4 ``Minisuperspace-Quenched Propagator''

At this stage in our discussion, we can explicitly define our approximation scheme. It consists of three main steps.
The first essential step, discussed in subsection IIB, consists in splitting the metric of the world-manifold according to Eq.(4). In a broad sense, this is a ``minisuperspace approximation'' to the extent that it restricts the general covariance of the action in parameter space. In other words, separating the center of mass proper time from the spatial coordinates on the world manifold $\Sigma_{p+1}$ effectively breaks the full invariance under general coordinate transformations into two symmetry groups:

$$\mathrm{General\quad Diffs} \longrightarrow (\mathrm{time})\mathrm{Rep} \otimes (\mathrm{spatial})\mathrm{Diffs}\ ,$$ (66)

so that the metric (4) shows a residual symmetry under independent time reparametrizations and spatial diffeomorphisms. By virtue of this operation, we were able to separate the center of mass motion from the bulk and boundary dynamics. This is encoded in the split form (10) of the total action for the $p$-brane.
The second and more strict interpretation of the minisuperspace approximation is that we now ``freeze'' the world metric into a background configuration $\overline{g}_{mn}$ where the space is a $p$-sphere, while $g_{00}$, or its square root $e(\, \tau\, )$, is free to fluctuate. In other words, we work in a minisuperspace of all possible world geometries.
The third and last step in our procedure is an adaptation of one of the most useful approximations to the exact dynamics of interacting quarks and gluons, namely, ``Quenched QCD''. There, the contribution of the determinant of the quark kinetic operator is set equal to one. In the same spirit we ``quench'' all the bulk oscillations:

$$Z_{\phi A}\left[\, e\ ;\Lambda\, \right]\longrightarrow \exp\left\{-i\, \Lambda\, \Omega_{p+1}\, \right\}\ .$$ (67)

Combining these three steps, we obtain from (65) the quenched-minisuperspace propagator:

$$G\left[\, x-x_0\ , \sigma\, \right] = N\, \int_0^\infty d \Omega_{p+1}\, \exp\left\{ \, i\, p\, \Omega_... ... {m_{p+1}\over i\, \pi\, \Omega_{p+1}} \right]^{{1\over 2}{D\choose p+1}}\times$$

$$\exp\left\{ \, {i\, m_{p+1}\over 2 \, \Omega_{p+1}}{\sigma^2 \ove... ...left[\, \int_0^T \!\!d\tau\, e(\, \tau\, )-{\Omega_{p+1}\over V_p}\, \right]\ .$$ (68)

The center of mass propagator $K_{cm}\left[\, x-x_0\ ; e(\, \tau\, )\, \right]$ can be computed as follows [27]. From the Lagrangian $L_{cm}$ we can define the center of mass momentum $p_\mu$ as follows

$$p_\mu\equiv {\partial L_{cm}\over \partial\dot x^\mu(\, \tau\, ) }= M_0\, {\dot x_\mu \over e(\, \tau\, )}\ .$$ (69)

By Legendre transforming $L_{cm}$ we obtain the center of mass Hamiltonian:

$$H_{cm}\equiv p_\mu\, \dot x^\mu -L_{cm} ={e(\tau)\over 2M_0}\left[\, p_\mu \, p^\mu + M_0^2\, \right]\ .$$ (70)

Moreover, the canonical form of $S_{cm}$ reads

$$S_{cm}= \int _0^T d\tau\, \left[\, p_\mu\, \dot x^\mu -{e(\,... ...2M_0} \, \left(\, p_\mu \, p^\mu + M_0^2\, \right)\, \right]\,$$ (71)

so that the quantum dynamics of the bulk center of mass is described by the path-integral

$$K_{cm}\left[\, x-x_0\ ; e(\, \tau\, )\, \right]\equiv \int \... ...t]\, e^{i S_{cm}\left[\, x\ , T\ ; e(\, \tau\, )\, \right]}\ .$$ (72)

This path-integral can be reduced to an ordinary integral over the constant four momentum $q_\mu$ of a point particle of mass $M_0$

$$K_{cm}\left[\, x-x_0\ ; e(\, \tau\, )\, \right]= \int { d^... ...M_0}\, \left( \, q_\mu\, q^\mu + M_0^2\, \right)\, \right]\ .$$ (73)

In order to get the explicit form of the center of mass propagator we have to integrate, in the ordinary sense, over $q_\mu$. However, before that we must integrate, in the functional sense, over the einbein field $e(\, \tau\, )$.

Using the properties of the Dirac-delta distribution, the einbein field can be integrated out:

$$\int \left[\, {\cal D }e\, \right]\delta\left[\, \int_0^T d\tau\, e(\, \tau\, ) - \Omega_{p+1}/V_p\, \right] \exp\left[ - i\int _0^T d\tau\, {e(\, \tau\, )\over 2M_0}\, \left(\, q_\mu\, q^\mu + M_0^2\, \right)\, \right]=$$

$$\exp\left[ - i{\Omega_{p+1}\over 2M_0 V_p}\, \left(\, q_\mu\, q^\mu +\, M_0^2\, \right)\, \right]\,$$ (74)

and

$$\int { d^D q\over (2\pi)^D }\, e^{i\, q_\mu \, (\, x^\mu-x^... ...\over 2\Omega_{p+1} }\,\left(\, x - x_0\right)^2\, \right]\ .$$ (75)

Using the above results, the ``QCD-QC combined approximation'' leads to the following expression for the propagator

$$G\left[\, x- x_0\ ,\sigma\, \right] = N\, \int_0^\infty d\Omega_{p+1}\, \exp\left\{ \, i\, \Omega_{p+1}... ...t\} \left[\, {m_{p+1}\over i\pi \Omega_{p+1}}\right]^{{1\over 2}{D\choose p+1}}$$

$$\times \exp\left\{ \, {i m_{p+1}\over 2\Omega_{p+1}} {\sigma^2\ov... ...[\, i{M_0 V_p\over 2\Omega_{p+1} }\, \left(\, x - x_0\, \right)^2 \, \right]\ .$$ (76)

The amplitude (76) can be cast in a more familiar form in terms of the Schwinger-Feynman parametrization

$$s\equiv {\Omega_{p+1}\over 4 V_p}, \quad d\Omega_{p+1}= 4V_p\, ds\ .$$ (77)

Using the above parametrization, the quantum propagator takes its final form in the minisuperspace-quenched approximation

$$G\left[\, x- x_0\ ,\sigma\ ; M_0\, \right] = N\, V_p \int_0^\infty ds\, \left(\, {\pi M_0 \over is}\, \right)^{D/2} \exp\left[\, i {M_0 \over 2s }\left(\, x - x_0\, \right)^2 \, \right]\times$$

$$\exp\left\{\,i\, s {M_0\over 2}\, (p+1)\, \right\} \left[\, {M_0\... ...oose p+1}} \exp\left\{ \, {i M_0\over 2s V_p^2}{\sigma^2\over(p+1)!}\, \right\}$$

$$= {i\over 2M_0}\int_0^\infty ds\, \left(\, {\pi M_0 \over is}\, \ri... ...2}{D\choose p+1}}\exp\left\{ \, i\, s \, {M_0\over 2}\, (p+1)\, \right\} \times$$

$$\exp\left\{\, i\, {M_0 \over 2s }\,\left[\, \left(\, x - x_0\, \right)^2 + {1\over V_p^2}{\, \sigma^2\over (p+1)!}\,\right] \, \right\}\ .$$ (78)

Here we have set $N=i/2M_0\, V_p$ in order to match the form of the point particle propagator.
The formula (78) represents the main result of all previous calculations and holds for any $p$ in any number of spacetime dimensions.