Noncommutative space-time of the relativistic equations with a Coulomb potential using Seiberg–Witten map

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 $\mathrm{2P}$ level to extract a bound on the parameter of noncommutativity, 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.