Non-commutative space time of the relativistic equations with a Coulomb potential using Seiberg-Witten map

We present an important contribution to the non-commutative 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 non-commutativity parameter using the Seiberg-Witten maps. We thus find the non-commutative modification of the energy levels and by comparing with the the current experimental results on the Lamb shift of the 2P level to extract a bound on the parameter of non-commutativity, we show that the fundamental length ($\sqrt{\Theta}$) is compatible with the value of the electroweak length scale ($l$). Phenomenologically, this effectively confirms the presence of gravity at this level.