5.1 Correspondence Principle, Uncertainty Principle and the Fractalization of Quantum Spacetime

If history of physics is any guide, conflicting scientific paradigms, as outlined in Figure 1, generally lead to broader and more predictive theories. Thus, one would expect that a synthesis of general relativity and quantum theory will provide, among other things, a deeper insight into the nature and structure of space and time. Thus, reflecting on those two major revolutions in physics of this century, Edward Witten writes [16], ``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.''.

If spacetime is a derived concept, then is seems natural to ask, ``what is the main property of the fuzzy stuff, let us call it quantum spacetime, that replaces the smoothness of the classical spacetime manifold, and what is the scale of distance at which the transition takes place?''. Remarkably, the celebrated Planck length represents a very near miss as far as the scale of distance is concerned. The new source of fuzziness comes from string theory, specifically from the introduction of the new fundamental constant 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.''.

That new quantum mechanical framework may well constitute the core of the yet undiscovered $M$-Theory, and the non perturbative functional quantum mechanics of string loops that we have developed in recent years may well represent a first step on the long road toward a matrix formulation of it. If this is the case, a challenging testing ground is provided by the central issue of the structure of quantum spacetime. This question was analyzed in Ref. [6] and we limit ourselves, in the remainder of this subsection, to a brief elaboration of the arguments presented there.

The main point to keep in mind, is the already mentioned analogy between ``loop quantum mechanics'' and the ordinary quantum mechanics of point particles. That analogy is especially evident in terms of the new areal variables, namely, the spacelike area enclosed by the string loop, given by Eq. (18), and the timelike, proper area of the string manifold, given by Eq. (4). With that choice of dynamical variables, the reparametrized formulation of the Schild action principle leads to the classical energy per unit length conservation $${H = {\mathcal{E}}}$$. Then, the loop wave equation can be immediately written down by translating this conservation law in the quantum language through the Correspondence Principle

$$P _{\mu \nu} (s) \longrightarrow \frac{i}{\sqrt{x ^{\prime 2} (s)}} \frac{\delta}{\delta \sigma ^{\mu\nu} (s)}$$ (34)

$$H \longrightarrow - i \frac{\partial}{\partial A} \quad .$$ (35)

Thus, we obtain the Schrödinger equation anticipated in the introduction,

$$\frac{1}{4 m ^{2} l _{C}} \oint _{C} d \mu (s) \frac{\del... ...\nu} (s)} = i \frac{\partial}{\partial A} \Psi[\sigma ; A]$$ (36)

where we have introduced the string wave functional $\Psi [\sigma ; A]$ as the amplitude to find the loop $C$ with area elements $\sigma ^{\mu \nu} [C]$ as the only boundary of a two-surface of internal area $A$. When no $A$-dependent potential is present in loop space, the wave functional factors out as

$$\Psi [\sigma ; A] = \psi [\sigma] e^{-i A{\mathcal E}}$$ (37)

and Eq. (36) takes the Wheeler-DeWitt form

$$\frac{1}{4} \oint _{C} d \mu (s) \frac{\delta ^{2} \psi [... ...} - m ^{2} l _{C} {\mathcal{E}} \psi[\sigma] = 0 \quad .$$ (38)

Alternatively, one can exchange the area derivatives with the more familiar functional variations of the string embedding through the tangential projection

$$x ^{\prime \mu} (s) \frac{\delta}{\delta \sigma ^{\mu \nu} (s)} = \frac{\delta}{\delta x ^{\nu} (s)} \quad ,$$ (39)

where $x ^{\prime \mu} (s)$ is the tangent vector to the string loop. As a consistency check on our functional equation, note that, if one further identifies the energy per unit length ${\mathcal{E}}$ with the string tension by setting $${{\mathcal{E}} = m ^{2}}$$, then, Eq.(38) reads

$$\frac{1}{l _{C}} \int _{0} ^{1} \frac{ds}{\sqrt{x ^{\prime... ...mu} (s) \delta x _{\mu} (s)} = m ^{4} \psi [\sigma] \quad ,$$ (40)

which is the string field equation proposed several years ago by Marshall and Ramond [17]. Note also that the Schrödinger equation is a ``free'' wave equation describing the random drifting of the string representative ``point'' in loop space. Perhaps, it is worth emphasizing that this ``free motion'' in loop space is quite different from the free motion of the string in physical space, not only because the physical string is subject to its own tension, i.e. elastic forces are acting on it, but also because drifting from point to point in loop space corresponds physically to quantum mechanically jumping from string shape to string shape. Any such ``shape shifting'' 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 found to be $${D _{H} = 3}$$. In order to reproduce this result, we need the gaussian form of a string wave-packet, which was constructed in Ref. [6] as an explicit solution of the functional Schrödinger equation for loops. For our purposes Eq. (38) is quite appropriate: rather than Fourier expanding the string coordinates $x ^{\mu} (s)$ and decomposing the functional wave equation into an infinite set of ordinary differential equations, we insist in maintaining the ``wholeness'' of the string and consider exact solutions in loop space, or adopt a minisuperspace approximation quantizing only one, or few oscillation modes, freezing all the other (infinite) ones. In the first case, it is possible to get exact ``free'' solutions, such as the plane wave

$$\Psi [\sigma ; A] \propto \exp i \left\{ {\mathcal{E}}... ... \oint _{C} x ^{\mu} d x ^{\nu} P _{\mu \nu} (x) \right\}$$ (41)

which is a simultaneous eigenstate of both the area momentum and Hamiltonian operators. This wave functional describes a completely delocalized state: any loop shape is equally likely to occur and therefore the string has no definite shape. A physical state of definite shape is obtained by superposition of the fundamental plane wave solutions (41). The quantum analogue of a classical string is the Gaussian wave packet:

$$\Psi [\sigma , A] = \left[ \frac{1}{2 \pi (\Delta \sigma) ^{2}} \right] ^{3/4} \left( 1 + \frac{i A}{m ^{2} (\Delta \sigma) ^{2}} \right) ^{-3/2} \times$$

$$\quad \times \exp \left\{ \frac{1}{1 + (i A / m ^{2} (\Delta\sigm... ...ma) ^{2}} + i \oint _{C} x ^{\mu} d x ^{\nu} P _{\mu \nu} (x) + \right. \right.$$

$$\qquad \left. \left. - \frac{i A}{4 m ^{2} l _{C}} \oint _{C} d \mu (s) P _{\mu \nu} P ^{\mu \nu} \right] \right\}$$ (42)

where the width $\Delta\sigma$ of the packet at $A = 0$ represents the area uncertainty. From the solution (42) one can derive some interesting results. First, we note that the center of the wave packet drifts through loop space according to the stationary phase principle

$$\sigma ^{\mu \nu} [C] - \frac{A}{m ^{2}} P ^{\mu \nu} [C] = 0$$ (43)

and spreads as $A$ increases

$$\Delta \sigma \longrightarrow \Delta \sigma (A) = \Delt... ... ^{2}}{4 m ^{4} (\Delta \sigma) ^{4}} \right) ^{1/2} \quad .$$ (44)

Thus, as the areal time $A$ increases, the string ``decays'', in the sense that it loses its sharply defined initial shape, but in a way which is controlled by the loop space uncertainty principle

$$\frac{1}{2} \Delta \sigma ^{\mu \nu} [C] \Delta P _{\mu\nu} [C] \gtrsim 1 \quad , \qquad \hbox{in natural units}$$ (45)

involving the uncertainty in the loop area, and the rate of area variation.

The central result that follows from the above equations, is that the classical world-sheet of a string, a smooth manifold of topological dimension two, turns into a fractal object with Hausdorff dimension three as a consequence of the quantum areal fluctuations of the string loop [6]. With historical hindsight, this result is not too surprising. Indeed, Abbott and Wise, following an earlier insight by Feynman and Hibbs, have shown in Ref. [18] that the quantum trajectory of a point-like particle is a fractal of Hausdorff dimension $2$. Accordingly, one would expect that the limiting Hausdorff dimension of the world-sheet of a string increases by one unit since one is dealing with a one parameter family of $1$-dimensional quantum trajectories. 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. As a matter of fact, the Gaussian wave packet (42) allows us to extend the Abbott and Wise calculation to the string case. By introducing the analogue of the ``DeBroglie wavelength $\Lambda$''as the inverse modulus of the loop momentum⁷

$$\frac{1}{2} P ^{\mu \nu} P _{\mu \nu} = \Lambda ^{-2}$$ (46)

one finds:

1. at low resolution, i.e., when the area uncertainty $\Delta \sigma \gg \Lambda$, the Hausdorff dimension matches the topological value, i.e., $D _{H} = 2$;
2. at high resolution, i.e., when the area uncertainty $\Delta \sigma \ll \Lambda$, the Hausdorff dimension increases by one unit, i.e., $D _{H} = 3$.

Hence, quantum string dynamics can be described in terms of a fluctuating Riemannian $2$-surface only when the observing apparatus is characterized by a low resolution power. As smaller and smaller areas are approached, the graininess of the world-sheet becomes manifest. Then a sort of de-compactification occurs, in the sense that the thickness of the string history comes into play, and the ``world-surface'' is literally fuzzy to the extent that its Hausdorff dimension can be anything between its topological value of two and its limiting fractal value of three.