2.5 ``Classical Quenching'' $\to$ Volume Dynamics

The Jacobi equation derived in the previous subsection takes into account both the intrinsic fluctuations $\delta y^{\mu}(\, \vec s\, )$ and the normal boundary deformations $dN^m\, \partial_m\, \phi^\mu$ induced by the bulk field $\phi^\mu$. The problem is that, even neglecting the boundary fluctuations induced by the bulk harmonic mode, the $p$-loop space Jacobi equation is difficult to handle [24], [25], [26]. In order to make some progress, it is necessary to forgo the local fluctuations of the brane in favor of the simpler, global description in terms of hyper-volume variations, without reference to any specific point on the $p$-brane where a local fluctuation may actually occur.
Thus, in our formulation of $p$-brane dynamics, classical quenching means having to relinquish the idea of describing the local deformations $\phi^\mu (\sigma)$ of the brane, and to focus instead on the collective mode of oscillation. In turn, by ``collective dynamics,'' we mean volume variations with no reference to the local fluctuations which cause the volume to vary. In this approximation we can write a ``global'', i.e., non-functional, $p$-brane wave equation.

The effective action that encodes the volume dynamics, say $S _{0}$, is obtained from $S_J + S_B$ by ``freezing'' both the harmonic and Kalb-Ramond bulk modes. The simplification is that the general action reduces to the following form

$$S_J+ S_B\longrightarrow S_0 = { 1\over (p+1)!}\, \sigma^{\mu\mu_2\dots\mu_{p+1}}\, P^{0)}{}_{\mu\mu_2\dots\mu_{p+1}}$$

$$+ \Omega_{p+1}\left[\, { 1\over 2 m_{p+1}(p+1)!}\, P^{0)}{}_{\mu\... ...u_{p+1}} P^{0)}{}^{ \mu\mu_2\dots\mu_{p+1}} - \frac{m_{p+1}}{2}\, p\, \right]\,$$ (36)

so that the functional equation reduces to a partial differential equation

$${1\over 2 m_{p+1}}\,{ \partial S_0\over \partial \sigma_{\m... ...m_{p+1}}{2}\, p = {\partial S_0\over\partial \Omega_{p+1}}\ .$$ (37)

The collective-mode dynamics is much simpler to handle. In fact, the functional derivatives that describe the shape variation of the brane have been replaced by ``ordinary'' partial derivatives that take into account only hyper-volume variations, rather than local distortions. In other words, while the original equation (35) describes the shape dynamics, the global equation (37) accounts for the collective dynamics of the brane. The advantage of the Jacobi equation (37) is that the partial derivative is taken with respect to a matrix coordinate $\sigma_{\mu_1\dots\mu_{p+1}}$ instead of the usual position four-vector.
The similarity with the point particle case $(p=0)$ suggests the following ansatz for $S_0$

$$S_0\left( \, \sigma\ ;\Omega\, \right)\equiv {B\over 2 \Omeg... ...ts\mu_{p+1}}_0\, \right)^2 -\frac{m_{p+1}}{2}\, p\, V_{p+1}\ ,$$ (38)

where $\sigma^{\mu_1\dots\mu_{p+1}}_0$ represents a constant ( matrix ) of integration to be determined by the ``initial conditions'', while the value of the $B$ factor is fixed by the equation (37). Indeed

$${ \partial S_0\over \partial \sigma^{\mu_1\dots\mu_{p+1} }}=... ...\mu_1\dots\mu_{p+1}}- \sigma_{0)\mu_1\dots\mu_{p+1}}\, \right)$$ (39)

and

$${\partial S_0\over\partial \Omega_{p+8}}= -{B\over 3\Omega_... ...ots\mu_{p+1}}\right)^2-\frac{m_{p+1}}{2}\, p\, \Omega_{p+1}\,$$ (40)

so that

$$B= -m_{p+1}\,$$ (41)

$$S_0\left(\sigma;V\right)= - { m_{p+1} \over 2 V_{p+1}}\, {1\over ... ...igma^{\mu_1\dots\mu_{p+1}}_{9)}\, \right)^2 -\frac{m_{p+1}}{2}\, p\, V_{p+1}\ .$$ (42)

For the sake of simplicity, one may choose the integration constant to vanish, i.e., one may set $\sigma^{\mu_1\dots\mu_{p+1}}_{0)}=0$. Thus, in the quenching approximation, we have obtained the classical Jacobi action for the hyper-volume bynamics of a free $p$-brane.