4 Discussion

The extension of the relation (19) beyond the saddle point approximation is currently under investigation, and requires a proper treatment of the $p$-brane degrees of freedom at the quantum level. As is well known, bosonic branes viewed as non-linear $\sigma$-models are non-renormalizable perturbative quantum field theories when $p>1$. However, we can look at $p$-branes not as $\sigma$-models but as elements of a more fundamental theory, say $M$-Theory, which is in essence a non-perturbative theory. This approach is not new. For example the Einstein-Hilbert action in three and four dimensions is not perturbatively renormalizable; neverthless, three dimensional Einstein-Hilbert gravity can be reformulated as a Chern-Simon gauge theory which can be be exactly solved at the quantum level [14]. In a similar way, four dimensional General Relativity can be written in terms of Ashtekar variables which provides an exact formulation of non-perturbative canonical quantum gravity [15]. From the same point of view, we think that perturbation theory is not the ultimate way to approach the problem of brane quantization. Moreover, a supersymmetric membrane in $D=11$ spacetime dimensions is expected to be a finite quantum model [16], where both ultraviolet and infrared divergences are kept under control. For a general discussion of quantum super-membranes we refer to [17], and limit our considerations to the semi-classical level. Hopefully, a proper understanding of $M$-Theory will provide a background independent formulation of string/brane theory where the quantum path-integral will be well defined. In the meanwhile, we shall work in the WKB approximation where one can choose the action (3), in place of (1) or (2), as a starting point. The non-linearity and reparametrization invariance of the Nambu-Goto action make difficult, if not impossible, to implement the original Feynman construction of the path-integral as a sum of phase space trajectories[18]. One is forced, almost unavoidably, to resort to standard perturbative approaches, e.g. normal modes expansion, or sigma-model effective field theory. Any perturbative approach captures some dynamical feature and misses all the other ones. The impressive results obtained in string theory through duality relations of several kind [19] show that what is not accessible in a given perturbation scheme can be obtained through a different one. With this in mind, we hope that an action of the type (3), where the brane variables enter polynomially, and the tension is brought in by a generalized gauge principle, can be more appropriate to implement the Feynman's original proposal, or, at least, to provide a different ``perturbative'' quantization scheme for the Nambu-Goto model itself. In such a, would be, ``new regime'' of the Nambu-Goto brane both massive and massless objects are present at once and correspond to different values of the world manifold strength $F_{m_1... m_{p+1}}$. It would be tempting to assign $F$ the role of ``order parameter'' and describe the dynamical generation of the brane tension as a sort of phase transition. Furthermore, it would be interesting to extend the action (3) in order to include negative tension branes as well. This kind of objects appear to play an important role in the realization of the brane world scenario [20], [21].
All these problems, are currently under investigation and eventual results will be reported in future publications.