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Pole prescription that preserves both renormalisability and complete positivity

Determine whether there exists a pole prescription for renormalisation in the classical–quantum gravity path-integral formulation that simultaneously renders the theory renormalisable and preserves complete positivity of the dynamics after renormalisation.

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Background

The authors show a formal correspondence between their classical–quantum gravitational path integral and quadratic gravity, suggesting renormalisability. However, unlike a quantum amplitude, the path integral here computes probabilities, imposing a complete-positivity (CP) requirement on the dynamics.

They provide tree-level evidence (for the scalar mode) that a pole prescription can yield positive semidefinite correlators, but do not establish a general proof that a single prescription guarantees both renormalisability and CP after renormalisation.

References

"While the formal correspondence of the path integral of to that of quadratic gravity, shows that there is a pole-prescription such that the theory is renormalisable, we stop short of claiming that the theory has been shown to be renormalisable, because we have not proven that their is a pole prescription such that the theory is simultaneously renormalisable, and satisfies the property that the dynamics remains completely positive (CP) after renormalisation."

— Renormalisation of postquantum-classical gravity (2402.17844 - Grudka et al., 27 Feb 2024) in Introduction