Recover full Einstein equations while preserving complete positivity in covariant CQ gravity

Determine whether a covariant classical–quantum (CQ) path-integral model of classical gravity interacting with quantum matter can reproduce all components of Einstein’s equations while maintaining completely positive dynamics, and, if possible, construct such a CP-preserving CQ theory that is consistent with observations.

Background

Prior work on covariant CQ dynamics for classical gravity coupled to quantum matter established a framework that reproduces only part of Einstein’s equations (e.g., the trace) while ensuring complete positivity of the evolution. Extending this framework to recover the full set of Einstein equations without sacrificing complete positivity remains a key theoretical challenge.

Resolving this issue would clarify whether fully consistent, observation-ready CQ models of gravity are viable, and would sharpen the comparison between CQ approaches and standard semiclassical or fully quantum gravity frameworks.