Hausdorff dimension of a quantum string

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.