Abstract:
We explain 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). The conditions are are equivalent to positivity constraints on gravitational amplitudes, respectively Hilbert space positivity (i) and statistical positivity (ii).
Requiring a statistical interpretation places additional constraints on all wormhole amplitudes, giving a concrete diagnostic of whether the GPI computes an average or not. To exemplify the difference between the two constraints, we present a class of gravitational toy models that satisfy are Hilbert space positive but fail statistical positivity; 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.
Requiring a statistical interpretation places additional constraints on all wormhole amplitudes, giving a concrete diagnostic of whether the GPI computes an average or not. To exemplify the difference between the two constraints, we present a class of gravitational toy models that satisfy are Hilbert space positive but fail statistical positivity; 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.
The constraints can be leveraged into a new form of Gravitational bootstrap for gravitational effective theories. We discuss checks and applications of gravitational bootstrap, in particular to 3d gravity and to axion wormholes which lead to a proof of the axion Weak Gravity Conjecture. We will present ongoing and future work on other applications of the gravitational bootstrap.