What is the Gravitational Field of a Mass in a Spatially Nonlocal Quantum Superposition?

The study of the gravitational field produced by a spatially non-local, superposed quantum state of a massive particle is a thrilling area of modern physics. One question to be answered is whether the gravitational field behaves as the classical superposition of two particles separated by a spatial distance with half the mass located at each position or as a quantum superposition with a far more interesting and subtle behaviour for the gravitational field. Quantum field theory is ideally suited to probe exactly this kind of question. We study the scattering of a massless scalar on such a spatially nonlocal, quantum superposition of a massive particle. We compute the differential scattering cross section corresponding to the interaction coming from the exchange of one graviton. We find that the scattering cross section is not at all represented by the Schr\"odinger-Newton picture of potential scattering from two localized sources with half the mass at each source. We discuss how our result would be lethal to the Schr\"odinger-Newton description of gravitation interacting with quantum matter and would be conducive to considering the gravitational field to be quantized. We comment on the experimental feasibility of observing such effects.

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