Quantizing the Full Einstein Equations by Invoking a Scalar Field Map

In the previous chapter we could prove that the temporal eigenfunctions were indeed eigenfunctions of a self-adjoint differential operator provided the spatial dimension n was large enough, namely, \(n\ge 17\) , the cosmological constant \(\varLambda <0\) and provided the eigenvalue \(\mu _0\) of the gravitational Hamiltonian was not too large. In the present chapter we shall prove this result without any restrictions on \(n, \mu _0\) and \(\varLambda \) except \(3\le n\) and \(\varLambda \not =0\) , though we have to assume \(n\not =4\) if \(\varLambda >0\) , by introducing an additional scalar field map in the action functional.