Quantizing the Full Einstein Equations

We quantized the full Einstein equations in a globally hyperbolic spacetime \(\textit{N} = \textit{N}^{n+1}, n \ge 3\) , and found solutions of the resulting hyperbolic equation in a fiber bundle E which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second order ordinary differential equation in \(\mathbb {R}_+\) . In case \(\textit{n} \ge 17\) and provided the cosmological constant \(\Lambda \) is negative the temporal eigenfunctions are eigenfunctions of a self-adjoint operator \(\hat{H}_0\) such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space.