2. Functional Jacobi equation

As a starting point for the study of string dynamics one can choose either the Nambu-Goto action, or the Schild action: both functionals lead to the same classical dynamics [5]. It is not clear, however, whether or not such an equivalence persists unrestricted at the quantum level. This is because, in a quantum theory, the string propagation kernel reflects the different weight assigned to the string trajectories in the two classical frameworks. Another potential source of inequivalence stems from the fact that the Nambu-Goto action is reparametrization invariant but non linear with respect to the generalized velocity, whereas the Schild action is linear but at the expense of reparametrization invariance. Apart from all this, the standard procedure to construct the path integral in quantum mechanics applies to quadratic actions, which is not the case for relativistic systems. One way to deal with the problem would be to follow the Dirac quantization procedure for constrained systems. But then, a canonical evolution of the system does not make sense because of the vanishing of the Hamiltonian. To preserve a Hamiltonian-type evolution, it is necessary to start with a non-reparametrization invariant theory. Even in this case, the resulting dynamics for an extended object is non-canonical.

All of the above arguments converge to the focal question: if one insists on a Hamiltonian, albeit non canonical formulation of string dynamics, is there an evolution parameter which plays the role of ``time variable'', and is this choice consistent with reparametrization invariance?

Previous attempts to deal with those questions lead to seemingly conflicting conclusions. For instance, the string propagator obtained in ref.[6], has been criticized in ref.[7]. From our vantage point, the critical issue is that of reparametrization invariance. Both authors are led to a string diffusion equation which is manifestly dependent on the string parameter s, leaving us with the impression that the lack of reparametrization invariance of the classical action manifests itself even at the quantum level. However, in ref.[6], the physical Green function is obtained by averaging over all the possible values of the proper evolution parameter. In ref.[7], instead, it is claimed that the parametric dependence of the propagation kernel is only apparent because the action is insensitive to the location of the area increment along the world-sheet boundary. The resulting wave equation is local, in the sense that it is defined at a single, representative, point on the string loop, and does not apply to the string as a whole. Evidently, in this approach the string is treated as a collection of constituent points, and this may well be a viable interpretation. However, inspired by our previous work on the classical dynamics of p-branes [8], [9], [10], we believe that the dynamics of each individual point on the string does not give a consistent account of the dynamics of the whole string.
The alternative point of view is that ``the whole string is more than the sum of its parts'', and in this paper we wish to suggest a different approach which, in our view, addresses directly the question of the choice of dynamical variables and the related issue of reparametrization invariance in the classical theory as well as in the quantum theory. The stipulation is also made that the classical theory must emerge as a well defined limiting case of the quantum theory. In order to fulfil this condition, we invoke a single dynamical principle encompassing both areas of string-dynamics, namely the Jacobi variational principle suitably adapted to the case in which the physical system is a relativistic extended object. Thus, the dynamical variables are restricted to vary within the family of string trajectories which are solutions of the classical equation of motion. In other words, the variational procedure applies only to the final configuration of the string, rather than to its spacetime history.

Against this conceptual backdrop, the formalism developed in this paper, largely inspired by the work of Nambu [11], [12] and Migdal [13], fully reflects our emphasis on the global structure of the string: our action functional is a reparametrized form of the Schild action, manifestly invariant under general coordinate transformation in the string parameter space, while preserving the polynomial structure in the dynamical variables; the natural candidate for the role of time variable is the proper area of the string world-sheet (equation (8)), i.e., the invariant measure of the model manifold representing the evolution of the string. The final outcome is a manifestly reparametrization invariant Schrödinger equation which has the same form of the corresponding equation obtained from the Nambu-Goto action using a lattice approximation, and admits gaussian type wave packets as solutions.
Our starting point is the Schild string action in Hamiltonian form

 

$$S[ x ( \xi ) , p ( \xi ) ] \frac{1}{2} \int _{X ( \xi ) } p _{\mu \nu} d x ^{\mu} \wedge d x ^{\nu} - \int _{\Sigma} d ^{2} \xi \, H(p)$$ = (1)

$$\equiv \frac{1}{4 m ^{2}} p _{\mu \nu} p ^{\mu \nu} \ ,$$ H(p) (2)

where $m ^{2} = 1/2 \pi \alpha '$ is the string tension, $\Sigma$ represents the model manifold of the string in parameter space, and $X ( \xi )$ represents its image in Minkowski space. Then, $p _{\mu \nu}$ stands for the linear momentum canonically conjugated to the world-sheet tangent element $\dot{x} ^{\mu \nu} \equiv \epsilon ^{ab} \partial _{a} x ^{\mu} \partial _{b} x ^{\nu}$. Both variables were originally introduced by Nambu [11], [12]. More recently, the same variables were used to formulate a gauge theory for the dynamics of strings and higher dimensional extended objects [8], [9], [10].
In order to cast the action (2) in a reparametrization invariant form, we introduce a new pair of world-sheet coordinates $( \sigma ^{0} , \, \sigma ^{1} )$ through the boundary preserving transformation $\xi ^{a} \rightarrow \sigma ^{a} = \sigma ^{a}( \xi )$, and promote the original pair $(\xi ^{0} , \, \xi ^{1})$ to the role of dynamical variables. Then, $S [ x ( \xi ) , p ( \xi ) ]$ transforms into

$$S [ x ( \sigma ) , p ( \sigma ) , \xi ( \sigma ) ] = \frac{... ...Sigma ( \sigma ) } d \xi ^{a} \wedge d \xi ^{b} \, H(p) .$$ (3)

The new action (3) is numerically equivalent to (2) and leads to the same equation of motion for $x ^{\mu}$, $p _{\mu \nu}$. Furthermore, variation with respect to the new fields $\xi ^{a} ( \sigma )$ leads to the energy-balance equation

$$\epsilon _{ab} \epsilon^{mn} \partial _{m} \xi ^{a} \parti... ..., \Longrightarrow H _{\mathrm{cl.}} = {\rm const.}\equiv E$$ (4)

which, in our case, correctly shows that the Hamiltonian is constant along a classical solution. The action (3) is linear with respect to the ``velocities'' $\dot{x} ^{\mu \nu}$ and $\dot{\xi} ^{ab} ( \equiv \epsilon ^{mn} \partial _{m} \xi ^{a} \partial _{n} \xi ^{b} )$. Hence, if one interprets $\pi _{ab} \equiv \epsilon _{ab} H$ as the momentum canonically conjugated to $\xi ^{a} ( \sigma )$, then (3) acquires the form of a reparametrization invariant theory in six dimensions [11].
The Jacobi equation for the string is obtained by varying $S[ x ( \sigma ) , p ( \sigma ) , \xi ( \sigma ) ]$ within the family of world-sheets which solve the string equations of motion. We emphasize that this type of variation corresponds to a deformation of the only free boundary of the world-sheet, i.e. C, and corresponds to the more familiar variation of the world-line end-point in the case of a particle. Then, the steps leading to the Jacobi equation are as follows. First, the contribution from the variation of the world-sheet itself vanishes by definition, and we obtain:

$$\delta S _{\mathrm{cl.}}[ \partial X ; A] = \int _{\partial... ... \, \delta x ^{\mu} - H _{\mathrm{cl.}} \delta A \quad .$$ (5)

Next, we note that in view of the constancy of the Hamiltonian over a classical trajectory, we can vary the area of the $\Sigma$ domain without reference to the specific point along the boundary $\partial \Sigma$ where the infinitesimal variation takes place. In other words, we can move ``$\delta$''in front of the area integral and then trade the functional variation $\delta A$ for an ordinary differential variation d A, and define

$$p _{\mu} ( s ) \equiv \frac{\delta S _{\mathrm{cl.}}}{\delta x ^{\mu} ( s )} = p _{\mu \nu} \, x ^{\, \prime \, \nu}$$ (6)

as the boundary momentum density. Similarly, the area-energy density E can be written as the partial derivative of the classical action with respect to the invariant measure of the $\Sigma$ domain in parameter space:

 

$$- \frac{\partial S _{\mathrm{cl.}}}{\partial A}$$ E = (7)

$$\equiv \frac{1}{2} \epsilon _{ab} \int _{\Sigma} d \xi ^{a} \wedge d \xi ^{b} \quad .$$ A (8)

Hence, the Jacobi variational principle in the form of equation (5) shows that $p _{\mu} ( s )$ is conjugated to the spacetime world-sheet boundary variation, while H _cl. describes the response of the classical action to an arbitrary area variation in parameter space. Thus, if we consider string dynamics from the loop space point of view [13], then A and $x ^{\mu} ( s )$ can be interpreted as the ``time'' and ``space'' positions of the final string C with respect to the initial one C ₀, which we assume to be fixed at the outset. In this perspective, H _cl. is the area-hamiltonian, or generator of the classical evolution from the ``initial time'' T = 0 to the final time T = A. Accordingly, $p _{\mu} ( s )$ is the generator of infinitesimal ``translations'' in loop space, which are perceived as infinitesimal deformations $x ^{\mu} ( s ) \rightarrow x ^{\mu} ( s ) + \delta x ^{\mu} ( s )$ of the string shape in Minkowski space.
Finally, note that in this formulation, E represents the energy per unit area associated with an extremal world-sheet of the action (3), while $p _{\mu} ( s )$ is the momentum per unit length of the string loop C. Therefore, the energy-momentum dispersion relation can be written either as an equation between densities

$$\frac{1}{2 m ^{2}} p _{\mu} p ^{\mu} = \frac{1}{4 m ^{2}} ... ...\nu} x ^{\, \prime \, 2} ( s ) = E x^{\, \prime \, 2} ( s )$$ (9)

or, as an integrated relation

$$\frac{1}{2m^2} \int _{0} ^{1} \frac{ds}{\sqrt{x^{\,\prime\,}... ...{\mu} = E \int _{0} ^{1} ds \sqrt{x^{\,\prime\,}} \quad .$$ (10)

The above equation, once written in terms of S _cl., turns into the promised functional Jacobi equation for the string:

$$\left(\int_0^1ds\,\sqrt{x^{\,\prime\,}}\right)^{-1} \int _{0... ...m ^{2} \frac{\partial S _{\mathrm{cl.}}}{\partial A} \quad .$$ (11)

Looking in more detail at this equation, we observe that the covariant integration over s takes into account all the possible locations of the point, along the contour C, where the variation can be applied. But, in this way, every point of C is overcounted a ``number of times'' equal to the string proper length. The first factor, in round parenthesis, is just the string proper length and removes such overcounting. In other words, we sum over all the possible ways in which one can deform the string loop, and then divide by the total number of them. The net result is that the l.h.s. of equation (11) is insensitive to the choice of the point where the final string C is deformed. Therefore the r.h.s. is a genuine reparametrization scalar which describes the system's response to the extent of area variation, irrespective of the way in which the deformation is implemented. With hindsight, the wave equation proposed in [6], [7] appears to be more restrictive than equation (11), in the sense that it requires the second variation of the line fuctional to be proportional to $x ^{\prime \, 2} ( s )$ at any point on the string loop, in contrast to equation (11) which represents an integrated constraint on the string as a whole.
Equation (11) is the starting point in the first quantization program via the Correspondence Principle: one introduces the reparametrization invariant operators

$$\widehat p_\mu(s)\equiv i\hbar {1\over\sqrt{x^{\,\prime\,}}} ... ...dehat{H} \equiv - i \hbar \frac{\partial}{\partial A} \ ,$$ (12)

and imposes the operatorial form of the dispersion relation (10) on the string wave functional $\Psi [ C ; A ]$. Alternatively, one can focus directly on the string propagation kernel $K \left[ x (s) , x _{0} (s) ; A \right]$, in which case we turn to Feynman's ``sum over histories'' method since this is probably the most natural and effective way to define $K \left[ x (s) , x _{0} (s) ; A \right]$ in quantum string-dynamics.