3 Quantum Equivalence

In the second part of this note we shall discuss the above equivalence at the quantum level. The basic quantity encoding the $p$-brane quantum dynamics is the boundary wave functional, or vacuum--one-brane amplitude

$$Z\equiv Z[\widehat Y ,\widehat A , \widehat g ] = \int^{\... ..., DA ] \exp\left( i S[ Y , g , A ] \right) \quad ,$$ (10)

where the sum is over all bulk fields configurations inducing ``hatted'' fields on the boundary of the brane. We are assuming that the brane world manifold has a single, $p$-dimensional boundary, parametrized as $\sigma^m=\sigma^m(s^a)$, $a=1 ,\dots p$, which is mapped into the physical brane $\widehat Y^\mu(s)$; $\widehat g$ and $\widehat A$ are the induced metric and gauge potential over $\widehat Y^\mu(s)$. The integration variables in $Z$ ``live'' in the brane bulk, while we let free the fields induced on the boundary, i.e. we do not assign an independent classical action to the hatted fields.

The first field to be integrated out is the gauge $p$-form $A$. The standard routine goes through a lengthy procedure of gauge fixing and Fadeev-Popov compensation to invert the classical kinetic operator and define an appropriate quantum propagator. On the other hand, one knows that a gauge $p$-form over a $(p+1)$-dimensional manifold has no dynamical degrees of freedom and can describe only a static interaction. In such a limiting case the Fadeev-Popov procedure leaves no propagating degree of freedom at the quantum level. To shorten the whole gauge fixing procedure of ghost terms with different rank [9], we shall provide an alternative ``recipe'' to kill all the apparent degrees of freedom. We write the path-integral in the first order version, where the gauge potential $A$ and field strength $F$ are introduced as independent integration variables [10] and we integrate away the gauge part of $A$ after inserting gauge fixing Dirac delta's and the corresponding ghost determinants in the functional measure. The remaining, gauge invariant part of $A$ enforces $F$ to be a classical solution of the field equations, which is a constant background field. No propagating degrees of freedom survive at the quantum level. A formal proof of the equivalence between second order and first order quantization procedures, in the general case of a $p$-form in $p+1$ dimensions, is beyond the purpose of this short note. Rather, we will briefly consider the simplest, non trivial case which is $p=1$ gauge form over a two-dimensional, flat manifold without boundary, and then translate the result to the case we are studying. The first order, gauge fixed and Fadeev-Popov compensated path integral is

$$Z_{p=1}=\int[ DF ] [ DA ]\delta\left[ \partial^m A... ...2}F^{mn} \partial_{[ m} A_{n ]} \right] \right\} \quad .$$ (11)

By splitting $A_m$ into the sum of a ``transverse'' vector $A_m^T$ and a ``gauge part'' $\partial_m\phi$, the integration measure $[ DA ]$ turns into $[ DA^T ][ D\phi ] \left( \det\Box \right)^{1/2}$ and (11) reads

$$Z_{p=1} = \int[ DF ] [ DA^T ] [ D\phi ] \left( \det\Box \right)^{1/2} \delta\left[ \Box\phi \right] \times$$

$$\qquad \qquad \times \Delta_{FP}\exp\left\{ i\int d^2\sigma \left... ...n} F^{mn}-{1\over 2}F^{mn} \partial_{[ m} A^T_{n ]} \right] \right\}$$

$$= \int [ DF ][ DA^T ] \left( \det\Box \right)^{1/2} \ex... ...{mn} F^{mn}+{1\over 2}A^T_n \partial_m F^{mn} \right] \right\} \quad ,$$ (12)

where the the gauge part has been integrated away thanks to the Fadeev-Popov determinant $\Delta_{FP}= \det \Box$ and only the gauge invariant $A^T$ vector remains in the classical action. The extra Jacobian, coming from the change of the integration measure, will be cancelled in a while when integrating $F$. The pay off for relaxing the relationship between $A$ and $F$ and getting rid of the gauge part is that $A^T$ linearly enters the first order action, i.e. $A^T$ plays the role of Lagrange multiplier imposing $F$ to satisfy the classical field equations

$$Z_{p=1} = \int[ DF ] \left( \det\Box \right)^{1/2} \delta\left[ \... ...\exp\left\{ i\int d^2\sigma \left[ {1\over 4}F_{mn} F^{mn} \right] \right\}$$

$$= \int[ DF ] \delta\left[ F^{mn}-\Lambda \epsilon^{mn} \ri... ...t\{ i\int d^2\sigma \left[ {1\over 4}F_{mn} F^{mn} \right] \right\} \quad .$$ (13)

Equation (13) shows that the first order formulation of a limiting rank, abelian gauge theory, and the Fadeev-Popov prescription lead to a ``trivial'' path integral for $F$. The Dirac-delta picks up the classical configurations of the world tensor $F$ and the whole path-integral ``collapses'' around the classical trajectory⁵. Thus, $F$ is ``frozen'' to a constant value $\Lambda$ and no degrees of freedom are left free to propagate. The same result can be obtained, with some additional work, for $p>1$ as well. Accordingly, we get

$$Z = \exp\left\{ - {1\over p!}\int_{\partial \Sigma} dN_{k_1} \sqrt{... ..., \widehat F^{k_1\dots k_{p+1}} \widehat A_{k_2\dots k_{p+1}} \right\}\times$$

$$\qquad \quad \times \int [ DF] \left( \det\Box \right)^{1/... ...rtial_{m_1 }\left( \sqrt{-g} F^{m_1 m_2\dots m_{p+1}} \right) \right]\times$$

$$\qquad \qquad \qquad \times \exp\left( {i\over 2(p+1)!} \int_\Sigma d^{p+1}\sigma \sqrt{- g} F_{m_1\dots m_{p+1}}^{ 2} \right)$$

$$= \exp\left\{ -{i\Lambda\over p!}\int_{\partial\Sigma} \widehat A_{... ...ft( -{i\Lambda^2\over 2} \int_\Sigma d^{p+1}\sigma \sqrt{-g} \right) \quad .$$ (14)

The first term in (14) is a pure boundary quantity produced by a partial integration of the term $F\partial_{[\cdot}A_{\cdot]}$. A similar term arises in string theory when boundary and bulk quantum dynamics are properly split [11].
After integrating out the gauge degrees of freedom the resulting path-integral reads

$$Z = \int^{\widehat g} [ Dg_{mn} ] \int^{\widehat Y} [ DY^\mu ] ... ...hat A_{k_1\dots k_{p+1}} ds^{k_1}\wedge \dots\wedge ds^{k_p} \right) \times$$

$$\qquad\qquad\times\exp\left( -i\int_\Sigma d^{p+1}\sigma \left[... ... 2} {(-\gamma)\over \sqrt{-g}} +{\Lambda^2\over 2}\sqrt{-g} \right] \right)$$

$$\equiv \int^{\widehat g} [ Dg_{mn} ] \int^{\widehat Y} [ DY^\mu ]\... ...-\gamma)\over \sqrt{-g}} +{\Lambda^2\over 2}\sqrt{-g} \right] \right) \quad .$$ (15)

We remark that this integration procedure is exact and leads to a bulk action plus a boundary correction represented by the generalized Wilson factor $W_{\widehat A}[ \partial\Sigma ]$.
We also notice that the world metric enters the path-integral only through the world volume density. Accordingly, we can ``change'' integration variable

$$\int^{\widehat g} [ Dg_{mn} ] \quad \longrightarrow \quad... ...ehat e} [ De ] \delta\left[ e(\sigma) -\sqrt{-g} \right]$$ (16)

and write (15) as

$$Z=\int^{\widehat e} [ De ] \int^{\widehat Y} [ DY^\mu... ...gma)} +{\Lambda^2\over 2} e(\sigma) \right] \right) \quad .$$ (17)

The saddle point value for the auxiliary field $e(\sigma)$ is defined by:

$${ \delta S\over \delta e(\sigma) } = 0 \quad \longrightarr... ...ma)={1\over\Lambda} \sqrt{m_{p+1}} \sqrt{-\gamma } \quad .$$ (18)

By expanding $Z$ around the saddle point $e_{cl.}(\sigma)$ we obtain the Dirac-Nambu-Goto path-integral. Correspondingly, we get the following semi-classical equivalence relation

$$Z = \int^{\widehat g} [ Dg_{mn} ] \int^{\widehat Y} [ DY^\mu ] \int^{\widehat A}[ DA ]\times$$

$$\qquad\qquad\times\exp\left[ -i m_{p+1} \int_\Sigma d^{p+1}\sig... ...int_\Sigma d^{p+1}\sigma \sqrt{- g} F_{m_1\dots m_{p+1}}^{ 2}(A) \right]$$

$$\approx \int [ DY^\mu ] W_{\widehat A} [ \partial\Sigma ]\exp\left[ - i\rho_p \int_\Sigma d^{p+1}\sigma \sqrt{-\gamma} \right] \quad .$$ (19)