: usages of the Schengen Agreement in France

The creation of a common European space following the integration of the Schengen Agreement into the acquis communautaires through the Amsterdam Treaty in 1997, and the subsequent treaties and summits, lead Member States to consider border control as a common issue. One could have thought that the lifting of the internal borders within the Schengen space would have threatened the border guard corps at the national level. This is not the case. I will show that, thanks to a change in the model of French border guards, their power and influence have in fact risen in the second part of the 1990’s. In response to the fear of a drastic cut in the workforce, French border guards mobilize to define a new model of border guard: the alien police model, which aimed at fighting against illegal immigration.administrative adaptation; Europeanization; France; free movement; immigration policy; national parliaments; policy analysis; public administration; Schengen

Algebras of generalized functions with smooth parameter dependence

We show that spaces of Colombeau generalized functions with smooth parameter dependence are isomorphic to those with continuous parametrization. Based on this result we initiate a systematic study of algebraic properties of the ring $\widetilde{\mathbb{K}}_{sm}$ of generalized numbers in this unified setting. In particular, we investigate the ring and order structure of $\widetilde{\mathbb{K}}_{sm}$ and establish some properties of its ideals.Comment: 19 page

A relativistic variant of the Wigner function

The conventional Wigner function is inappropriate in a quantum field theory setting because, as a quasiprobability density over phase space, it is not manifestly Lorentz covariant. A manifestly relativistic variant is constructed as a quasiprobability density over trajectories instead of over phase space.Comment: v3: as accepted by Phys. Lett.