3. The Shape Uncertainty Principle

In this section we discuss the new form that the uncertainty principle takes in the functional theory of string loops. Let us consider a Gaussian momentum wave function

$$\Phi (P) = \frac{1}{\left[ \pi (\Delta P) ^{2} \right] ^{3/... ...) \left(P _{\mu \nu} (s) - K _{\mu \nu} \right) ^{2} \right]$$ (27)

where

$$K _{\mu \nu} (C) = \langle P _{\mu \nu} (C) \rangle$$ (28)

represents the string momentum mean value around which the Gaussian wavepacket is centered, and $\Delta P$ is a measure of the momentum dispersion in the wavepacket.
Thus, $K _{\mu \nu} (C)$ is a reparametrization invariant functional of the loop C representing the string ``drift'' through loop space.
In order to arrive at the shape uncertainty principle, we first evaluate the quantum average of the loop squared momentum

$$\frac{1}{2} \langle P _{\mu \nu} (C) P ^{\mu \nu} (C) \ran... ... _{\mu\nu} (C) K _{\mu \nu} (C) + 3 (\Delta P) ^{2} \quad .$$ (29)

Therefore, the string momentum mean square deviation, or uncertainty squared, is

 

$$\Delta \Sigma _{p} ^{2} \equiv \frac{1}{2} \langle P _{\mu \nu} (C) P ^{\mu \nu} (C) \rangle - \frac{1}{2} \langle P _{\mu \nu} (C) \rangle \langle P ^{\mu \nu} (C) \rangle$$

$$3 (\Delta P) ^{2} \quad .$$ = (30)

The wave functional in configuration space is obtained by Fourier transforming Eq. (27):

 

$$\Psi (C) \frac{1}{(2 \pi) ^{3/2}} \int [ {\mathcal{D}} P _{\rho \sigma}] \Phi( P - K ) \exp \left( i \oint _{C} x ^{\mu} dx ^{\nu} P _{\mu \nu} (x) \right)$$ =

$$\left[ \frac{(\Delta P) ^{2}}{2 \pi} \right] ^{3/4} \exp \left[ - \frac{i}{2 (\Delta P) ^{2}} \oint _{C} x ^{\mu} dx ^{\nu} K_{\mu \nu} + \right.$$ =

$$\qquad \qquad \qquad \qquad \qquad + \left . - \frac{(\Delta P) ^{2}}{4} \sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C) \right] \quad .$$ (31)

This is again a Gaussian wavepacket, whose ``center of mass'' moves in loop space with a momentum $K _{\mu \nu}$. Accordingly, the loop probability density still has a Gaussian form

 

$$\vert \Psi (C) \vert ^{2} \left[ \frac{(\Delta P) ^{2}}{2 \pi} \right] ^{3/2} \exp \left[ - \frac{(\Delta P) ^{2}}{4} \sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C) \right]$$ =

$$\equiv \left[ \frac{1}{4 \pi (\Delta \sigma) ^{2}} \right] ^{3/2} \exp\l... ...{1}{4 (\Delta \sigma) ^{2}} \sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C) \right]$$ (32)

centered around the vanishing loop with a dispersion given by $\Delta \sigma$. By means of the density (32), we obtain

 

$$\langle \sigma ^{\mu \nu} (C) \rangle$$ = 0 (33)

$$\frac{1}{2} \langle \sigma ^{\mu \nu} (C) \sigma _{\mu\nu} (C) \rangle = 3 (\Delta \sigma) ^{2} \frac{3}{2 (\Delta P) ^{2}} \quad .$$ = (34)

Then, comparing Eq. (34) with Eq. (30), we find that the uncertainties are related by

$$\Delta \Sigma _{\sigma} \Delta \Sigma _{p} = \frac{3}{\sqrt{2}} \quad , \quad (\hbar = 1 \quad \mathrm{units}) \quad .$$ (35)

Equation (35) represents the new form that the Heisenberg principle takes when string quantum mechanics is formulated in terms of diffusion in loop space, or quantum shape shifting. Just as a pointlike particle cannot have a definite position in space and a definite linear momentum at the same time, a physical string cannot have a definite shape and a definite rate of shape changing at a given areal time. In other words, a string loop cannot be totally at rest neither in physical nor in loop space: it is subject to a zero-point motion characterized by

$$\langle P _{\mu \nu} (C) \rangle$$ = 0 (36)

$$\left[ \frac{1}{2} \langle P _{\mu \nu} (C) P ^{\mu \nu} (C) \rangle \right] ^{1/2} \sqrt{3} (\Delta P) = \Delta \Sigma _{p} \quad .$$ = (37)

In such a state a physical string undergoes a zero-point shape shifting, and the loop momentum attains its minimum value compatible with an area resolution $\Delta \sigma$.
To keep ourselves as close as possible to Heisenberg's seminal idea, we interpret the lack of a definite shape as follows: as we increase the resolution of the ``microscope'' used to probe the structure of the string, more and more quantum petals will appear along the loop. The picture emerging out of this is that of a classical line turning into a fractal object as we move from the classical domain of physics to the quantum realm of quantum fluctuations. If so, two questions immediately arise:

1.

a classical bosonic string is a closed line of topological dimension one. Its spacetime image consists of a smooth, timelike, two-dimensional world-sheet. Then, if a quantum string is a fractal object, which Hausdorff dimension should be assigned to it?

2.

Is there any critical scale characterizing the classical-to-fractal geometrical transition?

These two questions will be addressed in the next section.