Limits of fractality: Zeno boxes and relativistic particles

Physical fractals invariably have upper and lower limits for their fractal structure. Berry has shown that a particle sharply confined to a box has a wave function that is fractal both in time and space, with no lower limit. In this paper, two idealizations of this picture are softened and a corresponding lower bound for fractality obtained. For a box created by repeated measurements (à la the quantum Zeno effect), the lower bound is Δx Δt( /mL) with Δt the interval between measurements and L the size of the box. For a relativistic particle, the lower bound is the Compton wavelength, /mc. The key step in deriving both results is to write the propagator as a sum over classical paths.