Hausdorff dimension of a quantum string

TL;DR

This paper extends the concept of fractal trajectories in quantum mechanics to relativistic strings, showing their paths form fractal surfaces with Hausdorff dimension three, revealing a new uncertainty principle for strings.

Contribution

It introduces the Hausdorff dimension of quantum string trajectories as fractal surfaces and interprets the resulting fuzziness as a new form of string uncertainty principle.

Findings

Quantum string trajectories are fractal surfaces with Hausdorff dimension three.

The perceived fuzziness depends on the resolution of the detecting apparatus.

A transition from smooth to fractal phase of the string surface is studied.

Abstract

In the path integral formulation of quantum mechanics, Feynman and Hibbs noted that the trajectory of a particle is continuous but nowhere differentiable. We extend this result to the quantum mechanical path of a relativistic string and find that the ``trajectory'', in this case, is a fractal surface with Hausdorff dimension three. Depending on the resolution of the detecting apparatus, the extra dimension is perceived as ``fuzziness'' of the string world-surface. We give an interpretation of this phenomenon in terms of a new form of the uncertainty principle for strings, and study the transition from the smooth to the fractal phase.