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## Conformal invariance of the Dirac equation

We are now in the situation that with an appropriate scaling of $S^4(1)$ we can can treat the hydrogen atom by an appropriate scaling of the metric so that the scaling of $C_{nl}$ reproduce the energy levels of the hydrogen atom with a Lamb shift difference.  We certainly have the situation where a single scaling by a constant $1/B$ would fix the principal energy level differences and we would obtain the variations in the fine structure by using $C_{nl}/B$.  This is a step in the right direction but it begs the question of why the universe has radius $1/h$.

CONSISTENCY QUESTION

The empirically chosen scale $B$ is extremely unsatisfactory because although numerically it is a reasonable model that does explain the Lamb shift, it is not yet clear how the natural Dirac operator of $S^4(1/h)$ is involved in describing hydrogen.  They differ by a conformal transformation of $S^4(1)$ so we naively assume that Dirac equation is conformally invariant then we have one answer to the question.  But this is not yet a satisfactory fundamental geometric explanation using the specific geometry of $S^4(1/h)$.

IMPLICATIONS OF ORDER OF MAGNITUDE MATCH TO LAMB SHIFT

Despite not yet having produced the ‘right’ Dirac theory of hydrogen-like electrons with the arbitrary scaling $B$, we can produce a situation where the geometric Dirac operator on a 4-sphere explains the Lamb shift and the principal energy level differences, we do have a reasonable explanation of the Lamb shift directly from the geometry of the universe and the Dirac equation.  In particular this explanation of the Lamb shift did not require any form of quantization beyond what is already produced by the 4-sphere geometry.  Therefore this unsatisfactory theory is still sufficient to show that it is possible to explain the empirical hydrogen energy spectrum without the need for a complex quantum field theory.  In particular, we can mark off ‘Lamb shift’ as a fundamental reason to seek a quantum field theory.  Our explanation of the energy levels of hydrogen can be described by classical spinor fields.