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.
A note on Penrose's Spin-Geometry Theorem and the geometry of `empirical quantum angles'
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abstract
In the traditional formalism of quantum mechanics, a simple direct proof of (a version of) the Spin Geometry Theorem of Penrose is given; and the structure of a model of the `space of the quantum directions', defined in terms of elementary $SU(2)$-invariant observables of the quantum mechanical systems, is sketched.
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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.