1 Introduction

The purpose of this paper is to introduce a new approximation scheme to the quantum dynamics of extended objects. Our approach differs from the more conventional ones, such as the normal modes expansion or higher dimensional gravity, in that it is inspired by two different quantization schemes: one is the minisuperspace approach to Quantum Cosmology (QC), the other is the quenching approximation to $QCD$. Even though these schemes apply to different field theories, they have a common rationale, that is, the idea of quantizing only a finite number of degrees of freedom while freezing, or ``quenching,'' all the others. In QC this idea amounts, in practice, to quantizing a single scale factor (thereby selecting a class of cosmological models, for instance, the Friedman-Robertson-Walker spacetime) while neglecting the quantum fluctuations of the full metric. The effect is to turn the exact, but intractable, Wheeler-DeWitt functional equation [1] into an ordinary quantum mechanical wave equation [2]. As a matter of fact, the various forms of the ``wave function of the universe'' that attempt to describe the quantum birth of the cosmos are obtained through this kind of approximation [3], or modern refinements of it [4]. On the other hand, in $QCD$, the dynamics of quarks and gluons cannot be solved perturbatively outside the small coupling constant domain. The strong coupling regime is usually dealt with by studying the theory on a lattice. However, even in that case, the computation of the fermionic determinant by Montecarlo simulation is actually impossible. Thus, in the ``quenched approximation,'' the quark determinant is set equal to unity, which amounts to neglecting the effect of virtual quark loops. In other words, this extreme approximation in terms of heavy-quarks with a vanishing number of flavors assumes that gauge fields affect quarks while quarks have no dynamical effect on gauge fields [5].

Let us compare the above two situations with the basic problem that one faces when dealing with the dynamics of a relativistic extended object, or $p$-brane, for short. Ideally, one would like to account for all local deformations of the object configuration, i.e., those deformations that may take place at a generic point on the $p$-brane. However, any attempt to provide a local description of this shape-shifting process leads to a functional differential equation similar to the Wheeler-DeWitt equation in QC. The conventional way of handling that functional equation, or the equivalent infinite set of ordinary differential equations [6], is through perturbation theory. There, the idea is to quantize the small oscillations about a classical configuration and assign to them the role of ``particle states'' [7]. Alternative to this approach is the quantization of a brane of preassigned geometry. This (minisuperspace) approach, pioneered in Ref.([8]), was used in Refs.([9], [10]) in order to estimate the nucleation rate of a spherical membrane. Presently, the minisuperspace approximation is introduced for the purpose of providing an exact algorithm for computing two specific components of the general dynamics of a $p$-brane: one is the brane collective mode of oscillation in terms of global volume variations, the other is the evolution of the brane center of mass.

In broad terms, this paper is divided into two parts: Section II deals with classical dynamics; Section III deals with quantum dynamics in terms of the path-integral, or ``sum over histories.''
Since the action for a classical $p$-brane is not unique, we start our discussion by providing the necessary background with the intent of justifying our choice of action. We then take the first step in our approximation scheme in order to separate the center of mass motion from the bulk and boundary dynamics. In subsections IIC and IID we derive an effective action for the bulk and boundary evolution, while in subsection IIE we discuss the meaning of the ``quenching approximation'' at the classical level.
An approximation scheme for a dynamical problem is truly meaningful and useful only when the full theory is precisely defined, so that the technical and logical steps leading to the approximate theory are clearly identified. Thus, in Section III we first tackle the problem of computing the general quantum amplitude for a $p$-brane to evolve from an initial configuration to a final one. The full quantum propagator is obtained as a sum over all possible histories of the world-manifold of the relativistic extended object. In subsection IIIA, we show in detail what that ``sum over histories'' really means, both mathematically and physically, in order to explain why the bulk quantum dynamics cannot be solved exactly. What can be calculated, namely, the boundary and center of mass propagator, is discussed in subsections IIIB and IIIC. The final expression for the quantum propagator and the generalized ``tension-shell'' condition in the quenched-minisuperspace approximation is given in subsection IIID. Finally, subsection IIIE checks the self-consistency of the result against some special cases of physical interest.