3.2 The Boundary Propagator

The main point of the whole discussion in the previous subsection is this: even though the bulk quantum dynamics cannot be solved exactly, since there is no way to compute the bulk fluctuations in closed form, the boundary propagator can be evaluated exactly. This is because the integral (51) is gaussian in $P^{0)}$:

$$K\left[\, \sigma\ ;\Omega_{p+1}\, \right]=\left[\, {m_{p+4... ...dots\mu_{p+1}} \, \sigma_{\mu_1\dots\mu_{p+1}}\, \right\} \ .$$ (58)

Moreover, one can check through an explicit calculation that the kernel $K$ solves the (matrix) Schroedinger equation

$$\left[\, {1\over 2m_{p+1} (p+1) !}\, {\partial^2 \over \p... ...+1}}\, K\left[ \, \sigma-\sigma_0\ ;\Omega_{p+1}\, \right]$$ (59)

with the boundary condition

$$\lim_{\Omega\to 0} K\left[\,\sigma-\sigma_0\ ;\Omega_{p+1} ... ... =\delta\left[\, \vert \sigma-\sigma_0\vert \, \right] \ .$$ (60)

Notice that the average proper volume $\Omega_{p+1}$ enters the expression of the kernel $K$ only through the combination $m_{p+1}/\Omega_{p+1}$. Thus, the limit (60) is physically equivalent to the infinite tension limit where $\Omega_{p+1}$ is kept fixed and $m_{p+1}\to\infty$:

$$\lim_{m_{p+1}\to \infty} K\left[ \, \sigma-\sigma_0\ ;\Omega... ...ght] =\delta\left[ \, \vert \sigma-\sigma_0\vert \, \right]\ .$$ (61)

In the limit (61) the infinite tension shrinks the brane to a pointlike object.
From the above discussion we infer that the quantum dynamics of the collective mode can be described either by the zero mode propagator (58), or by the wavelike equation

$$\left[ \, {1\over 2m_{p+1} (p+1) !}\, {\partial^2 \over \partial ... ...al \over \partial \Omega_{p+1}} \!\Psi_0\left[ \, \sigma;\Omega_{p+1}\, \right]$$

$$\Psi_0\left[ \,\sigma\ ;\Omega_{p+1}\, \right]=\int \!\!\left[\, ... ...igma-\sigma_0\ ;\Omega_{p+1}\, \right]\, \phi\left[\, \sigma_0\ ;0\, \right]\ ,$$ (62)

where $\phi\left[ \, \sigma_0\ ;0 \, \right]$ represents the initial state wave function. Comparing the ``Schroedinger equation'' (59) with the Jacobi equation (35) suggests the following Correspondence Principle among classical variables and quantum operators:

$$P^{0)}_{\mu_1\dots\mu_{p+1}} \quad\longrightarrow\quad i\, {\partial\over \partial\sigma^{\mu_1\dots\mu_{p+1}}} \ ,\qquad\left(\, p\ge 1\,\right)$$ (63)

$$E \quad\longrightarrow\quad -i{\partial\over\partial \Omega_{p+1}} \ .$$ (64)

In summary, the main result of this section is that the general form of the quantum propagator for a closed bosonic $p$-brane can be written in the following form

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

$$\times\exp\left\{\, {i m_{p+1}\over 2 (p+1)!\Omega_{p+1}}\, \sigm... ...\left[\, x-x_0\ ; e(\, \tau\, )\, \right]\, Z\left[\, e\ ;\Lambda\, \right] \ .$$ (65)