Euclidean Maxwell Theory

Abstract

This chapter, motivated by quantum cosmology, studies the quantization of the electromagnetic field on a flat Euclidean background bounded by one three-sphere, or two concentric three-spheres. The conformal anomaly is evaluated by using zeta-function regularization, jointly with Faddeev-Popov formalism for quantum amplitudes. Magnetic boundary conditions set to zero at the boundary the tangential components A k of the electromagnetic potential, the gauge-averaging functional and the ghost zero-form. Electric boundary conditions set instead to zero at the boundary the normal component of the electromagnetic potential, the normal derivative of A k and the normal derivative of the ghost zero-form. The gauges considered are the Lorentz gauge, or the Coulomb gauge, or a more general family of gauges, which reduce to the above choices for particular values of some dimensionless parameters. This means that, for problems with boundaries, the gauge-averaging functional may take explicitly into account the effect of the extrinsic-curvature tensor. In the Lorentz gauge, the mode-by-mode analysis leads to a ζ(0) value in agreement with the geometric results on heat-kernel asymptotics for minimal operators and mixed boundary conditions. To achieve this, it is essential to take into account the contributions of longitudinal, normal and ghost modes.