We use a one-dimensional model system to compare the predictions of two different yardsticks to compute the position of a particle from its quantum field theoretical state. Based on the first yardstick (defined by the Newton-Wigner position operator), the spatial density can be arbitrarily narrow, and its time evolution is superluminal for short time intervals. Furthermore, two spatially distant particles might be able to interact with each other outside the light cone, which is manifested by an asymmetric spreading of the spatial density. The second yardstick (defined by the quantum field operator) does not permit localized states, and the time evolution is subluminal.
Wagner, R E.; Ware, M R.; Stefanovich, E V.; Su, Qichang; and Grobe, Rainer, "Local and nonlocal spatial densities in quantum field theory" (2012). Faculty publications – Physics. Paper 9.