Operator ordering and consistency of the wave function of the Universe

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 wave function can only be consistently defined for particular choices of operator ordering in the Wheeler-DeWitt equation. The requirement of regularity of the wave function has the particular consequence that the probability amplitude, which has been used previously in the literature in discussions of issues such as the prediction of inflation, is likewise ill defined for certain choices of operator ordering with Vilenkin’s boundary condition. By contrast, the Hartle-Hawking no-boundary wave function can be consistently defined within these models, independently of operator ordering. The significance of this result is discussed within the context of the debate about the predictions of semiclassical quantum cosmology. In particular, it is argued that inflation cannot be confidently regarded as a “prediction” of the tunneling wave function, for reasons similar to those previously invoked in the case of the no-boundary wave function. A synthesis of the no-boundary and tunneling approaches is argued for.