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.
