4.1 Green Function Equation $\to$ Tension-Shell Condition

For completeness of exposition, in this subsection we derive the master equation satisfied by the Green function (78) in the quenched-minisuperspace approximation. Then, by formally inverting that equation we arrive at an alternative expression for the Green function in momentum space. The advantage of this procedure is that it provides a useful insight into the structure of the $p$-brane propagator.
To begin with, it seems useful to remark that the propagation kernel in (78) is the product of the center of mass kernel $K_{cm}\left(\, x-x_0\ ; s\, \right)$ and the volume kernel $K\left(\, \sigma\ ; s\,\right)$; each term carries a weight given by the phase factor $\exp i\, M_0\, s$ and $\exp i\,p\, M_0\, s$, respectively. Finally, we integrate over all the values of the Feynman parameter $s$:

$$G\left[\, x- x_0\ ; M_0\, \right]= {i\over 2M_0}\, \int_0^\... ...(\, x-x_0\ ; s\, \right)\, K\left(\, \sigma\ ; s\, \right)\ .$$ (79)

As is customary in the Green function technique, we may add an infinitesimal imaginary part to the mass in the exponent, that is, $(M_0/2)\to (M_0/2) +i\epsilon$, so that the oscillatory phase turns into an exponentially damped factor enforcing convergence at the upper integration limit. The ``$i\epsilon$'' prescription in the exponent allows one to perform an integration by parts leading to the following expression

$$G\left[\, x- x_0\ ; M_0\, \right] = {1\over M_0^2\, (p+1)}\, \lef... ...eft(\, x-x_0\ ; s\, \right)\, K\left(\, \sigma\ ;s\, \right)\, \right]^\infty_0$$

$$- {1\over \, M_0^2\, (p+1)}\, \int_0^\infty ds \, \exp\left\{ \, ... ...cm}\left(\, x-x_0\ ; s\, \right)\, K\left( \, \sigma\ ;s\, \right)\, \right)\ .$$ (80)

Convergence of the integral enables us to express the partial derivative $\partial/\partial s$ by means of the diffusion equations for $K_{cm}$ and $K$ and to move the differential operators $\partial_\mu \, \partial^\mu$, $\partial^2/\partial\sigma^2$ out of the integral:

$$G\left[\, x- x_0\ ; M_0\, \right] = {1\over \, M_0^2\, (p+1)}\left[\, \delta\left(\, x-x_0\, \right)\... ...p+1)} \int_0^\infty ds\, \exp\left\{ \, i\, {M_0\over 2}\, s\, (p+1)\, \right\}$$

$$\times \left[\, \left(\, K\left(\, \sigma\ ;s\, \right)\, \partial^\mu\,\partial_\mu\, K_{cm}\left(\, x-x_0\ ; s\, \right)\, \right) \right.$$

$$\qquad \left. \left(\, K_{cm}\left(\, x-x_0\ ;s\, \right){V_p\ove... ...^{\mu_1\dots\mu_{p+1}}\, \partial\sigma_{\mu_1\dots\mu_{p+1}}}\right)\, \right]$$

$$= -{1\over M_0^2\, (p+1)}\, \delta\left(\, x-x_0\, \right)\, \delta\left(\, \sigma\ ;s\, \right)$$

$$+ {1\over M_0^2(p+1)}\, \left(\, \partial_\mu\,\partial^\mu + {V_p\... ... ^{\mu_1\dots\mu_{p+1}}} \, \right)\, G\left(\, x-x_0\ , \sigma\ ; s\, \right),$$ (81)

from which we deduce the desired result,

$$\left[\, \partial_\mu\, \partial^\mu + {V_p\over (p+1)!}\, ... ...\left(\, x-x_0\, \right)\, \delta\left(\, \sigma\, \right)\ .$$ (82)

This is the Green function equation for the non-standard differential operator $\partial^\mu\partial_\mu +V_p\, \partial^2/\partial\sigma^2$. Finally, we ``Fourier transform'' the Green function by extending the momentum space to a larger space that includes the volume momentum as well:

$$G(\, x-x_0\ , \sigma\, ) = \int {d^Dq\over (2\pi)^D}\int [\, dk_{\mu_1\dots\mu_{p+1}}\, ] \e... ...er (p+1)!}\, k_{\mu_1\dots\mu_{p+1}}\, \sigma^{\mu_1\dots\mu_{p+1}} }\, \right)$$

$$\times {1\over q^2 + {V_p^2\over (p+1)!}\, k^2_{\mu_1\dots\mu_{p+1}} +(p+1)\, M_0^2}\ .$$ (83)

The vanishing of the denominator in (83) defines a new tension-shell condition:

$$q^2 + {V_p^2\over (p+1)!} k^2_{\mu_1\dots\mu_{p+1}} +(p+1)\, M_0^2=0\ .$$ (84)

Real branes, as opposed to virtual branes, must satisfy the condition (84) which links together center of mass and volume momentum squared. Equation (84) represents an extension of the familiar Klein-Gordon condition for point-particles to relativistic extended objects.
Some physical consequences of the ``tension-shell condition'', as well as the mathematical structure of the underlying spacetime geometry [28], are currently under investigation and will be reported in a forthcoming letter. In the next subsection we limit ourselves to check the consistency of our results against some familiar cases of physical interest.