2 Classical Equivalence

The main purpose of this note is to investigate how the dynamical generation of the brane tension and the equivalence between diverse action functionals can be extended at the quantum level in the WKB approximation of a ``sum over histories'' approach. However, before considering the path-integral it is instrumental to review how classical dynamics leads to the action (1) as an effective, on-shell action.

The guiding principle to assign the $p$-brane tension the role of a dynamical variable is borrowed from modern cosmology, where the cosmological constant can be represented by a maximal rank gauge $p$-form [8]. Thus, we introduce the following action functional

$$S[ Y , g , A ] = -\int_\Sigma d^{p+1}\sigma \sqrt{- g}\left[ {m_{p+1}\over 2}\... ...(-g)} -{1\over 2(p+1)!} F_{m_1\dots m_{p+1}} F^{m_1\dots m_{p+1}} \right]$$

$$= -\int_\Sigma d^{p+1}\sigma \left[ {m_{p+1}\over 2} {(-\gamm... ...\over 2(p+1)!} F_{m_1\dots m_{p+1}} F^{m_1\dots m_{p+1}} \right] \quad ,$$ (3)

where the world manifold $\Sigma$ has a space-like boundary $\partial\Sigma$ whose target space image will represent a closed, $p$-dimensional, relativistic object. Moreover, we introduce on the world manifold a maximal rank gauge field, $A _{m_2\dots m_{p+1}}( \sigma )$, with field strength $F_{m_1\dots m_{p+1}} \equiv \partial_{[ m_1 } A_{m_2\dots m_{p+1} ]}( \sigma )$. To preserve gauge invariance under $\delta A_{m_2\dots m_{p+1}}=\partial_{[ m_2} \Lambda_{m_3\dots m_{p+1} ]}$ in the presence of a boundary we must give up a current-potential interaction term and consider only a gravitational coupling $A$-$g$. By a suitable rescaling of the brane coordinates the dimensional constant $m_{p+1}$ can be washed out, and the classical action (3) written without any dimensional scale. Our goal is to show that the Dirac-Nambu-Goto functional can be obtained as an effective action from (3) once the classical field equations for the $p$-form gauge potential are solved.
Varying the action (3) with respect to $A$ we get

$${\delta S[ Y , g , A ]\over \delta A_{m_2\dots m_{p+1}}... ...eft( \sqrt{- g} F^{m m_2\dots m_{p+1}} \right)=0 \quad ,$$

which, since $A$ is maximal on the $p$-brane, has the solution

$$F^{m m_2\dots m_{p+1}}= \Lambda \epsilon^{m m_2\dots m_{p+1}... ...{1\over \sqrt{- g} }\delta^{[ m m_2\dots m_{p+1} ]} \quad ,$$ (4)

where $\Lambda$ is an arbitrary integration constant. By inserting the solution (4) back into (3) we obtain

$$S[ Y , g ] = -\int_\Sigma d^{p+1}\sigma \left[ {m_{p+... ...over \sqrt{-g}} +{\Lambda^2\over 2}\sqrt{-g} \right] \quad ,$$ (5)

where, the world manifold cosmological constant $\Lambda^2$ shows up as the on-shell value of the gauge field kinetic term.
The on-shell action (5) depends from the world metric only through the volume density $\sqrt{-g}$. Hence, variations with respect to $g_{mn}$ reduce to variations with respect $\sqrt{-g}$:

$${\delta S[ Y , g ]\over\delta g_{mn}(\sigma)}=0 \quad \... ...quad {\delta S[ Y , g ]\over\delta \sqrt{-g}}=0 \quad .$$ (6)

By inserting the solution (4) into (6) we get

$$m_{p+1}{(-\gamma)\over (-g) } = \Lambda^2 \quad \Rightarrow ... ... -g} = {1\over\Lambda} \sqrt{ m_{p+1}} \sqrt{-\gamma} \quad$$ (7)

and

$$S = - \Lambda \sqrt{ m_{p+1} } \int_\Sigma d^{p+1}\sigma ... ...equiv -\rho_p\int_\Sigma d^{p+1}\sigma \sqrt{-\gamma} \quad .$$ (8)

After solving for the world metric in terms of the brane coordinates, the action (3) turns out to be equivalent to a Dirac-Nambu-Goto action with a dynamically induced brane tension given by $${\rho_p\equiv \Lambda \sqrt{m_{p+1} } }$$. Let us remark that $\Lambda$ can take any value including zero. Accordingly, null branes, corresponding to the action

$$S_{null}[ Y , g ] = -{m_{p+1}\over 2} \int_\Sigma d^{p+1}\sigma {(-\gamma)\over \sqrt{-g}} \quad ,$$ (9)

are included in our description as well. This special case stresses how the parameter $m_{p+1}$ is not necessarily the brane tension, but only a dimensional constant needed to allow the various dynamical fields in the action to keep their canonical dimensions⁴.