Graviton corrections to Maxwell’s equations

We use dimensional regularization to compute the one loop quantum gravitational contribution to the vacuum polarization on flat space background. Adding the appropriate Bogoliubov-Parsiuk-Hepp-Zimmermann counterterm gives a fully renormalized result which we employ to quantum correct Maxwell’s equations. These equations are solved to show that dynamical photons are unchanged, provided the free state wave functional is appropriately corrected. The response to the instantaneous appearance of a point dipole reveals a perturbative version of the long-conjectured, “smearing of the light cone”. There is no change in the far radiation field produced by an alternating dipole. However, the correction to the static electric field of a point charge shows strengthening at short distances, in contrast to expectations based on the renormalization group. We check for gauge dependence by working out the vacuum polarization in a general 3-parameter family of covariant gauges.

Figure 1

A two loop contribution to the self-mass-squared in $\lambda {\varphi }^4$ theory.

Figure 2

Lowest order gravitational effect on light due to quantum fluctuations of matter. Photon lines are wavy, graviton lines are winding, and matter lines are solid.

Figure 3

Graviton contributions to the one loop vacuum polarization.

Figure 4

Vertex correction included (along with many other diagrams) in the Bjerrum-Bohr result [8], but not in either our result (101) or that of Radkowski [7]. Charged scalar lines are solid with an arrow, photon lines are wavy and graviton lines are winding.