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## Consequences of matching Lamb shift to 2 orders of magnitude

The Lamb shift of empirical hydrogen is $\delta = 4.3 \times 10^{-7}$.  Solving the Dirac equation on $S^4(1/h)$ with approximate Coulomb potential $F = e /\sin(h t)$ where $(t,\Omega)$ are geodesic polar coordinates, we can get an estimate $\delta_{theory} = \frac{e}{h}( \frac{1}{2^2-3} - \frac{2}{3^2 - 3} ) (\delta_{1,0}-\delta_{1,1})$ (see this).

Now

$\delta_{1,0} - \delta_{1,1} = -1.6$ by numerical calculation gives $\delta_{theory} = 4.13 \times 10^{-5}$.  This is two orders of magnitude off.

So bad is this error?  First of all, the relative difference from the $n=1$ and $n=2$ in the empirical hydrogen is $0.185$.  Therefore although numerically the theoretical Lamb shift is two orders of magnitude off from the experimentally determined one, it is small relative to the energy level differences between the $n=1$ and $n=2$ levels of hydrogen.  Therefore we have a useful estimate of the Lamb shift.

A nonzero theoretical value for Lamb shift is thus predicted by a different theory than quantum field theory: the different theory is produced by beginning with $S^4(1/h)$ as the base space of the universe and then solving the Dirac equation with the approximate Coulomb potential using the mass of the hydrogen nucleus $m_0 c^2$.  This implies that the Dirac theory without additional assumptions in $S^4(1/h)$ can explain the Lamb shift (with some error).  The ingredients different from the classical Dirac equation on flat three-space are the 4-sphere geometry with fixed scale and we can consider the Coulomb potential to be positioned orthogonal to the physical hypersurface but this does not affect the validity of this new theoretical estimate of the Lamb shift.  In particular, this provides an alternative explanation for the Lamb shift (with 2 orders of magnitude error) from quantum field theory.  In this explanation, we do not use quantization by any method other than that naturally occuring from the S4 geometry.