Operator ordering and consistency of the wavefunction of the Universe

TL;DR

This paper investigates how operator ordering affects the consistency of the tunneling wavefunction in quantum cosmology, showing that certain orderings lead to ill-defined probabilities, unlike the no-boundary wavefunction.

Contribution

It demonstrates that Vilenkin's tunneling wavefunction is only well-defined for specific operator orderings, contrasting with the robustness of the Hartle-Hawking no-boundary wavefunction.

Findings

Tunneling wavefunction's consistency depends on operator ordering.

Probability amplitude is ill-defined for some orderings with tunneling boundary conditions.

No-boundary wavefunction remains well-defined regardless of operator ordering.

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. 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 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 wavefunction 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 wavefunction, for reasons similar to those previously invoked in the case of the no-boundary wavefunction. A synthesis of the no-boundary and tunneling approaches is argued for.