Wheeler-DeWitt equation in five dimensions and modified QED

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 Schrödinger equation for a wave functional, Ψ, that depends on the matter degrees of freedom, and on the 5D gravity degrees of freedom. After the Kaluza-Klein splitting, the functional Schrödinger 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.