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, Combinatorial quantum theory and quantized directions, in Advances in Twistor Theory, Eds
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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.