1. Introduction and Synopsis

A classical string is a one-dimensional, spatially extended object, so that its timelike orbit in spacetime is described by a smooth, two-dimensional manifold. However, since the advent of quantum theory and general relativity, the notion of spacetime as a preexisting manifold in which physical events take place, is undergoing a process of radical revision. Thus, reflecting on those two major revolutions in physics of this century, Edward Witten writes [1], ``Contemporary developments in theoretical physics suggest that another revolution may be in progress, through which a new source of ``fuzziness'' may enter physics, and spacetime itself may be reinterpreted as an approximate, derived concept.''. The new source of fuzziness comes from string theory, specifically from the introduction of the new fundamental constant, ($\alpha '$), which determines the tension of the string. Thus, at scales comparable to $(\alpha ') ^{1/2}$, spacetime becomes fuzzy, even in the absence of conventional quantum effects ($\hbar = 0$). While the exact nature of this fuzziness is unclear, it manifests itself in a new form of Heisenberg's principle, which now depends on both $\alpha '$ and $\hbar$. Thus, in Witten's words, while ``a proper theoretical framework for the [new] uncertainty principle has not yet emerged, [...] the natural framework of the [string] theory may eventually prove to be inherently quantum mechanical.''.
The essence of the above remarks, at least in our interpretation, is that there may exist different degrees of fuzziness in the making of spacetime, which set in at various scales of length, or energy, depending on the nature and resolution of the Heisenberg microscope used to probe its structure. In other words, spacetime becomes a sort of dynamical variable, responding to quantum mechanical resolution just as, in general relativity, it responds to mass-energy. The response of spacetime to mass-energy is curvature. Its response to resolution seems to be ``fractalization''. This, in a nutshell, is the central thesis of this paper.
Admittedly, in the above discussion, the term ``fuzziness'' is loosely defined, and the primary aim of this paper is to suggest a precise measure of the degree of fuzziness of the quantum mechanical path of a string. In order to achieve this objective, we need two things, a) the new form of the uncertainty principle for strings, and b) the explicit form of the wave-packet for string loops. Then, we will be able to compute the Hausdorff dimension of a quantum string and to identify the parameter which controls the transition from the smooth phase to the fractal phase.
There are some finer points of this broadly defined program that seem worth emphasizing at this introductory stage, before we embark on a technical discussion of our results. The main point is that, unlike superstring theory, our formulation represents an attempt to construct a quantum mechanical theory of (closed) strings in analogy to the familiar case of point-particles. The ground work of this approach was developed by the authors in two previous papers, in which we have extended the Hamilton-Jacobi formulation and Feynman's path integral approach to the case of classical and quantum closed strings [2], [3]. That work was largely inspired by the line functional approach of Carson and Hosotani [4], and by the non-canonical quantization method proposed by Eguchi [5], and this is reflected by our unconventional choice of dynamical variables for the string, namely, the spacelike area enclosed by the string loop and its 2-form conjugate momentum. Furthermore, central to our own quantum mechanical approach, is the choice of ``time variable'', which we take to be the timelike, proper area of the string manifold, in analogy to the point-particle case.
Presently, we are interested primarily in the analysis of the quantum fluctuations of a string loop. By quantum fluctuations, we mean a random transition, or quantum jump, between different string configurations. Since in any such process, the shape of the loop changes, we refer to it as a ``shape shifting'' process. We find that any such process, random as it is, is subject to an extended form of the uncertainty principle which forbids the exact, simultaneous knowledge of the string shape and its area conjugate momentum. The main consequence of the Shape Uncertainty Principle is the ``fractalization'' of the string orbit in spacetime. The degree of fuzziness of the string world-sheet is measured by its Hausdorff dimension, whose limiting value we find to be $${D _{H} = 3}$$. In order to obtain this result, we need the gaussian form of a string wave-packet, which we construct as an explicit solution of the functional Schrodinger equation for loops. Next, we try to quantify the transition from the classical, or smooth phase, to the quantum, or fractal phase. Use of the Shape Uncertainty Principle, and of the explicit form of the loop wave-packet, enables us to identify the control parameter of the transition with the DeBroglie area characteristic of the loop. Accordingly, the paper is organized as follows: in Section 2, we discuss the basic solutions of the loop Schrödinger equation. These solutions represent the analogue of the plane wave and gaussian wave-packet in ordinary quantum mechanics. In Section 3, we introduce the Shape Uncertainty Principle which governs the shape shifting processes in loop quantum mechanics. Section 4 is devoted to the fractal properties of the string quantum path. We compute its Hausdorff dimension and study the classical-to-fractal transition. Section 5 concludes the paper with a summary of our results and some final remarks on the formal aspects of this work.