Quantum superpositions of Minkowski spacetime

Within any anticipated unifying theory of quantum gravity, it should be meaningful to combine the fundamental notions of quantum superposition and spacetime to obtain so-called "spacetime superpositions": that is, quantum superpositions of different spacetimes not related by a global coordinate transformation. Here we consider the quantum-gravitational effects produced by superpositions of periodically identified Minkowski spacetime (i.e.\ Minkowski spacetime with a periodic boundary condition) with different characteristic lengths. By coupling relativistic quantum matter to fields on such a spacetime background (which we model using the Unruh-deWitt particle detector model), we are able to show how one can in-principle "measure" the field-theoretic effects produced by such a spacetime. We show that the detector's response exhibits discontinuous resonances at rational ratios of the superposed periodic length scale.