4.2.1 Infinite Tension Limit

When probed at low energy (resolution) an extended object effectively looks like a point-particle. In this case, ``low energy'' means an energy which is small compared with the energy scale determined by the brane tension. In natural units, the tension of a $p$-brane has dimension: $\left[\, T_p\,\right]=(\, energy\,)^ {p+1}$. Thus, when probing the brane at energy $E<< (\,T_p\,)^{1/p+1}$ one cannot resolve the extended structure of the object. From this perspective, the ``point-like limit'' of a $p$-brane is equivalent to the ``infinite tension limit''. In either case, no higher vibration modes are excited and one expects the brane to appear concentrated, or ``collapsed'', in its own center of mass. This critical limit can be obtained from the general result (78) by setting $p=0$ and performing the limit $V_p\to 0$ using the familiar representation of the Dirac-delta distribution:

$$\delta(x)\equiv \lim_{\epsilon \to 0}\left(\,{1\over \pi\epsilon}\, \right)^{d/2}\exp \left(\, -x^2/\epsilon\,\right)\ .$$ (85)

In our case: $x^2\rightarrow \sigma^2/(p+1)!$, $\epsilon \rightarrow -i M_0/4s V_p^2$, $d\rightarrow{D\choose p+1}$, and the whole dependence on the volume coordinates of the brane reduces to a delta function which is different from zero only when $\sigma = 0$. In this case, $G\left[\, x- x_0\ ; M_0\, \right]$ reduces to the familiar expression for the Feynman propagator for a point particle of mass $M_0$,

$$G\left[\, x- x_0\ ; M_0\, \right] = \delta\left[\, \sigma^2\, \right]\, {i\over 2M_0}\,\int_0^\infty ... ... \exp\left\{\, i\, {M_0 \over 2s }\, \left(\, x - x_0\, \right)^2\, \right\}\ .$$ (86)