On spin coherence and minimum length in recombination experiments

In this work, we have inquired about the effects of the minimum length hypothesis on the recombination of a delocalized mass. The question stems from a class of gedankenexperiments involving the gravitational interaction of such a delocalized system with another distant body, while the former is being recombined. These experiments are aimed at investigating potential violations of causality if gravity is quantum in nature. We focus specifically on the case of the system being split in a superposition of spatially separated spin states, which bears a strong resemblance to the Humpty-Dumpty effect studied by Schwinger and collaborators. We find that by taking the physical distance between the two parts of the system to be bounded from below by a minimum value (usually of the order of the Planck length), the ability to restore coherence by reconstructing the original spin state is bound by the same conditions found in previous works on the subject where, to avoid tensions between causality and complementarity, emission of gravitons has been invoked as a mean to carry away the coherence of the state. We also find that the velocities experienced by the delocalized system during recombination can approach the speed of light when the aforementioned bounds are reached and, therefore, we propose a relativistic generalization, finding a more complex dependence on mass and velocity for the spin coherence. We conclude by positing an intriguing equivalence between graviton emission and minimal length in accounting for the expected loss of coherence.