2.1 Background

The action of a classical $p$-brane is not unique. The first (mem)brane action, dates back to 1962 and was introduced by Dirac in an attempt to resolve the electron-muon puzzle [11]. The Dirac action was reconsidered in Ref.([8]) and quantized following the pioneering path traced by Nambu-Goto in the lower dimensional string case. The Dirac-Nambu-Goto action represents the world volume of the membrane trajectory in spacetime. Thus, it can be generalized to higher dimensional $p$-branes as follows

$$S_{\mathrm{DNG}}\left[\, Y\, \right] = -m_{p+1}\int_{\Sigma ... ...eft(\, \partial_m \, Y^\mu \, \partial_n \, Y_\mu\,\right) \ ,$$ (1)

where $m_{p+1}$ represents the ``$p$-tension'' ( we denote with ``$p$'' the spatial dimensionality of the brane ) and the coordinates $\sigma^m$, $m=0, 1, \dots{}, p$, span the $(p+1)$-dimensional world-manifold $\Sigma$ in parameter space. On the other hand, the embedding functions $Y ^\mu(\sigma)$, $\mu=0, 1, \dots{}, D-1$, represent the brane coordinates in the target spacetime.
An alternative description that preserves the reparametrization invariance of the world-manifold is achieved by introducing an auxiliary metric $g_{ m n}(\sigma)$ in parameter space together with a ``cosmological constant'' on the world-manifold[12], [13]

$$S_{\mathrm{HTP}}\left[\, Y\ , g\, \right]= -{m_{p+1}\over 2}... ... n}\partial_m\, Y^\mu\, \partial_n\, Y_\mu -(p-1)\,\right] \ ,$$ (2)

where $g\equiv \det\, g_{ m n}$. The two actions (1) and (2) are classically equivalent in the sense that the ``field equations'' ${\delta S/\delta g^{ m n}(\sigma)}=0$ require the auxiliary world metric to match the induced metric, i.e., $g_{ m n}=\gamma_{ m n}=\partial_m\, Y^{\mu}\, \partial_n\, Y_{\mu}$. The two actions are also complementary: $S_{\mathrm{DNG}}$ provides an ``extrinsic'' geometrical description in terms of the embedding functions $Y^\mu(\, \sigma\, )$ and the induced metric $\gamma_{mn}$, while $S_{\mathrm{HTP}}$ assigns an ``intrinsic'' geometry to the world manifold $\Sigma$ in terms of the metric $g_{ m n}$ and the ``cosmological constant'' $m_{p+1}$, with the $Y^\mu(\, \sigma\, )$ functions interpreted as a ``multiplet of scalar fields'' that propagate on a curved $(p+1)$-dimensional manifold.
Note that in both functionals (1) and (2), the brane tension $m_{p+1}$ is a pre-assigned parameter. More recently, new action functionals have been proposed that bridge the gap between relativistic extended objects and gauge fields[14], [15], [16]. The brane tension itself, or world-manifold cosmological constant, has been lifted from an a priori assigned parameter to a dynamically generated quantity that may attain both positive and vanishing values. Either a Kaluza-Klein type mechanism [17] or a modified integration measure have been proposed as dynamical processes for producing tension at the classical [18] and semi-classical level [19].
For our present purposes, the form (2) of the $p$-brane action is the more appropriate starting point. There are essentially two reasons for this choice:

1. Unlike the Nambu-Goto-Dirac action, or the Schild action [20], Eq. (2) is quadratic in the variables $\partial_m X^\mu$. As we shall see in the following subsections, this property, together with the choice of an appropriate coordinate system on $\Sigma_{p+1}$, facilitates the factorization of the center of mass motion from the deformations of the brane.
2. Equation (2) can be interpreted as a scalar field theory in curved spacetime. From this point of view, the minisuperspace quantization approach is equivalent to a quantum field theory in a fixed background geometry, at least as far as the auxiliary metric is concerned.