4.2 Classical-to-fractal geometric transition

The role of area resolution, as pointed out in the previous sections, leads us to search for a critical area characterizing the transition from classical to fractal geometry of the string stack. To this end, we consider tha case in which the string possesses a non vanishing average momentum, in which case we may use a gaussian wave packet of the form (27). The corresponding wave functional, in the $\sigma$-representation, is

$$\Psi _{K} (C ; A) = \frac{[ (\Delta \sigma) ^{2} / 2 \pi] ^{3/4}} {[ (\Delta \sigma) ^{2} + i A / 2 m ^{2}] ^{3/2}} \cdot$$

$$\quad \cdot \exp \left[ - \frac{ \left( \sigma ^{\mu \nu} \sigma ... ... \nu} / 4 m ^{2} \right) } {2 [ (\Delta \sigma) ^{2} + i A / 2 m ^{2}]} \right]$$ (44)

where we have used equation (34) to exchange $\Delta P$ with $\Delta \sigma$. The corresponding probability density ``evolves'' as follows

 

$$\vert \Psi _{K} (C ; A) \vert ^{2} = \frac{(2 \pi) ^{-3/2}} { [ (\Delta \sigma) ^{2} + A ^{2} / 4 (\Delta\sigma) ^{2} m ^{4} ] ^{3/2} } \cdot$$

$$\qquad \qquad \qquad \qquad \cdot \exp \left[ - \frac{ \left( \si... ...\sigma) ^{2} + A ^ {2} / 4 (\Delta \sigma) ^{2} m ^{4} \right]} \right] \quad .$$ (45)

Therefore, the average area variation $\langle \Delta S \rangle$, when the loop wave packet drifts with a momentum $K _{\mu \nu} (C)$, is

$$\langle \Delta S \rangle \equiv \left[ \int [ {\mathcal{D}... ...(C) \vert \Psi _{K} (C ; \Delta A) \vert ^{2} \right] ^{1/2}$$ (46)

For our purpose, there is no need to compute the exact form of the mean value (46), but only its dependence on $\Delta \sigma$. This can be done in three steps:

1.

introduce the adimensional integration variable

$$Y ^{\mu \nu} (C) \equiv \frac{\sigma ^{\mu \nu} (C)}{\Delta \sigma} \quad ;$$ (47)

2.

shift the new integration variable as follows

$$Y ^{\mu \nu} (C) \rightarrow \bar{Y} ^{\mu \nu} (C) \equiv... ...Delta A K ^{\mu \nu}}{2 m ^{2} (\Delta \sigma) ^{2}} \quad ;$$ (48)

3.

rescale the integration variable as

$$\bar{Y}^{\mu \nu} (C) \rightarrow Z ^{\mu \nu} (C) \equiv ... ...) ^{2}}{4 m ^{4} (\Delta \sigma) ^4} \right] ^{1/2} \quad .$$ (49)

Then, we obtain

 

$$\langle \Delta S \rangle \frac{\Delta A}{\sqrt 2 \Lambda _{DB} 2 m ^{2} (2 \pi) ^{3/4}} \cdot$$ =

$$\quad \cdot \left[ \int [{\mathcal{D}} Z] \left( \frac{\Lambda _{... ...} K ^{\mu \nu} \right) ^{2} e ^{- Z ^{\mu \nu} Z _{\mu \nu} / 2} \right] ^{1/2}$$ (50)

where,

 

$$\Lambda _{DB} ^{-1} \equiv \sqrt{\frac{1}{2} K ^{\mu \nu} K _{\mu \nu}}$$ (51)

$$\beta \equiv \frac{\Delta A}{2 m ^{2} (\Delta \sigma) ^{2}} \quad .$$ (52)

The parameter $\beta$ measures the ratio of the ``temporal'' to ``spatial'' uncertainty, while the area $\Lambda _{DB}$ sets the scale of the surface variation at which the string momentum is $K _{\mu \nu}$. Therefore, always with the particle analogy in mind, we shall call $\Lambda _{DB}$ the loop DeBroglie area. Let us assume, for the moment, that $\Delta A$ is independent of $\Delta \sigma$, so that either quantity can be treated as a free parameter in the theory. A notable exception to this hypothesis will be discussed shortly. Presently, we note that taking the limit $(\Delta \sigma) \rightarrow 0$, affects only the first term of the integral (50), and that its weight with respect to the second term is measured by the ratio $\Lambda _{DB} / (\Delta \sigma)$. If the area resolution is much larger than the loop DeBroglie area, then the first term is negligible: $\langle \Delta S \rangle$ is independent of $\Delta \sigma$ and $\langle S \rangle$ scales as

$$\Lambda _{DB} \ll (\Delta\sigma) \quad : \quad {\mathcal{S}} _{H} \approx (\Delta\sigma) ^{D _{H} - 2} \quad .$$ (53)

In this case, independence of $(\Delta \sigma)$ is achieved by assigning D _H = 2. As one might have anticipated, the detecting apparatus is unable to resolve the graininess of the string stack, which therefore appears as a smooth two dimensional surface.
The fractal, or quantum, behavior manifests itself below $\Lambda _{DB}$, when the first term in (50) provides the leading contribution

$$\Lambda_{DB} \gg (\Delta \sigma) \quad : \quad {\mathcal{S}... ...+ \frac{4 m^4 (\Delta \sigma) ^{4}}{(\Delta A) ^{2}}} \quad .$$ (54)

This expression is less transparent than the relation (53), as it involves also the $\Delta A$ resolution. However, one may now consider two special subcases in which the Hausdorff dimension can be assigned a definite value.
In the first case, we keep $\Delta A$ fixed, and scale $\Delta \sigma$ down to zero. Then, each $\langle \Delta S \rangle \propto (\Delta \sigma) ^{-1}$ diverges, because of larger and larger shape fluctuations, and

$${\mathcal{S}} _{H} \approx \frac{A}{\Delta \sigma} (\Delta \sigma) ^{D _{H} - 2}$$ (55)

requires D _H = 3.
The same result can be obtained also in the second subcase, in which both $\Delta \sigma$ and $\Delta A$ scale down to zero, but in such a way that their ratio remains constant,

$$\frac{2 m ^{2} (\Delta\sigma) ^{2}} {(\Delta A)} _{\Delta \s... ...ightarrow 0} = \mathrm{const.} \equiv \frac{1}{b} \quad .$$ (56)

The total interior area $A = N \Delta A$ is kept fixed. Therefore, as $\Delta A \sim (\Delta \sigma) ^{2} \rightarrow 0$, then $N \rightarrow \infty$ in order to keep A finite. Then,

$$\langle \Delta S \rangle \propto \frac{\Delta A}{\Delta \si... ... \sqrt{ 1 + \frac{1}{b ^{2}}} \propto \Delta \sigma \quad ,$$ (57)

and

$${\mathcal{S}} _{H} \propto A (\Delta \sigma) ^{D _{H} - 2} \frac{1}{\Delta \sigma} \sqrt{1 + \frac{1}{b ^{2}}} \quad ,$$ (58)

which leads to D _H = 3 again. In the language of fractal geometry, this interesting subcase corresponds to self-similarity. Thus, the condition (57) defines a special class of self-similar loops characterized by an average area variation which is proportional to $\Delta \sigma$ at any scale.