6. Conclusions

The main results of this paper can be summarized as follows. Starting with the canonical form of the Schild action for a closed, bosonic string, it is possible to formulate the Hamilton-Jacobi theory of string dynamics in loop space, with the proper area of the string manifold playing the role of evolution parameter. The conjugate dynamical quantity is an area-Hamiltonian which is quadratic in the corresponding momenta so that it is possible to extend to strings, indeed to any p-brane, many of the results which are applicable to a relativistic point particle. The Feynman path integral for quantum strings is then equivalent to the functional wave equation (29) which was derived without recourse to a lattice approximation. For the kernel (42), the path integral, or the corresponding wave equation can be solved exactly.
If, as it is normally done, one starts from a reparametrization invariant path integral over the string coordinates, the corresponding amplitude describes the propagation of strings with fixed ``energy'' $E =1/2 \pi \alpha '$. The relation between the two amplitudes is given by equation (24).
Three generalizations of the above results are almost straightforward:

1.

the wave equation for a closed string coupled to a Kalb-Ramond field can be obtained from (29) through the replacement

$$\frac{\delta}{\delta x ^{\mu} (s)} \longrightarrow \frac{\... ... i \kappa B _{\mu \nu} (x) \, x ^{\prime \, \nu} (s) \ .$$ (48)

Unfortunately, there is no straightforward way to solve the wave equation, or to integrate the path integral, for an arbitrary gauge potential;

2.

some more work is required to extend the above formalism to the case of a closed p-brane imbedded into a D-dimensional spacetime. However, no essential difficulties arise in treating higher dimensional extended objects;

3.

once the quantum mechanical propagator is known, then one can second quantize the system. If we introduce the string wave function $\Psi [ x (s) ]$ as an element of a functional space of string states, then we can write (29) as

$$i \hbar \frac{\partial}{\partial A} \vert \Psi \rangle = H \, \vert \Psi \rangle \ ,$$ (49)

where H is the area-hamiltonian operator. Then, the corresponding Green function can be written as follows

$$G [ x (s) , x _{0} (s) ; m ^{2} ] = \frac{i}{\hbar} \int _... ... \langle x(s) \vert \frac{1}{H} \vert x _{0} (s) \rangle \ ,$$ (50)

or

$$G [ x (s) , x _{0} (s) ; m ^{2}] = \mathcal{N} \int [D \Ps... ...s) ] \exp \left( i S[\Psi ^{*} , \Psi ] / \hbar \right) \ ,$$ (51)

where, $\Psi ^{*} [ x (s) ]$, $\Psi [ x _{0} (s) ]$ constitute a pair of complex functional fields, and

 

$$S [ \Psi ^{*} , \Psi ] \int [D x ^{\mu} (s)] \, \Psi ^{*} [ x (s) ] \left[ - \hbar ^{2} \left( \int _{0} ^{1} ds \sqrt{x ^{\, \prime \, 2}} \right) ^{-1} \cdot \right .$$ =

$$\qquad \qquad \left . \cdot \int _{0} ^{1} \frac{ds}{\sqrt{x^{\, ... ...{2}}{\delta x ^{\mu} (s) \ \delta x _{\mu} (s)} + m ^{4} \right] \Psi [ x (s) ]$$ (52)

is the Marshall-Ramond [16] action for the dual string model. Therefore, by extending the classical Hamilton-Jacobi formulation of string dynamics into the quantum domain, one arrives at a functional field theory in loop space.

As a final speculative remark, it seems worth observing that the functional approach to the quantum mechanics of strings is analogous, in several ways, to the functional approach to quantum cosmology. For instance, the `` wave function of the universe '' is defined in the functional space of all possible 3-geometries, and we suggest that spatial loop configurations in string theory play the same role as spatially closed 3-geometries in quantum cosmology. Likewise, the functional Schrödinger equation for strings, at fixed areal-time, plays the same role as the Wheeler-DeWitt equation in quantum cosmology. If this analogy is more than coincidental, then quantum string theory in loop space may shed some light on the many dark areas of quantum cosmology.