Gauge-invariant semidiscrete Wigner theory

A gauge-invariant Wigner quantum mechanical theory is obtained by applying the Weyl-Stratonovich transform to von Neumann equation for density matrix. The reduces Weyl in electrostatic limit, when vector potential and thus magnetic field are zero. Both cases involve a center-of-mass followed Fourier integral on relative coordinate introducing momentum variable. latter continuous if limits of infinite or, equivalently, coherence length infinite. However, involves transforms electromagnetic components, which imposes conditions their behavior at infinity. Conversely, systems bounded often very small, as is, instance, case modern nanoelectronics. This implies finite length, avoids need regularize non-converging integrals. Accordingly, space becomes discrete, giving rise quantization semi-discrete equation. To gain insights into peculiarities this one needs analyze specific conditions. We derive evolution linear show that it significantly simplifies limit dictated long behavior, derivatives. In discrete picture these derivatives presented difference quantities which, together with further approximations, allow develop computationally feasible model offers physical involved processes. particular, Fredholm second kind obtained, where "power" kernel measuring rate modification evolution, can be evaluated.