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Coulomb potential in S4 universe

Consider the approximate Coulomb potential $F = e/\sin(t)$ where $e$ is the elementary charge.  We will consider first the case of the radius 1 universe $S^4(1)$ and then later scale by $R = 1/h$.  Now consider the equation $(D + F)\psi = \mu\psi$ where $D$ is the Dirac operator.  Expanding the equation in eigenfunctions $\Phi_{nl}$ of $D$ and taking inner product we obtain

$\lambda_n + \frac{e}{\lambda_n^2 - S/4} \frac{C_{nl}}{B_{nl}} = \mu$ (1)

where $S$ is the scalar curvature so $S/4=3$.  Now $C_{nl}$ we have examined before and $B_{nl}$ are the normalization constants for the eigenfunctions.  According to Camporesi and Higuchi,

$B_{nl} = \frac{|\Gamma(2+n)|^2}{(n-l)!(3+n+l)!}$

One can check by numerical calculations that $\delta_{nl} = \frac{C_{nl}}{B_{nl}}$ is different for different $l$.

Now we consider (1) for scaling by $R$.  The eigenvalue scales to $R^{-1}\lambda_n$ and the integrals scale by $R^{4}$, and the scalar curvature by $R^{-2}$.  The equation corresponding to (1) is

$R^{-1} \lambda_n + \frac{e}{R^{-2}(\lambda_n^2-3)} \delta_{nl} = R^{-1} \mu$ (2)

We want to consider this equation for $R=1/h$ and consider this to be the model of a Dirac hydrogen atom in $S^4(1/h)$.

THE GAP TO THE ACTUAL HYDROGEN ATOM

The equation (2) would be an excellent model of the hydrogen energy spectrum using $(n,l)$ mapping directly to the orbital levels so long as there is a reasonable agreement in the fine structure.  For a given $n$, we will always have different values for $\delta_{nl}$ so there cannot be any degeneracy and therefore no anomaly with Lamb shift.  The energy levels as a function of $n$ has the familiar inverse-square relation with a precise adjustment due to the scalar curvature of $S^4$.  The factor $e/R^2$ in the second term of (2) needs to be understood.  Here again it is tempting to fit $R$ empirically but this will evade the hard question of matching nature as we know from other evidence that the slope of the redshift is explained extremely accurately by an $S^4(1/h)$ universe.

One idea is to consider the relative differences between the second term in (2) with respect to the first term.  So consider the coefficient of $\latex \delta_{nl}$ to be
$\frac{e R^{-1} \lambda_n}{R^{-2}(\lambda_n^2-3)} = \frac{ e R \lambda_n}{\lambda_n^2-3}$

This us quite useful because $e R = e/h$ has the right order of magnitude for relative empirical Lamb shift for $2p1$ and $2s1$ orbitals corresponding to $n=0$ and $n=1$.

The elementary charge is $latex e = 1.602 \times 10^{-19}$ Coulombs.  We can use units of electron volts for $e$ in our case with the same numerical value.  We want to use $1/h$ in length units,  but using usual units $h = 4.135 \times 10^{-15} eV\cdot s$.  In usual units, $e/h = 3.87 \times 10^{-5}$ and the unit is seconds.  We will get energy units if we use $1/h$ in units of meters.  Now in (2) we have $\delta_{1,0} = 12.4$ and $\delta_{1,1} = 62$ so

$\frac{e}{h} \frac{1}{6} = 6.457 \times 10^{-6}$ (2)

This is not an exact fit to the Lamb shift however it is close in order of magnitude.

ANOTHER DIRECT ATTEMPT First, let $p$ denote the frequency of the 2p orbital and $s$ the frequency of the 2s orbital of hydrogen, measured.  Now let $A=1/p$ denote the scaling necessary to map $n=1$ eigenvalue of the Dirac operator to the observed second orbital.  Now $A = 1.216\times 10^{-5}$.  This is positive because this experimental value corresponds to the theoretical value in (2).  So we are off by a factor of around 2 for this value if we use (2).Let $B = (p-s)/p = -4.25 \times 10^{-7}$ be the experimental relative difference between the orbitals.  We have to multiply (2) by $\delta{1,1}-\delta_{1,0} = 49.6$ and we will be off by a couple of orders in the magnitude for the match.

TRANSITION ESTIMATE

Instead of (2) which uses a single value of $n$, let us consider the transition value for $n=1 \rightarrow n=2$ (recall that the eigenvalues of the Dirac are $2+k$ which explains the constants in the following).  So

$\frac{e}{h}( \frac{ 1}{ 3^2 - 3 } - \frac{2}{4^2-3} = 4.97 \times 10^{-7}$ (3)

With $\delta_{2,1} - \delta_{2,0} = - 37.56$ we would be a factor off from the Lamb shift still but gap is narrower.  Since we are seeking an exact match, it is significant that the value in (3) is by itself quite close to the relative difference of frequencies between 2p1 and 2s1 energy levels of empirical hydrogen which is $q = 4.3 \times 10^{-7}$ but this is not yet a correct mapping because there are terms from $\delta_{nl}$ that need to be multiplied.