Wheeler-DeWitt Equation in Five Dimensions and Modified QED

TL;DR

This paper derives a five-dimensional Wheeler-DeWitt equation, leading to modified quantum electrodynamics models with different treatments of time and vacuum energy issues.

Contribution

It introduces a five-dimensional framework for quantum gravity and electrodynamics, providing two versions of modified QED with distinct handling of time and vacuum energy.

Findings

One version includes an emergent time parameter.

Another version faces issues with infinite vacuum energy.

The framework unifies gravity and electrodynamics in 5D.

Abstract

We consider the ADM splitting of the Einstein-Hilbert action in five dimensions in the presence of matter that can be either a "point particle", or a set of scalar fields. The Hamiltonian, being a linear superposition of constraints, is equal to zero. Upon quantization, we obtain the Schr\"odinger equation for a wave functional, \Psi, that depends on the matter degrees of freedom, and on the 5D gravity degrees of freedom. After the Kaluza-Klein splitting, the functional Schr\"odinger equation decomposes so that it contains a part due to 4D gravity, a part due to electrodynamics, and a part due to matter. Depending on choice of the matter term, we obtain two different versions of a modified quantum electrodynamics. In one version, time automatically appears, and there is no problem with infinite vacuum energy density of matter fields, whereas in the other version such problems exist.