Time-of-flight fuzziness from deformed relativistic symmetries

Abstract

A general expectation in quantum gravity is that quantum properties of spacetime manifest themselves as fuzziness, causing an irreducible uncertainty in the measurement of spacetime-related observables. In this work, we derive the time-of-flight fuzziness for free particles within a noncommutative spacetime model with $\kappa$-Poincaré deformed relativistic symmetries. To retain the full quantum structure of the theory we work in the corresponding noncommutative space of worldlines. This provides a convenient framework for constructing quantum states and deriving the probability distributions for the parameters characterizing particle trajectories, allowing us to obtain the probability distribution of particle times of flight. Our results reproduce the well-known systematic time-of-flight correction, leading to energy-dependent departures from the special-relativistic expectation, and simultaneously predict a novel stochastic contribution to the time of flight. While the systematic contribution scales with the ratio between the particle energy and the quantum-gravity energy scale and is amplified by the propagation distance, the stochastic contribution scales as the square root of the product of the quantum-gravity length scale and the travel time, multiplied by a prefactor which depends on the particle velocity.