A Note on Hamilton-Jacobi Formalism and D-brane Effective Actions

We first review the canonical formalism with general space-like hypersurfaces developed by Dirac by rederiving the Hamilton-Jacobi equations which are satisfied by on-shell actions defined on such hypersurfaces. We compare the case of gravitational systems with that of the flat space. Next, we remark as a supplement to our previous results that the effective actions of D-brane and M-brane given by arbitrary embedding functions are on-shell actions of supergravities.


Introduction
We showed in [1,2] that the effective actions of D-brane and M-brane are on-shell actions of supergravities. We derived the Hamilton-Jacobi (H-J) equations of supergravities, which are satisfied by on-shell actions, regarding a radial direction as time, and solved those equations.
We also found that these solutions to the H-J equations are the on-shell actions around the supergravity solutions that are conjectured to be dual to various gauge theories, which in particular include noncommutative super Yang Mills. In the gauge/gravity correspondence, the on-shell actions in gravities are considered to be generating functionals of correlation functions in gauge theories. Therefore, our results in [1,2] should be useful for going beyond the AdS/CFT correspondence and studying the more general gauge/gravity correspondence.
However, it is not clear why the D-brane effective actions which includes the all-order contributions in the α ′ expansion are obtained within supergravities, which are just the lowest order approximation for string theories in the α ′ expansion. In order to clarify this reason, we must establish the exact correspondence between our calculations and the derivations of the D-brane effective action in string theory. On one hand, the worldvolumes of the D-brane effective actions in our solutions are fixed-time hypersurfaces, since the ordinary H-J formalism gives on-shell actions defined on the boundary hypersurfaces specified by the final time.
On the other hand, the worldvolumes of D-branes in string theories are defined by embedding functions which can specify arbitrary hypersurfaces in the target space [3]. Therefore, before trying to establish the above correspondence, we should first investigate whether the D-brane effective action whose worldvolume is defined by such embedding functions is an on-shell action of supergravity or not.
Hence, this brief note is concerned with the on-shell action that is obtained by substituting into the action the classical solution which satisfies a boundary condition given on not a fixedtime hypersurface but an general hypersurface. (See Fig.1.) We need a generalization of the H-J formalism that gives such on-shell actions. Indeed, this generalization was studied by Dirac around fifty years ago [4]. The generalization in gravitational systems is much simpler. In fact, by performing a general coordinate transformation that transforms the general hypersurface defined by the embedding functions to a fixed-time hypersurface, one can show that the answer to the above question is positive. However, it is interesting and useful for further developments to compare the case of gravitational system with that of the flat space, so that we will give a heuristic argument below. We will first derive Dirac's result in the flat space in a different way, and next make the comparison. Finally, we remark that the effective actions of D-brane and M-brane given by arbitrary embedding functions are on-shell actions of supergravities PSfrag replacements As we mentioned, we pay attention to on-shell actions with the boundary values of fields given on general space-like hypersurfaces. Let us consider a four-dimensional scalar field theory in the flat space as an example. The action is given by where η M N = diag(−1, 1, 1, 1). The equation of motion is The space-like hypersurface which specifies the boundary is parametrized by the X M (σ i ), where i = 1, 2, 3. (See Fig.1.) In principle, we can obtain the solutionφ(x) to the equation of motion (2.2) which satisfies a boundary condition By substituting this solution into I, we obtain the on-shell action in which we are interested, It is in general difficult to solve the equation of motion in the above situation and to obtain the on-shell action directly. Instead, we seek for the Hamilton-Jacobi equations, which are the differential equations satisfied by the on-shell action of this kind.
In order to develop the generalized Hamilton-Jacobi formalism, we consider an action, where σ α = (τ, σ i ) (α = 0, 1, 2, 3, i = 1, 2, 3, σ 0 = τ ). In this action, let both the induced scalarφ(σ) and the coordinates x M (σ) be dynamical variables as in [4]. We regard τ as time in this system. Moreover, let the boundary values of the x M (σ) parametrize the same space-like hypersurface, where T is the boundary value of τ that is the final time.Ĩ is invariant under the reparametrization of σ by which bothφ(σ) and the x M (σ) are transformed as scalars. If this diffeomorphism is fixed by the gauge fixing condition x M (σ) = σ M ,Ĩ reduces to I. ThereforeĨ and I are equivalent. We will actually see below that the on-shell action ofĨ is equal to that of I.

The equations of motions ofĨ are
Let us consider the classical solutionx M (σ) andφ(σ) which satisfies the boundary condition, By substituting this classical solution intoĨ, we obtain the on-shell action, where T 0 is the initial time. If we defineφ(x) byφ(σ) =φ(x(σ)) and x =x(σ),φ(x) satisfies (2.2) and (2.3). Therefore the coordinate transformation x =x(σ) gives the desired relation S(X, Φ(X)) =S(T,Φ, X). (2.10) Let us derive the Hamilton-Jacobi equations satisfied byS. By solving these equations we obtainS and hence S as well.S is a functional of the final time T and the boundary values,Φ and X. The variation ofS with respect to T ,Φ, and X is given by We have used the equations of motion (2.7) and δφ(T 0 , σ i ) = δx(T 0 , σ i ) = 0. Moreover, the Then, (2.12) Here we define L M and γ ij for convenience: From (2.12) and (2.13), noting that we obtain the Hamilton-Jacobi equations It is instructive to rederive the above Hamilton-Jacobi equations in the canonical formalism following Dirac [4].Ĩ is rewritten in the canonical formalism as follows:

Application to gravitational systems
In this section, we consider the same problem as the previous section in gravitational systems.
Let us consider as an example a four-dimensional gravity given by As before we consider the classical solutionḡ M N (x) andφ(x) which satisfies the boundary where the X M (σ) parametrize an space-like hypersurface. 2 Note that this boundary condition is invariant under general coordinate transformations of x. We also consider the on-shell which is an analogue of (2.4).
Note that we can obtain (2.5) from (2.1) by performing a general coordinate transformation x = x(σ). Similarly, we obtain from (3.1) Then the on-shell action ofĨ is one can obtain the on-shell action with those given on an arbitrary space-like hypersurface.
Note that this consequence directly comes from the facts that I is invariant under general coordinate transformations and that an arbitrary space-like hypersurface can be transformed to a fixed-time hypersurface by a general coordinate transformation.

Discussion
As we have seen in the example of the D3-brane case in the previous section, all we can do at present is to reduce ten-dimensional supergravities or eleven-dimensional supergravity to (p+2)-dimensional gravity and obtain the p-brane effective action, whose (p+1)-dimensional worldvolume is given by arbitrary embedding functions, as an on-shell action of the (p + 2)dimensional gravity. In other words, we can only consider hypersurfaces whose codimension is one. Thus our results in this note are not completely satisfactory, since in string theories one can consider D-branes whose codimension is larger than one. Hence, an issue we should next study is an 'on-shell action' that one obtains when one takes a hypersurface whose codimension is larger than one, specifies the 'boundary' values on the hypersurface and substitutes into the action the classical solution satisfying the 'boundary' condition. We need to develop a formalism that gives such an 'on-shell action' and see whether the p-brane effective action is an 'on-shell action' of a (p + k)-dimensional gravity which is obtained by reducing ten-dimensional supergravities or eleven-dimensional supergravity, where 2 < k ≤ 10 − p (or 11 − p). 3 We would like to thank T. Yoneya for bringing our attention to Ref. [4].