Quantum Gravity by the Complex Canonical Theory and its Application to Cosmology

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The results of the research during the period of two years are summarized as follows.1) We can construct a consistent framework for canonical quantum theories of totally constrained systems including general relativity only in terms of quantum constraints by extending the conventional framework of quantum mechanics.2) The canonical equation of motion is in general inconsistent with the reality condition if one formally quantizes the complex canonical theory. In the approach explained in 1), however, it is possible to construct a consistent theory by regarding the complex canonical theory as a special representation of the real triad formalism.3) I found an exact solution to all the quantum constraint equations in the complex canonical theory for the case of pure gravity with non-vanishing cosmological constant. This solution is the first exact solution that describes the quantum behavior of generic spacetimes. Though it corresponds to the Hartle-Hawking's wavefunction in the spatially homogeneous sector, it has one important new feature that it extends to the classically forbidden region and falls off there.4) The DeSitter spacetime is the unique Lorentzian WKB orbit described by this solution. In the cosmological context this suggests a picture that the universe described by a Lorenzian metric is created from an ensemble of Eulidean spacetimes.