Limits of fractality: Zeno boxes and relativistic particles

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 $\Delta x\sim \Delta t(\hslash /mL)$ 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, $\hslash /mc$. The key step in deriving both results is to write the propagator as a sum over classical paths.

References in corpus (1)

• Zeno dynamics yields ordinary constraints

Cited by in corpus (3)

• Relativistic probability waves
• Bound Eigenstate dynamics under a sudden shift of the well's wall
• Contrary Inferences for Classical Histories within the Consistent Histories Formulation of Quantum Theory