5. Conclusions, some final remarks, and outlook

Classical string dynamics is based on the simple and intuitive notion that the world-sheet of a relativistic string consists of a smooth, two-dimensional manifold embedded in a preexisting spacetime. Switching from classical to quantum dynamics changes this picture in a fundamental way. In the path integral approach to particle quantum mechanics, Feynman and Hibbs were first to point out that the trajectory of a particle is continuous but nowhere differentiable [7]. This is because, according to Heisenberg's principle, when a particle is more and more precisely located in space, its trajectory becomes more and more erratic. Abbott and Wise were next to point out that a particle trajectory, appearing as a smooth line of topological dimension one, turns into a fractal object of Hausdorff dimension two, when the resolution of the detecting apparatus is smaller than the particle DeBroglie wavelength [8]. Likewise, extending the path integral approach to the string case, one must take into account the coherent contributions from all the world-sheets satisfying some preassigned boundary conditions, and one might expect that a string quantum world-sheet is a fractal, non-differentiable surface. However, to give a quantitative support to this expectation is less immediate than it might appear at first glance. From an intuitive point of view, each string bit can be localized only with a finite resolution in position and momentum. Thus, a string loses its classical, well defined geometric shape, and its world-sheet appears fuzzy as a consequence of quantum fluctuations. However, the notion of a string world-sheet is fully relativistic, and the implementation of fractal geometry in high energy physics is an open field of research with many aspects of it still under discussion [9]. In this paper, we have argued that it is advantageous to work with a first quantized formulation of string dynamics. The canonical quantization approach consists in Fourier analyzing the string vibrations around an equilibrium configuration and assign the Fourier coefficients the role of ladder operators creating and annihilating infinitely many vibration modes. Based on our previous work [2], [3], we have suggested an alternative view of a closed string quantum vibrations, and we readily admit that the exact relationship between the two quantization schemes is unclear and requires a more exhaustive investigation. However, the conceptual difference between the two approaches is sharp: we are quantizing the string motion not through the displacement of each point on the string, but through the string shape. The outcome of this novel approach is a quantum mechanics of loops, whose spacetime interpretation involves quantum shape shifting transitions, i.e., quantum mechanical jumps among all possible string geometric shapes. The emphasis on string shapes, rather than points, represents a departure from the canonical formulation and requires an appropriate choice of dynamical variables, namely, the string configuration tensor $\sigma (C)$, and the areal time A. This functional approach enables us to extend the quantum mechanical discussion by Abbott and Wise to the case of a relativistic (closed) string. It may seem somewhat confusing, if not contradictory, that we deal with a relativistic system in a quantum mechanical, i.e., non relativistic framework, as opposed to a quantum field theoretical framework [10]. However, this is not new in theoretical physics [11]. In any case, one has to realize that there are two distinct levels of discussion in our approach. At the spacetime level, where the actual deformations in the string shape take place, the formulation is fully relativistic, as witnessed by the covariant structure of our equations with respect to the Lorentzian indices. However, at the loop space level, where each ``point'' is representative of a particular loop configuration, our formulation is quantum mechanical, in the sense that the string coordinates $\sigma$ and A are not treated equally, as it is manifest, for instance, in the loop Schrödinger equation (7). As a matter of fact, this is the very reason for referring to that equation as the ``Schrodinger equation'' of string dynamics: the timelike variable A enters the equation through a first order partial derivative, as opposed to the functional ``laplacian'', which is of second order with respect to the spacelike variables $\sigma$. Far from being an artifact of our formulation, we emphasize that this spacetime covariant, quantum mechanics of loops, is a direct consequence of the Hamilton-Jacobi formulation of classical string dynamics. This is especially evident in the form of the classical area- Hamiltonian, equation (11), from which, via the correspondence principle, we have derived the loop Schrödinger equation.
Incidentally, our formulation raises the interesting question as to whether it is possible to ``covariantize'' the Schrödinger equation in loop space, i.e., to treat the spacelike and timelike generalized coordinates on the same footing. At the moment this is an open problem. However, if history is any guide, one might think of generalizing the loop Shrodinger equation into a functional Klein-Gordon equation, or, to follow Dirac's pioneering work and take the ``square root'' of the functional laplacian in order to arrive at a first order functional equation for string loops. This second route would involve an extension of Clifford's algebra along the lines suggested, for instance, by Hosotani in the case of a membrane [4]. Be that as it may, with our present quantum mechanical formulation, we have given a concrete meaning to the fractalization of a string orbit's in spacetime in terms of the shape uncertainty principle. We have concluded that the Hausdorff dimension of a quantum string's world-surface is three, and that two distinct geometric phases exist above and below the loop De Broglie area. We must emphasize that D <sub>H</sub> = 3 represents a limiting value of the Hausdorff dimension, in the sense specified in Section 4. In actual fact, the world-surface of a quantum string is literally ``fuzzy'' to a degree which depends critically on the parameter $\beta$, the ratio between temporal and spatial resolution. Self similarity, we have shown, corresponds to a constant value of $\beta$, with D _H = 3. In such a case, the shape shifting fluctuations generated by petal addition, effectively give rise to a full transverse dimension in the string stack.
As a final remark, we note that the quantum mechanical approach discussed in this paper is in no way restricted to string-like objects. In principle, it can be extended to any quantum p-brane, and we anticipate that the limiting value of the corresponding fractal dimension would be D _H = p + 2. Then, if the above over all picture is correct, p-brane fuzziness not only acquires a well defined meaning, but points to a fundamental change in our perception of physical spacetime. Far from being a smooth, four-dimensional manifold assigned ``ab initio'', spacetime is, rather, a ``process in the making'', showing an ever changing fractal structure which responds dynamically to the resolving power of the detecting apparatus.