Quantum cosmology with electromagnetism

We investigate the semiclassical quantum cosmology of the spatially homogeneous k=+1 Friedmann-Robertson-Walker minisuperspace model with the electromagnetic field included as a small perturbation on the gravitational background. The wave function is chosen adopting the Hartle-Hawking proposal. Owing to conformal invariance, the electromagnetic field remains in its ground state for all times. At small three-geometries the field contributes to the semiclassical prefactor of the wave function a factor of $a^{\zeta \mathrm{}\left(0\right)\mathrm{}}$, where a is the scale factor and ζ(0)=-(77/180 is the value at zero of the associated generalized ζ function. The semiclassical expansion of the Wheeler-DeWitt equation acquires in the next-to-leading order a divergent sum of the same kind as the corresponding sums coming from fields of spin 0, (1/2, and 2. This suggests that the semiclassical expansion of the analogue of the Wheeler-DeWitt equation might be naturally finite to the next-to-leading order in a theory containing symmetries between fields of different spins.