We extend the three-dimensional noncommutative relations of position and momentum operators to those in four dimension. Using Seiberg-Witten (SW) map, we give Heisenberg representation these algebras endow parameters associated with Planck constant, length cosmological constant. As an analog electromagnetic gauge potential, effect can be interpreted as effective field, which depends on Plank co...

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism togethe...

The Coulomb potential is derived in " one space-one time " dimension, and introduced into Dirac and Klein-Gordon equations. The equations are solved, and somewhat surprising result-nonexistence of bound state solutions in the lower dimension-discussed and identified as another fine example of the " Klein paradox " .