When the gravity path integral describes a statistical average
Abstract
We find the necessary and sufficient conditions for the gravitational path integral (GPI) to admit an interpretation in terms of (i) an inner product in the Hilbert space of open/closed universes or (ii) a statistical average of boundary observables. The conditions needed for (i) are necessary but not sufficient for (ii). Requiring the stronger condition (ii) places additional constraints on all wormhole amplitudes, giving a concrete diagnostic of whether the GPI computes an average or merely a pseudo-average. To exemplify the difference between the two constraints, we present a class of gravitational toy models that satisfy (i) but fail the stronger condition (ii); to emphasize the power of the stronger condition, we initiate a bootstrap study of statistical ensembles based on condition (ii), finding that the resulting constraints substantially sharpen the bounds on the moments of the ensemble obtained by imposing (i). More broadly, both of these positivity conditions impose nontrivial constraints on wormhole amplitudes and thereby restrict which gravitational effective field theories can be embedded in consistent theories of quantum gravity.
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BibTeX
@misc{ubaldo2026when,
title = {{When the gravity path integral describes a statistical average}},
author = {Gabriele Di Ubaldo and Luca V. Iliesiu and Henry W. Lin and Cynthia Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16113},
eprint = {2609.16113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16113},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - When the gravity path integral describes a statistical average
AU - Gabriele Di Ubaldo
AU - Luca V. Iliesiu
AU - Henry W. Lin
AU - Cynthia Yan
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16113
UR - https://arxiv.org/abs/2609.16113
AB - We find the necessary and sufficient conditions for the gravitational path integral (GPI) to admit an interpretation in terms of (i) an inner product in the Hilbert space of open/closed universes or (ii) a statistical average of boundary observables. The conditions needed for (i) are necessary but not sufficient for (ii). Requiring the stronger condition (ii) places additional constraints on all wormhole amplitudes, giving a concrete diagnostic of whether the GPI computes an average or merely a pseudo-average. To exemplify the difference between the two constraints, we present a class of gravitational toy models that satisfy (i) but fail the stronger condition (ii); to emphasize the power of the stronger condition, we initiate a bootstrap study of statistical ensembles based on condition (ii), finding that the resulting constraints substantially sharpen the bounds on the moments of the ensemble obtained by imposing (i). More broadly, both of these positivity conditions impose nontrivial constraints on wormhole amplitudes and thereby restrict which gravitational effective field theories can be embedded in consistent theories of quantum gravity.
ER -