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Next: 4. Fractal Strings Up: Hausdorff Dimension of a Previous: 2.3 Gaussian Loop-wavepacket

   
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

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

where

 \begin{displaymath}K _{\mu \nu} (C) = \langle P _{\mu \nu} (C) \rangle
\end{displaymath} (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

 \begin{displaymath}\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 .
\end{displaymath} (29)

Therefore, the string momentum mean square deviation, or uncertainty squared, is
 
$\displaystyle \Delta \Sigma _{p} ^{2}$ $\textstyle \equiv$ $\displaystyle \frac{1}{2}
\langle
P _{\mu \nu} (C) P ^{\mu \nu} (C)
\rangle
-
\frac{1}{2}
\langle
P _{\mu \nu} (C)
\rangle
\langle
P ^{\mu \nu} (C)
\rangle$  
  = $\displaystyle 3 (\Delta P) ^{2}
\quad .$ (30)

The wave functional in configuration space is obtained by Fourier transforming Eq. (27):
 
$\displaystyle \Psi (C)$ = $\displaystyle \frac{1}{(2 \pi) ^{3/2}}
\int [ {\mathcal{D}} P _{\rho \sigma}]
\Phi( P - K )
\exp
\left(
i
\oint _{C} x ^{\mu} dx ^{\nu}
P _{\mu \nu} (x)
\right)$  
  = $\displaystyle \left[
\frac{(\Delta P) ^{2}}{2 \pi}
\right] ^{3/4}
\exp
\left[
-
\frac{i}{2 (\Delta P) ^{2}}
\oint _{C} x ^{\mu} dx ^{\nu} K_{\mu \nu}
+
\right.$  
    $\displaystyle \qquad \qquad \qquad \qquad \qquad +
\left .
-
\frac{(\Delta P) ^{2}}{4}
\sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C)
\right]
\quad .$ (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
 
$\displaystyle \vert \Psi (C) \vert ^{2}$ = $\displaystyle \left[ \frac{(\Delta P) ^{2}}{2 \pi} \right] ^{3/2}
\exp
\left[
-
\frac{(\Delta P) ^{2}}{4}
\sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C)
\right]$  
  $\textstyle \equiv$ $\displaystyle \left[
\frac{1}{4 \pi (\Delta \sigma) ^{2}} \right] ^{3/2}
\exp\l...
...{1}{4 (\Delta \sigma) ^{2}}
\sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C)
\right]$ (32)

centered around the vanishing loop with a dispersion given by $\Delta \sigma$. By means of the density (32), we obtain
 
$\displaystyle \langle \sigma ^{\mu \nu} (C) \rangle$ = 0 (33)
$\displaystyle \frac{1}{2}
\langle \sigma ^{\mu \nu} (C) \sigma _{\mu\nu} (C) \rangle
=
3 (\Delta \sigma) ^{2}$ = $\displaystyle \frac{3}{2 (\Delta P) ^{2}}
\quad .$ (34)

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

 \begin{displaymath}\Delta \Sigma _{\sigma}
\Delta \Sigma _{p}
=
\frac{3}{\sqrt{2}}
\quad , \quad
(\hbar = 1 \quad \mathrm{units})
\quad .
\end{displaymath} (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
$\displaystyle \langle P _{\mu \nu} (C) \rangle$ = 0 (36)
$\displaystyle \left[
\frac{1}{2}
\langle P _{\mu \nu} (C) P ^{\mu \nu} (C) \rangle
\right] ^{1/2}$ = $\displaystyle \sqrt{3} (\Delta P)
=
\Delta \Sigma _{p}
\quad .$ (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.


next up previous
Next: 4. Fractal Strings Up: Hausdorff Dimension of a Previous: 2.3 Gaussian Loop-wavepacket

Stefano Ansoldi
Department of Theoretical Physics
University of Trieste
TRIESTE - ITALY