A Partition Function for Quantized Globally Hyperbolic Spacetimes with a Negative Cosmological Constant

In the Chapters [3, 4, 6, 7, 8] we applied our model of quantum gravity for the quantization of the full Einstein equations to globally hyperbolic spacetimes with an asymptotically Euclidean Cauchy hypersurface, to spacetimes with Cauchy hypersurfaces that are products of a Euclidean space with a compact manifold and to a Schwarzschild-AdS and a Kerr-AdS black hole, respectively. In Chapter [5] we quantized the interaction of gravity with the forces of the Standard Model by quantizing the Hamilton condition, i.e., the normal Einstein equation. The resulting quantum systems and the underlying spatial and temporal self-adjoint operators all have similar structures such that we can now apply quantum statistics by first proving, for the temporal Hamiltonian \({H}_o\) , that \(e^{-\beta H_o}, \beta > 0\) ,is of trace class and then, that this result is also valid for the spatial Hamiltonian \({H}_1\) which has the same eigenvalues but with larger multiplicities. Since the lowest eigenvalue is strictly positive the extension of \(e^{-\beta H_1}\) to the corresponding symmetric Fock space is also of trace class and we are thus able to define a partition function \({Z}\) , the operator density \(\rho \) , the entropy \(\textit{S} \) , and the average energy. \(\textit{E}\) We prove that \(\textit{S} \) and \(\textit{E}\) tend to infinity if the cosmological constant \(\varLambda \) tends to 0 and vanish if \(|{\varLambda}|\) tends to infinity. We also conjecture that \(\textit{E}\) is the source of the dark matter and that the dark energy density is a multiple of the eigenvalue of \(\rho \) with respect to the vacuum vector which is \({Z}^{-1}\)