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If a state can be a superposition of energy states, and mass equals energy (special relativity), and mass curves space-time (general relativity), then could we say that space-time around a quantum system that is in a superposition of states is also in a "superposition of curvatures"?

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Good question. Absolutely, save for the fact that as of yet we haven't found or interpreted experimental results about gravity in the quantum realm. –  kηives Jul 20 '12 at 16:57
Somebody did this experimentally--- they did a Cavendish experiment triggered by quantum atomic decay. Of course they didn't see gravity from the other universe. Of course, objective collapse proponents would say that the macroscopic object collapsed. –  Ron Maimon Jul 21 '12 at 3:14
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You would ask most physicists (except sir Roger Penrose) and they will tell you that you need planck scale energies to measure quantum gravity

I would dare to suggest instead a gravitational generalization of the schr¨odinger cat and the cavendish experiment mashup:

  • take in vacuum space, some mass $M$ of the same order as the one used by cavendish do estimate $G$. Now have some quantum system of two states coupled to a system (the tricky experimental part of the setup) that will provide a thrust to $M$ or not depending of the measured eigenvalue of the quantum system

  • if the thrust system really does not decohere significantly with the environment, you should have the mass (just like the cat) in a superposition of states of different position. So the spacetime curvature must be in a superposition as well

  • now place test masses nearby. Does the phase in the different eigenstates of $M$ affect the spacetime? well it does affect the electromagnetic field, otherwise we wouldn't see interference terms of light, so it should affect gravity as well.

  • We should see interference terms in the gravitational field. Not that hard to detect, if you think that Cavendish did this measurement (less the quantum superposition part) in 1797!!!

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"so it should affect gravity as well." Careful. The EM equations are linear but the GR equations aren't. I'm still thinking about this. Superposition holds only for linear systems, correct? –  Alfred Centauri Jul 20 '12 at 18:31
@AlfredCentauri, the linear field approximation should be entirely valid in this regime, where the gravitational $h_{\mu \nu}$ perturbation is expressed as a Green integral over the retarded sources, just like EM –  diffeomorphism Jul 20 '12 at 18:33
@AlfredCentauri, the nonlinearity of gravity to quantum superposition is just as irrelevant as the nonlinearity of dielectric nonlinear terms to superpositions of light states; in the hamiltonian view, such corrections just shift and transform the spectra, but the eigenstates of those shifted hamiltonians are always in linear superpositions –  lurscher Jul 20 '12 at 19:39
@lurscher, indeed and thanks. While out mowing for awhile, I worked things out. –  Alfred Centauri Jul 20 '12 at 22:36
@RonMaimon, I digested your comment whilst mowing and saw that I was indeed mixing contexts. Thanks. –  Alfred Centauri Jul 20 '12 at 22:46
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