Operator ordering and consistency of the wavefunction of the Universe

Recent developments concerning the possibility of open inflation in quantum cosmology [1,2] have revived an old debate about foundational issues of that subject [2–4]. It is therefore timely to raise an issue which has been largely overlooked previously, but which in our opinion has a direct bearing on these foundational issues. Much of the current debate originates from differing probability amplitudes calculated in an approximation which assumes that the transition amplitude for nucleation of a universe from “nothing” is dominated by a Euclidean instanton. Different probability amplitudes are assumed to correspond to the different boundary conditions implied by the Hartle-Hawking “no-boundary” proposal [5], and the “tunneling” proposals of Vilenkin [6,7] and Linde [2,8] which are themselves distinct. In particular, the nucleation probability for instanton-dominated transitions is assumed to be