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## Dirac spectrum on S4(1/h) and the hydrogen atom

Let us scale away the constant $h$ and consider the spectrum of the operator $D_A = D + A$ where $A = Ze/\sin(t)$ on $S^4(1)$ in order to understand clearly the issue of the spectrum of massive particles satisfying Dirac equation.  Let $m_0$ be the rest mass and $M = m_0 c^2/h$.  Clearly $D_A\psi = M\psi$ will not have any solutions unless $M$ belongs to the spectrum of $D_A$ and if $M$ is in the spectrum of $D_A$ (which will imply $M$ is an integer), then the energy spectrum of the particle should be identified with:

$\lambda + \frac{C_{nl}}{\lambda^2 - R/4}$ (1)

where $n$ is the eigenvalue level and $R$ is the scalar curvature equal to 12 for the 4-sphere and $C_{nl}$ are explicitly computable constants.

$C_{nl} = \int_0^{2\pi} \phi_{nl}(\cos(t)) \Delta_{S^4}(1/\sin(t)) \sin^3(t) dt$

The first factor is $\phi_{nl}(\cost(t)) = \cos^{l+1}(t/2)\sin^l(t/2) P^{(2+l,1+l)}_n(\cos(t))$ as per the eigenspinor calculation of Camperosi and Higuchi and the latter we had computed to find

$\Delta( 1/\sin(t)) = \sin^{-3}(t)[1 + 2 \sin(t)]$

Thus computation of $C_{nl}$ for all $l\le n$ can be accomplished by using integral tables (specifically 7.391 of Gradshteyn and Ryzhik).

Before we can return to the fine issue of Lamb shift, we should consider what the relation between (1) and the energy levels of a massive particle can be and whether an identification is sensible.  We suppose that $n$ is relatively large.  Then the energy levels corresponding to $M$ will correspond to $l=0,1,2,\dots,n$.  In other words, one should avoid confusion between the indexing of eigenvalues of $D$ and the energy levels of $D_A-M$.  By the time we are considering the energy levels of the particle, we have already fixed the eigenspace of $D$ for analysis.  One interesting feature from (1) is that whatever might be the interpretation, the energy levels are all distinct without any degeneracy whatsoever, since $C_{nl}$ are all distinct for differing $l$.  Therefore if these energy levels correspond to energy levels of hydrogen or some other massive particle, there cannot be an issue like the Lamb shift issue.  But we may still have some work to do to show that the identification matches the behavior of the hydrogen energy spectrum.

The trouble with the model above where we use only a single eigenspace of $D$ is that there is a clustering in the observed energy levels of hydrogen clearly seen here:

In order to explore the resolution, instead of starting with a large $n$ eigenspace of $D$ for the mass $M$ consider the ad hoc model of considering the shifted spectrum of $D$ to zero and assume that (1) still describes the energy levels.  Now we pretend that multiple eigenspaces of $D$ are used to analyze the spectrum of $D_A$.  In this case, we can use (1) for $C_{nl}$ for $n=1,2,\dots$.  The energy levels would still be given by (1) but now for multiple low $n$.  We still have the property that $C_{n0} \not= C_{n1}$ for any $n$ and therefore there is always a “Lamb shift” but now we are in a situation closer to the usually accepted theory of the orbitals with $n$ corresponding to principal quantum numbers and $l$ corresponding to the fine structure.  This has empirical sense and is also closer to the classical analysis of the Dirac equation for the hydrogen atom in flat space.  The energy spectrum of $D_A$ is then given by

$M-\lambda_n = \frac{C_{nl}}{\lambda_n^2 - R/4}$ (1)

The right hand side has $\lambda_n^2$ in the denominator where $\lambda_n$ are eigenvalues of $D$ rather than $D_A$ and correspond to the familiar factcor $1/n^2$ in the analysis of classical Dirac equation for hydrogen-like atoms.  So we have a model for hydrogen that has the right first order factor and the fine structure at every level are given by $C_{nl}$ for different $l$.  These are explicitly computable (although it is nontrivial to provide simple characterizations of the differences).  In particular this model of hydrogen spectrum using the spectrum of $D$ and $D_A$ is sane and reasonable even on a 4-sphere clearly up to first order.  This model encapsulates all the fine structure into $C_{nl}$.

In order to move closer to a realistic model with exact numerical fits, we have to analyze $C_{nl}$ in more detail but even without explicit calculation it is clear that the energy levels corresponding to $n=2$ and $l=0,1$ will be different in this model and therefore the Lamb shift will not be an issue if there is a match to the principal energy levels.