The paper constructs, for any local C^2 Lorentzian metric, six Poincaré-invariant quantum systems whose empirical distance observable reproduces the local geodesic lengths and thus the curvature, but only because the curvature is built into the chosen states.
Penrose, Theory of quantized directions, in Collected Works, vol 1, pp
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Curved spacetimes from quantum mechanics
The paper constructs, for any local C^2 Lorentzian metric, six Poincaré-invariant quantum systems whose empirical distance observable reproduces the local geodesic lengths and thus the curvature, but only because the curvature is built into the chosen states.