We present the theory of tunneling geometries, which describes in the language of analytic continuation the nucleation of the Lorentzian universe from the Euclidean spacetime. We reformulate the underlying no-boundary wave function in the manifestly unitary representation of true physical variables and calculate it in the one-loop approximation. For this purpose a special technique of complex extremals is developed, which reduces the formalism of complex tunneling geometries to real ones, and also the method of collective variables is applied, separating the macroscopic degrees of freedom from the perturbative microscopic modes. The quantum distribution of Lorentzian universes on the space of collective variables incorporates the probability conservation and boils down to the partition function of quasi-de Sitter gravitational instantons weighted by their Euclidean effective action. The over-Planckian behavior of their distribution is determined by the anomalous scaling of the theory, which serves as a criterion for the high-energy normalizability of the cosmological wave function and the validity of the semiclassical expansion. It also provides a calculational scheme for obtaining the quantum scale of inflation which was recently shown to establish a sound link between quantum cosmology, inflation theory, and particle physics in the model with a nonminimally coupled inflaton field.

• Received 13 June 1994

DOI:https://doi.org/10.1103/PhysRevD.50.5093