European Journal of Physics

Brownian representation of fractal quantum paths

Didier Sornette

Published Nov 1, 1990• 22 citations

Abstract

The analogy between quantum 'paths' and random walks of a Brownian particle is extended to the relativistic regime. This yields a coherent picture of the fine structure of quantum paths: these can be viewed as analogies of the trajectory of a random walker. The fractal dimension of quantum paths is df=2 at scales large compared with the Compton wavelength Lambda but small compared with the de Broglie wavelength lambda and becomes df=1 at scales smaller than Lambda . The transition between the two regimes is exhibited explicitly and is analogous to the cross-over from ballistic to diffusive motion of Brownian particles. The analogue of the collisions in Brownian motion is electron-positron pair creations and annihilations, and the unit thermodynamic free energy kT corresponds to the rest mass mc2 of the particle.

DOI: 10.1088/0143-0807/11/6/004

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