Canonical quantization of a closed Euclidean universe with a cosmological constant

We study canonical quantization of a closed Euclidean universe with a cosmo logical constant and a massless scalar field. Tile closed Euclidean univers e with art ordinary matter state can be matched at a finite radius only wit h tile closed Lorentzian universe with tile Wick-rotated exotic slate. The exotic state provides the Lorentzian universe with a potential barrier exte nding from the cosmological singularity tu tile classical turning point and corresponding to tile Euclidean geometry through tile Wick-rotation and av oids the singularity problem at the matching boundary. lie find analyticall y the approximate wave functions for quantum creation of tile Universe from nothingness. We prescribe the Hartle-Hawking's no-boundary wave function, the Linde's wave function, and tile Vilenkin's tunneling wave function. ill particular, we find the wave function for the Euclidean geometry, whose se miclassical solution is regular at the matching boundary with the Lorentzia n geometry but singular ai tile cosmological singularity.