Mesoscopic electrical transmission line in the chargeanticharge framework: Spectral properties and Casimir forces

In the chargeanticharge framework, we solve explicitly the nonlinear quantum evolution equation for the charge operator of the direct transmission line with discrete charge. The associated spectrum is completely consistent with the well-known limit of continuous charge. In the zero point charge fluctuations state, the attraction between plates is compared with the corresponding Casimir force (related to field fluctuations) which, now, could be interpreted in terms of virtual charge fluctuations. The spectrum of the dual transmission line (left-handed) is also found. Some aspects related to quantum dots (coulomb blockade), structure fine constant and thermodynamics properties are also touched upon.