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Limits of fractality: Zeno boxes and relativistic particles

L. S. Schulman
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Abstract

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 article, 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 $Δx\sim Δt (\hbar/{mL})$ with $\Dt$ the interval between measurements and $L$ is the size of the box. For a relativistic particle, the lower bound is the Compton wavelength, $\hbar/mc$. The key step in deriving both results is to write the propagator as a sum over classical paths.

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L. S. Schulman. 2001-09-28. Limits of fractality: Zeno boxes and relativistic particles. https://doi.org/10.1016/s0960-0779(02)00027-9

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