Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map

Abstract

We present an important contribution to the noncommutative approach to the hydrogen atom to deal with Lamb shift corrections. This can be done by studying the Klein{Gordon and Dirac equations in a non-commutative space-time up to first-order of the noncommutativity parameter using the Seiberg-Witten maps. We thus find the noncommutative modification of the energy levels and by comparing with the current experimental results on the Lamb shift of the 2P level to extract a bound on the parameter of noncommutativity, we show that the fundamental length (√Θ) is compatible with the value of the electroweak length scale (l). Phenomenologically, this effectively confirms the presence of gravity at this level.