We deal with the “nonrelativistic limit”, i.e. the limit c → ∞, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the “moving charge” of the spinor. This limit, sometimes also called “Post-Newtonian” limit, yields a SchrödingerPoisson system, where the spin and the m...

This setting can be generalized by replacing E by a differential 1-form ω, subject to the Maxwell-type equation (δd− k)ω = 0 [7]. Herein, d denotes the exterior derivative, and δ the co-derivative, its formal L adjoint [2,8]. The latter equation seamlessly extends to dimensions n other than 3. Moreover, it can be posed in Riemannian manifolds, while the original double curl equation relies on f...

We develop a spinor equation of the electromagnetic field, which is equivalent to the Maxwell equation and has a similar form as the Dirac equation. The spinor is the very conjugate momentum of the vector potential in the Lagrangian mechanics. In this framework the electromagnetic field described by the spinor exhibits the SU(2) internal symmetry. The quantization and the problem of longinal an...

2 Macroscopic modeling 5 2.1 Concentration measures, mixture velocity and diffusion fluxes . . . . . . . . 5 2.2 Species transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Fick model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Limits of Fick model: osmotic diffusion, reverse diffusion and diffusion barrier 8 2.5 Maxwell-Stefan mo...

The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell’s equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.