Unimodular Hartle-Hawking wave packets and their probability interpretation - Alexandre,B; Magueijo,J (Physical Review D, 2023)
We reexamine the Hartle-Hawking wave function from the point of view of a quantum theory whichstarts from the connection representation and allows for off-shell nonconstancy of Λ (as in unimodulartheory), with a concomitant dual relational time variable. By translating its structures to the metricrepresentation we find a nontrivial inner product rendering wave packets of Hartle-Hawking wavesnormalizable and the time evolution unitary; however, the implied probability measure differssignificantly from the naive jψj2. In contrast with the (monochromatic) Hartle-Hawking wave function,these packets form traveling waves with a probability peak describing de Sitter space, except near thebounce, where the incident and reflected waves interfere, transiently recreating the usual standing wave.Away from the bounce the packets get sharper both in metric and connection space, an apparentcontradiction with Heisenberg’s principle allowed by the fact that the metric is not Hermitian, eventhough its eigenvalues are real. Near the bounce, the evanescent wave not only penetrates into theclassically forbidden region but also extends into the a2 < 0 Euclidean domain. We work out thepropagators for this theory and relate them to the standard ones. The a ¼ 0 point (aka the “nothing”) isunremarkable, and in any case a wave function peaked therein is typically non-normalizable and/orimplies a nonsensical probability for Λ (which the Universe would preserve forever). Within this theory itmakes more sense to adopt a Gaussian state in an appropriate function of Λ, and use the probabilityassociated with the evanescent wave present near the time of the bounce as a measure of the likelihood ofcreation of a pair of time-symmetric semiclassical Universes.