Hydrogen Lamb Shift In Higher Dimensional Space-times

Theory of extra dimensions is a hot topic recently.An extra dimensional theory(ADD) was proposed by N.Arkani-Hamed et.al.in1998. According to their theory,the gravity will deviate the Inverse-Square-Lawto be of 1/r²⁺ⁿ,where n denotes the number of extra dimensions.We can see that this kind of gravity will be much stronger than the Newtonian gravity,even stronger than the electro-static force when the scale is small.Because the coupling constant of Newtonian gravity is much smaller than the fine structure constant of electromagnetic force,we can ignore the gravitational effects.Since the ADD gravity is much stronger than the Newtonian gravity in small scale,its effects will not be ignored in such micro-world as in atoms.The ADD gravity will cause the energy levels to move a distance which may be detected and we can use the experiment to decide its accommodation.Lamb shift comes from the self-energy of electrons and vacuum polarization.It is studied both in theoretical and experimental widely and deeply.Especially,the relative movement between energy levels of 2² S_1/2 and 2² P_1/2 is one of the most precise measurement in physics.The movement of energy levels caused by ADD gravity is not include in any theoretical calculations of Lamb shift.Set the number of extra dimensions,compare the energy levels movement with the theoretical and experimental values of Lamb shift,when the energy levels movement caused by ADD gravity is larger than the difference between the theory and experiment,we can decide the ADD theory with certain extra dimensions not proper.Based on the above considerations,we see the nucleus(proton) to ditribute uniformly in a finite ball,use the static perturbation theory to calculate the effect in Lamb shift caused by ADD gravity with 2 extra dimensions,compare the result with the theoretical calculations and experimental values.Our result is -1.1×10^-16 MHz.Comparing our result with recent experimental values(1057.845(9)MHz,1057.839(12)MHz) and theoretical value(1057.842(1)(4)MHz),we can see that our calculations make no decisions of weather ADD gravity is right.