2 A New Proposal and Its Implementation.

Keeping in mind the scenario outlined in the Introduction, in this brief report we wish to introduce a new kind of quantum fluctuation. Specifically, we conjecture that, at the Planck scale, not only shapes and topology fluctuate, but the very dimensionality ``$p$'' of a $p$-brane becomes a quantum variable.
This hypothesis is supported by an explicit calculation of the quantum propagator of a bosonic $p$-brane [6]. Let us recall that the world-history of an extended object is the mapping of a $p$-cell, i.e., a topological simplex of order ``$p$'', to a physical $p$-brane, and is usually encoded into an action of Dirac-Nambu-Goto type, or extensions of it. From the action, in principle, one may deduce the Green function by evaluating the ``Wilson loop'' associated with a given $p$-brane. In practice, the quantum propagator of a bosonic $p$-brane cannot be computed exactly in closed form. Thus, in a recent paper [6] we borrowed the minisuperspace quantization scheme from Quantum Cosmology [7] and the quenching approximation from QCD [8] in order to derive a new form of the bosonic $p$-brane propagator. Combining those two techniques leads to an expression for the propagator that describes the boundary dynamics of the brane [9] rather than the spectrum of the bulk small oscillations. This new approximation includes both the center of mass quantum motion in target spacetime as well as the collective deformation of the brane as described by transitions between different (hyper-)volume quantum states. The latter quantum jumps account for the volume variations induced by local deformations of the brane shape, and generalize the ``areal quantum dynamics'', originally formulated by Eguchi in the case of strings [10], to the case of higher dimensional objects.
The propagator calculated within the above approximation scheme has the form:

$$G ( y - y _{0} , {Y} - {Y} _{0} ) =\int \frac{d ^{D} k}{ (2 \pi) ... ...right]}{k ^{2}+ (p+1) V _{p} ^{2} m ^{2} _{p+1}+V ^{2} _{p}\, K ^{2}/(p+1)!}\ .$$ (1)

In the above expression, $\left(\, y^\mu\ , k_\mu\,\right)$ represent the brane center of mass position and momentum; the multivectors $\left(\, {Y}^{\mu _{1} \dots \mu _{p+1}}\ , K_{\mu_{1}\dots \mu _{p+1}}\,\right)$ stand for the brane volume tensor and the canonically conjugated volume momentum; $m_{p+1}$ is the brane tension and $V_p$ represents the brane spacelike $p$-volume. Inspection of the Feynman Green function (1) reveals a strong similarity to the propagator of a ``pointlike'' probe, if one interprets the volume variables $\left(\, Y\ , K \,\right)$ as new ``coordinates'' in an extended phase space.
It may be helpful, before proceding further with our discussion, to get a ``feeling'' of the quantum propagator by checking the consistency of our formulation against some well known cases of physical interest. The precise details of the derivation are in Ref. [6].