Two standard derivations of semiclassical gravity are equivalent as partial classical limits, so the equation predicts only the expectation value of the Einstein tensor.
This derivation further indicates that, strictl y speak- ing, only a statistical interpretation of the Einstein tens or as ⟨Gµν ⟩ψ is viable
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Structure and statistical properties of the semiclassical Einstein equations
Two standard derivations of semiclassical gravity are equivalent as partial classical limits, so the equation predicts only the expectation value of the Einstein tensor.