Quantum Cosmology and Wormhole

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(1) (2+1) dimensional quantum gravity1-1 It is shown that the trajectory of a torus universe is a semi-circle in the moduli space. The operator ordering ambiguity is resolved by imposing the modular invariance in the Wheeler-DeWitt equation. The wave function turns out to be a Maass function.1-2 In the Einstein gravity with a negative cosmological constant, we have explicitly constructed a topology changing process in the WKB approximation. The wave function is shown to be localized in the hyperbolic geometry. The particles as defects are also created.(2) We have completely classified topology (except H^3) and dynamical degrees of freedom of compact locally homogeneous universes.(3) It is shown that the effect of baby universes is unfavorable to the dimensional reduction a la Kaluza-Klein.