Canonical Quantization of a Closed Euclidean Universe with a Cosmological Constant

Abstract

We study canonical quantization of a closed Euclidean universe with a cosmological constant and a massless scalar field. The closed Euclidean universe with an ordinary matter state can be matched at a finite radius only with the closed Lorentzian universe with the Wick-rotated exotic state. The exotic state provides the Lorentzian universe with a potential barrier extending from the cosmological singularity to the classical turning point and corresponding to the Euclidean geometry through the Wick-rotation, and avoid the singularity problem at the matching boundary. We find analytically the approximate wave functions for quantum creation of the Universe from the {\it nothingness}. We prescribe the Hartle-Hawking's no-boundary wave function, Linde's wave function and Vilenkin's tunneling wave function. In particular, we find the wave function for the Euclidean geometry, whose semiclassical solution is regular at the matching boundary with the Lorentzian geometry but singular at the cosmological singularity.