Operator ordering and consistency of the wavefunction of the Universe
Abstract
We demonstrate in the context of the minisuperspace model consisting of a closed Friedmann-Robertson-Walker universe coupled to a scalar field that Vilenkin's tunneling wavefunction can only be consistently defined for particular choices of operator ordering in the Wheeler-DeWitt equation. This conclusion is based on the explicit investigation of the model with arbitrary operator ordering. The requirement of regularity of the wavefunction has the particular consequence that the probability amplitude, which has been used previously in the literature in discussions of issues such as the length of the period of inflation, is likewise ill-defined for certain choices of operator ordering with Vilenkin's boundary condition. These orderings include the ``d'Alembertian ordering'', which has been found to be the ``natural'' ordering in some previous studies. By contrast, the Hartle-Hawking no-boundary wavefunction can be consistently defined within these models, independently of operator ordering.Keywords
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