Feeds:
Posts

## On why there should be a Lamb shift for Dirac equation on S4

Let $F = Ze/\sin(t)$ be the approximate Coulomb potential on $S^4(1)$ in geodesic normal coordinates $(t,\Omega)$.  The eigenspinors of the Dirac operator can be written as a factor in $t$ multiplied by a factor in $\Omega$ that we ignore for the moment.  The first factor is

$\phi_{nl} = \cos^{l+1}(t/2)\sin^l(t/2) P_n^{(2+l,1+l)}(\cos t)$

Where $n$ is the eigenvalue index and $l=0,1,\dots,n$ are ordering of eigenfunctions.  The Lamb shift issue is that of an energy level value difference between the 2p and 2s levels of hydrogen.  Suppose now that we have found that for mass $m_0 c^2$ we have found that some element $\lambda_k + C_k(l=0,n) = m_0 c^2$ which will be the ‘ground state’ of the spectrum.  The issue of Lamb shift then is the issue of any difference between the inner products $\langle F, \phi_{n0} \rangle$ and $\langle F, \phi{n1} \rangle$.  That there is a difference between these integrals is clear but we can calculate the difference explicitly using the same technique as here.

The nice thing about $\latex \langle F, \phi_{n1} \rangle$ is the exact cancellation of $1/\sin(t)$ and the factors in front of the Jacobi polynomial term which means that the integration is easier than the $l=0$ case.  The ‘Lamb shift’ in this case will be the difference between integrating $\Delta_{S^4}(1/\sin(t))$ with respect to $\phi_{n0}$ and $\phi_{n1}$ and it is clearly nonzero.

I will return to this issue with more details in the next days or weeks.