4.2.2 Spherical Membrane Wave Function

From the results of the previous subsections we can immediately extract the generalized Klein-Gordon equation for a $2$-brane in four spacetime dimensions:

$$\left[ \, \partial_\mu\, \partial^\mu + {V_2\over 3!}\, {\... ...1)\, M_0^2\, \right]\, \Psi\left(\, x\ , \sigma\, \right)=0\ .$$ (87)

In the following we show briefly how this wave equation specializes to the case of a gauge fixed, or spherical, membrane of fixed radius [8], [9]. Moreover, since it is widely believed that $p$-brane physics may be especially relevant at Planckian energy, we assume that our $2$-brane is a fundamental object characterized by Planck units of tension and length. Thus, for a spherical $2$-brane of radius $R$ we have:

$$V_2=4\pi l_{\mathrm{Pl}}^2,$$ (88)

$$\sigma^{\mu_1\mu_2\mu_3 }=\delta^{[\, \mu_1 }_x \, \delta^{\mu_2 }_y \, \delta^{\mu_3 \, ] }_z \, { 4\pi\over 3}\, R^3,$$ (89)

$$d\sigma^{\mu_1\mu_2\mu_3 }=\delta^{[\, \mu_1 }_x \, \delta^{\mu_2 }_y\, \delta^{\mu_3 ] }_z \, 4\pi R^2\, dR,$$ (90)

$${\partial\over\partial\sigma^{\mu_1\mu_2\mu_3 }}= {1\over ... ...\mu_2 }^y\, \delta_{\mu_3\,]}^z \, {\partial\over\partial R}.$$ (91)

Since there is no mixing between ordinary and volume derivative, we can use the method of separation of variables to factorize the dependence of the wave function on $x$ and $\sigma$

$$\Psi\left(\, x\ , \sigma\, \right)=\phi(x)\, \psi_0\left(\,... ... \partial_\mu\, \partial^\mu - M_0^2\, \right]\,\phi(x) =0\ .$$ (92)

Here, $\phi_0( x )$ represents the center of mass wave function, while the ``relative motion wave function'' $\psi\left(\, R\, \right)$ must satisfy the following equation

$$\left[\, {1\over 4\pi R^2}\, {\partial\over \partial R }\,... ...0^2\over l_{Pl}^2}\, \right]\, \psi\left(\, R\, \right)=0 \ .$$ (93)

Apart from some ordering ambiguities, equation (93) is the wave equation found in Ref. [8],[9], [10], for the zero energy eigenstate, provided we identify the membrane tension $\rho$ through the expression

$$\rho^2 ={ p\, M_0^2\over 4\pi l_{\mathrm{Pl}}^2}\ .$$ (94)

It may be worth emphasizing that our approach preserves time reparametrization invariance throughout all computational steps, while the conventional minisuperspace approximation assumes a gauge choice from the very beginning.