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Recovering the full Einstein equations with a metric-proportional diffusion kernel

Establish whether the covariant classical–quantum path integral with diffusion kernel D_{0,μνρσ} proportional to (−g)^{−1/2} g_{μν} g_{ρσ} can recover the complete Einstein field equations in the classical limit, rather than only suppressing deviations from the trace of Einstein’s equations.

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Background

In the covariant classical–quantum path integral formalism, choosing D_{0, μνρσ} ∝ (−g){−1/2} g_{μν} g_{ρσ} ensures diffeomorphism invariance and suppresses departures from the trace of Einstein’s equations. This construction is sufficient to obtain a consistent classical–quantum Nordström gravity model, but it does not guarantee recovery of the full Einstein equations.

The authors note that, with this kernel choice, it remains unresolved whether one can obtain all components of Einstein’s equations in the classical limit, beyond the trace. This is a key point for matching general relativity in appropriate regimes.

References

"As noted in, if $D_{0, \mu \nu \rho \sigma}=D_0 (-g){-1/2}g_{\mu\nu}g_{\rho\sigma}$ with $D_0$ a positive constant, geometries which deviate too far from satisfying the trace of Einstein's equations are suppressed. This allows one to construct a diffeomorphism-invariant and self-consistent theory of classical-quantum Nordstrom gravity but whether one can recover the remainder of Einstein's equations has been an open question."

— Renormalisation of postquantum-classical gravity (2402.17844 - Grudka et al., 27 Feb 2024) in Section 3 (From the classical-quantum path integral to Quadratic Gravity)