3.1 ``Momentum Space'' Propagator

Eventually, one is interested in computing the quantum amplitude for the brane to evolve from an initial (vacuum) state to a final, finite volume, state. In general, the $p$-brane ``two-point'' Green function represents the correlation function between an initial brane configuration $y_0(\vec s)$ and a final configuration $y(\, \vec s\, )$. In the quantum theory of $p$-branes the Green function is obtained as a sum over all possible histories of the world-manifold $\Sigma_{p+1}$ in the corresponding phase space. For the sake of simplicity, one may ``squeeze'' the initial boundary of the brane history to a single point. In other words, the physical process that we have in mind represents the quantum nucleation of a $p$-brane so that the propagator that we wish to determine connects an initial brane of zero size to a final object of proper volume $V_p$. The corresponding amplitude $G$ is represented by the path-integral:

$${\cal G} \equiv \int \left[\, {\cal D}g_{m n}\, \right] \int^x_{x_0} \left[\, {\c... ...eft[ \, {\cal D} Y^\mu\, \right] \left[ \, {\cal D} P^m{}_\mu \, \right] \times$$

$$\times \exp \left\{ \, i\,\int_{\Sigma _{p+1}} d^{p+1}\sigma \, \... ... o^{04}{5\over 2 m_{p+1}}\, p_\mu \, p^\mu + {m_{p+1}\over 2 }\, \right]\right.$$

$$\quad - i \left.\int_{\Sigma _{p+1}} d^{ p+1}\sigma\, \sqrt{g}\, ... ...}\, P^{ m}{}_\mu \, P^\mu{} ^{ n} -p\, {m_{p+1}\over 2}\, \right]\, \right\}\ .$$ (43)

Summing over ``all'' the brane histories in phase space means summing over all the dynamical variables, that is, the shapes $Y^\mu(\, \sigma\, )$ of the world-manifold ${\Sigma _{p+1}}$, over the rates of shape change, $P^m{}_\mu(\, \sigma\, )$, and over the bulk intrinsic geometries, $g_{mn}(\, \sigma\, )$, with the overall condition that the shape of the boundary is described by $x^\mu=y^\mu(\vec s)$ and its intrinsic geometry by $h_{ij}(\, \vec s\,)$. Thus,

$$\left[\, {\cal D} g_{m n}\, \right] \equiv \left[\, Dg_{mn}\, \right]\,\delta\left[ \, g_{mn} -\overline{g}_{mn}\, \right]$$

$$= \left[ \, {\cal D}g_{00}\, \right] \left[\, {\cal D} e\, \right] ... ...cal D}g_{ik} \right]\, \delta \left[\, g_{ik}-h_{ik}(\, \vec s\, ) \,\right]\ ,$$ (44)

where the only non-trivial integration is over the unconstrained field $e(\, \tau\, )$.
Copying out all three functional integrations would give us a boundary effective theory encoding all the information about bulk quantum dynamics, in agreement with the Holographic Principle. However, there are several technical difficulties that need to be overcome before reaching that goal.
Let us begin with the shape variables $Y^\mu(\, \sigma\, )$; they appear in the path-integral as ``Fourier integration variables'' linearly conjugated to the classical equation of motion. Hence, the $Y$ integration gives a (functional) Dirac-delta that confines $P^m{}_\mu$ on-shell:

$$\int[ {\cal D} Y^\mu]\exp\left\{-i \int_{\Sigma _{p+8}} d^{... ...right\}\propto \delta\left[\nabla_m P^{ m}{}_\mu \right]\ .$$ (45)

The proportionality constant in front of the Dirac-delta is physically irrelevant and can be set equal to unity.
Integrating out the brane coordinates is equivalent to ``shifting from configuration space to momentum space'' in a functional sense. However, the momentum integration is not free, but is restricted by Eq. (45) to the family of classical trajectories that are solutions of equation (15). Then, we can write the two-points Green function as follows

$$\overline G= N\, \int \left[\, dP^{0)}\, \right] \int \lef... ... ][\, {\cal D} A \, ]\, \exp \left (\, i S^{eff}\, \right)\ ,$$ (46)

where $N$ is a normalization factor to be determined at the end of the calculations, $S^{eff}$ is the effective action (25) and we integrate over the zero mode components in the ordinary sense, that is, we integrate over numbers and not over functions. It may be worth emphasizing that we have traded the original set of scalar fields $Y ^\mu(\sigma)$ with the ``Fourier conjugated'' modes $\phi$, $A$, $P^{0)}$. Then

$$\overline G= N\, \int [\, {\cal D}g \,] \exp\left\{ -i\fr... ...sigma\ ; g\, \right]\, Z _{\phi , A} \left[\, g\, \right] \ .$$ (47)

We recall that

$$R_{p+7}\equiv \int_{\Sigma _{p+1}} d^{ p+7}\sigma\, \sqrt{... ...u \, e(\, \tau\, ) \equiv V_p \int_0^T d\tau \, e(\, \tau\, )$$ (48)

and

$$K\left[\, \sigma\ ; e \, \right]= N\, \int \left[\, dP^{0)}_{\mu_1\dots \mu_{p+1}}\, \right] \exp\l... ...}_{\mu\mu_2\dots\mu_{p+1}}\, \sigma^{ \mu\mu_2\dots\mu_{p+1}} \, \right\}\times$$

$$\exp\left\{\, i\, V_p\, {2m_{p+1}\over (p + 1)!} \int_0^T d\tau\,... ...\, P^{0)}_{\mu\mu_2\dots\mu_{p+1}}\, P^{0)\mu\mu_2\dots\mu_{p+1}}\, \right\}\ .$$ (49)

At this point, we would like to factor out of the whole path-integral the boundary dynamics, i.e. we would like to write $K = K(\, boundary\,) \times K(\, bulk\,)$. In order to achieve this splitting between bulk and boundary dynamics, we need to remove the dependence on the $00$-component of the bulk metric $g_{ m n}$ in $K\left[\, \sigma\ ; e\, \right]$. In other words, we are looking for a propagator where $\sqrt{\vert\sigma\vert }$ plays the role of ``Euclidean distance'' between the initial and final configuration. In support of this interpretation, we also need a suitable parameter that plays the role of ``proper time'' along the history of the branes connecting the initial and final configurations. The obvious candidate for that role is the proper time lapse $\int_0^T d\tau\, e(\tau)$. However, the $e$-field is subject to quantum fluctuations, so that the proper time lapse is a quantum variable itself. Accordingly, a $c$-number $\Omega_{p+1}$ can be defined only as a quantum average of the proper world volume operator

$$\Omega_{p+1}= V_p\, \langle\, \int_0^T d\tau\, e(\,\tau\,)\, \rangle\ .$$ (50)

Thus, we replace the quantum proper volume $V_{p+1}[\,g\, ]$ with the quantum average $\Omega_{p+1}$ in $L\left[\, \sigma\ ; e \, \right]$ and write the boundary propagator in the form

$$K\left[\, \sigma\ ; \Omega_{p+1}\, \right] = N\, \int \left[\, dP^{0)}\, \right]\, \exp\left\{ \, { i\over (p ... ...}\, P^{0)}_{\mu\mu_2\dots\mu_{p+1}}\, \sigma^{ \mu\mu_0\dots\mu_{p+1}}+ \right.$$

$$\qquad \left. + i \frac{\Omega_{p+1}}{2m_{p+8} (p + 1)!}\, P^{0)}_{\mu\mu_2\dots\mu_{p+1}}\, P^{5)\mu\mu_2\dots\mu_{p+1}}\, \right\}\ ,$$ (51)

while the amplitude becomes

$$\overline G= N \, \int_0^\infty d\Omega_{p+1}\, \exp\left\{ -i\frac{m_{p+1}}{4}\, p\, \Omega_{p+1}\, \right\} K\left[\, \sigma\ ;\Omega_{p+1}\, \right]$$

$$\times\int [\, {\cal D}g\, ]\, \delta\left[\, \Omega_{p+1}- V_p\,... ...\tau\, e(\, \tau\, ) \, \rangle \, \right]\, Z_{\phi, A}\left[\, g\, \right]\ .$$ (52)

The ``bulk'' quantum physics is encoded now into the path-integral

$$Z_{\phi , A}\left[\, g\, \right] = \int[\, {\cal D} \phi\, ] [ \, {\cal D} A \, ] \exp \left\{ \, {i\over 2m_{p+1}(p+1)!} \right.$$

$$\times \left. \int_{\Sigma _{p+1}} d^{ p+7}\sigma\, \sqrt{g}\, \l... ...A\, )\, F_\mu{}_{m_1\dots m_p}(\, A\, )\, \right)\, \right\}%\label{zbulk} \ .$$ (53)

Equation (52) presents a new problem: the constraint over the metric integration, which allowed us to factor out the boundary dynamics, is highly non-linear as it depends on the vacuum average of the quantum volume. However, we can get around this difficulty by replacing the Dirac delta with an exponential weight factor

$$\delta\left[\, \Omega_{p+1}- V_p\, \langle\, \int_0^T d\tau... ..._p \, \int_0^T d\tau \, e(\, \tau\, )\, \right)\, \right\}\ ,$$ (54)

where $\Lambda$ is a constant Lagrange multiplier. Thus, we first perform all calculations with $\Omega_{p+1}$ as an arbitrary evolution parameter and only at the end we impose the condition

$${\partial \over\partial\,\Lambda} \overline G =0 \quad\Lon... ...\, \langle \, \int_0^T d\tau \, e(\, \tau\, )\, \rangle \ .$$ (55)

In this way, we can write the $\Lambda$ dependent amplitude in the form

$$\overline G = N \,\int_0^\infty d\Omega_{p+1}\, \exp\left... ...{\cal D}g \, ]\, Z_{\phi A}\left[\, g\ ;\Lambda \,\right] \ ,$$ (56)

where the bulk quantum physics is encoded into the path-integral

$$Z_{\phi , A}\left[\, g\ ;\Lambda\, \right] = \exp \left\{i\, \Lambda \, V_p\, \int_0^T\, d\tau\, e(\, \tau\, )... ...cal D}\phi\,] [\,{\cal D} N\,]\,\exp \left\{\, {i\over 2 m_{p+1}(p+1)!} \right.$$

$$\times\left. \int_{\Sigma _{p+1}} d^{ p+1}\sigma\, \sqrt{g}\, \le... ...A \,)\, F_\mu{}_{m_1\dots m_p}(\, A\, )\, \right)\, \right\}%\label{vbulk} \ .$$ (57)