Abstract
We show that the usual Born–Oppenheimer type of approximation used in quantum gravity, in which a semiclassical time parameter emerges from a weak-coupling expansion of the Wheeler–DeWitt constraint, leads to a unitary theory at least up to the next-to-leading order in minisuperspace models. As there are no unitarity-violating terms, this settles the issue of unitarity at this order, which has been much debated in the literature. Furthermore, we also show that the conserved inner product is gauge-fixed in the sense that the measure is related to the Faddeev–Popov determinant associated with the choice of semiclassical time as a reparametrization gauge. This implies that the Born–Oppenheimer approach to the problem of time is, in fact, an instance of a relational quantum theory, in which transition amplitudes can be related to conditional probabilities.
Funding source: Università di Bologna
Award Identifier / Grant number: Unassigned
Acknowledgments
The author thanks Claus Kiefer, Manuel Krämer, Branislav Nikolić, and David Brizuela for useful discussions over the years; Alexander Y. Kamenshchik, Alessandro Tronconi, and Giovanni Venturi for interesting dialogues; and the Dipartimento di Fisica e Astronomia of the Università di Bologna as well as the I.N.F.N. Sezione di Bologna for financial support.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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