Quantum cosmology for Tunneling universes

In a quantum cosmological model consisting of a Euclidean region and a Lorentzian region, Hartle-Hawking's no-boundary wave function, and Linde's wave function and Vilenkin's tunneling wave function are briefly described and compared with each other. We put a particular emphasis on semiclassical gravity from quantum cosmology and compare it with the conventional quantum field theory in curved spacetimes. Finally, we discuss the recent debate between Rubakov and Vilenkin on catastrophic particle Production in a tunneling universe within semiclassical gravity.