5.2 Superconductivity and Quantum Spacetime

Quantum strings, or more generally branes of various kind, are currently viewed as the fundamental constituents of everything: not only every matter particle or gauge boson must be derived from the string vibration spectrum, but spacetime itself is built out of them.

If spacetime is no longer preassigned, then logical consistency demands that a matrix representation of $p$-brane dynamics cannot be formulated in any given background spacetime. The exact mechanism by which $M$-Theory is supposed to break this circularity is not known at present, but ``loop quantum mechanics'' points to a possible resolution of that paradox. If one wishes to discuss quantum strings on the same footing with point-particles and other $p$-branes, then their dynamics is best formulated in loop space rather than in physical spacetime. As we have repeatedly stated throughout this paper, our emphasis on string shapes rather than on the string constituent points, represents a departure from the canonical formulation and requires an appropriate choice of dynamical variables, namely the string configuration tensor and the areal time. Then, at the loop space level, where each ``point'' is representative of a particular loop configuration, our formulation is purely quantum mechanical, and there is no reference to the background spacetime. At the same time, the functional approach leads to a precise interpretation of the fuzziness of the underlying quantum spacetime in the following sense: when the resolution of the detecting apparatus is smaller than a particle DeBroglie wavelength, then the particle quantum trajectory behaves as a fractal curve of Hausdorff dimension $2$. Similarly we have concluded on the basis of the ``shape uncertainty principle'' that the Hausdorff dimension of a quantum string world-sheet is $3$, and that two distinct phases (smooth and fractal phase) exist above and below the loop DeBroglie area. Now, if particle world-lines and string world-sheets behave as fractal objects at small scales of distance, so does the world-history of a generic $p$-brane including spacetime itself [19], and we are led to the general expectation that a new kind of fractal geometry may provide an effective dynamical arena for physical phenomena near the string or Planck scale in the same way that a smooth Riemannian geometry provides an effective dynamical arena for physical phenomena at large distance scales.

Once committed to that point of view, one may naturally ask, ``what kind of physical mechanism can be invoked in the framework of loop quantum mechanics to account for the transition from the fractal to the smooth geometric phase of spacetime?''. A possible answer consists in the phenomenon of $p$-brane condensation. In order to illustrate its meaning, let us focus, once again, on string loops. Then, we suggest that what we call ``classical spacetime'' emerges as a condensate, or string vacuum similar to the ground state of a superconductor. The large scale properties of such a state are described by an ``effective Riemannian geometry''. At a distance scale of order $(\alpha ') ^{1/2}$, the condensate ``evaporates'', and with it, the very notion of Riemannian spacetime. What is left behind, is the fuzzy stuff of quantum spacetime.

Clearly, the above scenario is rooted in the functional quantum mechanics of string loops discussed in the previous sections, but is best understood in terms of a model which mimics the Ginzburg-Landau theory of superconductivity. Let us recall once again that one of the main results of the functional approach to quantum strings is that it is possible to describe the evolution of the system without giving up reparametrization invariance. In that approach, the clock that times the evolution of a closed bosonic string is the internal area, i.e., the area measured in the string parameter space $${A \equiv \frac{1}{2} \epsilon _{ab} \int _{D} d \xi ^{a} \wedge d \xi ^{b} }$$, of any surface subtended by the string loop. The choice of such a surface is arbitrary, corresponding to the freedom of choosing the initial instant of time, i.e., a fiducial reference area. Then one can take advantage of this arbitrariness in the following way. In particle field theory an arbitrary lapse of Euclidean, or Wick rotated, imaginary time between initial and final field configurations is usually given the meaning of inverse temperature

$$i \Delta t \longrightarrow \tau \equiv \frac{1}{\kappa _{B} T}$$ (47)

and the resulting Euclidean field theory provides a finite temperature statistical description of vacuum fluctuations.

Following the same procedure, we suggest to analytically extend $A$ to imaginary values, $i A \longrightarrow a$, on the assumption that the resulting finite area loop quantum mechanics will provide a statistical description of the string vacuum fluctuations. This leads us to consider the following effective (Euclidean) lagrangian of the Ginzburg-Landau type,

$$L (\Psi , \Psi ^{*}) = - \frac{1}{4 m ^{2}} \left( \oint _{C} dl (s) \right) ^{-1} \oint... ...ta \sigma ^{\mu\nu} (s)} - i g A _{\mu \nu} (x) \right) \Psi \bigg \vert ^{2} +$$

$$\qquad + \Psi ^{*} \frac{\partial}{\partial a } \Psi - V (\vert \Psi \vert ^{2}) - \frac{1}{2 \cdot 3!} H ^{\lambda \mu \nu} H _{\lambda \mu \nu}$$ (48)

$$V (\vert \Psi \vert ^{2}) \equiv \mu _{0} ^{2} \left( \frac{a _{c}}{a} - 1 \right) \vert \Psi \vert ^{2} + \frac{\lambda}{4} \vert \Psi \vert ^{4}$$

$$H _{\lambda \mu \nu} = \partial _{[ \lambda} A _{\mu \nu ]} \quad .$$ (49)

Here, the string field is coupled to a Kalb-Ramond gauge potential $A _{\mu \nu} (x)$ and $a _{c}$ represents a critical loop area such that, when $a \le a _{c}$ the potential energy is minimized by the ordinary vacuum $\Psi [C] = 0$, while for $a > a _{c}$ strings condense into a superconducting vacuum. In other words,

$$\vert \Psi[C] \vert ^{2} = \cases{ 0 & if $i a \le a ... ... \left( 1 - a _{c} / a \right) & if $a > a _{c}$ } \quad .$$ (50)

Evidently, we are thinking of the string condensate as the large scale, background spacetime. On the other hand, as one approaches distances of the order $(\alpha ') ^{1/2}$ strings undergo more and more shape-shifting transitions which destroy the long range correlation of the string condensate. As we have discussed earlier, this signals the transition from the smooth to the fractal phase of the string world-surface. On the other hand, 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, spacetime fuzziness acquires a well defined meaning. Far from being a smooth, $4$-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. At a distance scale of the order of Planck's length, i.e., when

$$a _{c} = G _{N}$$ (51)

then the whole of spacetime boils over and no trace is left of the large scale condensate of either strings or $p$-branes.
As a final remark, it is interesting to note that since the original paper by A.D.Sakarov about gravity as spacetime elasticity, $G _{N}$ has been interpreted as a phenomenological parameter describing the large scale properties of the gravitational vacuum. Eq. (51) provides a deeper insight into the meaning of $G _{N}$ as the critical value corresponding to the transition point between an ``elastic'' Riemannian-type condensate of extended objects and a quantum phase which is just a Planckian foam of fractal objects.