# Lab Notes for a Scientific Revolution (Physics)

## March 6, 2008

### Electrodynamic Potentials and Non-Linear QED in Kaluza-Klein

I have now added new sections 12, 13 and 14 to the Kaluza-Klein paper earlier posted.  These sections examine the relationship between the electrodynamic potentials and the gravitational potentials, and the connection to QED.  You may view this all at:

Electrodynamic Potentials and Non-Linear QED

Most significantly, these three new sections not only connect to the QED Lagrangian, but, they show how the familiar QED Lagrangian density

${\rm L}_{QCD} =-A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$

emerges in the linear approximation of 5-dimensional Kaluza-Klein gravitational theory.

Then, we go in the opposite direction, to show the QED Lagrangian density / action for non-linear theory, based on the full-blown apparatus of gravitational theory.

Expressed in terms of the electrodynamic field strength $F^{\sigma \tau }$ and currents $J_{\beta }$, this non-linear result is:

${\rm L}_{QCD} =0={\textstyle\frac{1}{8\kappa }} b\overline{\kappa }g^{5\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau } \approx -A^{\beta } J_{\beta } -{\textstyle\frac{1}{4}} F^{\sigma \tau } F_{\sigma \tau }$, (13.6)

where the approximation $\approx$ shows the connection to the linear approximation.  Re-expressed solely in terms of the fifth-dimensional gravitational metric tensor components $g_{5\sigma }$ and energy tensor source components $T_{\beta 5}$, this result is:

$\kappa {\rm L}_{QCD} =0={\textstyle\frac{1}{2}} g^{5\beta } \kappa T_{\beta 5} +{\textstyle\frac{1}{8}} g^{\sigma \alpha } \partial ^{\beta } g_{5\alpha } \left[\partial _{\sigma } g_{5\beta } -\partial _{\beta } g_{5\sigma } \right]$. (14.4)

You may also enjoy the derivations in section 12 which decompose the contravariant metric tensor into gravitons, photons, and the scalar trace of the graviton.

Again, if you have looked at earlier drafts, please focus on the new sections 12, 13 and 14.  Looking for constructive feedback, as always.