1. Introduction

There are at least two approaches to the quantum theory of relativistic strings. One way is to look at a string model as a field theory in two spacetime dimensions. In this case, the string coordinates $x ^{\mu} ( \tau , \sigma )$ are reinterpreted as a multiplet of scalar fields defined over the string manifold parametrized by a Lorentzian coordinate mesh $( \tau , \sigma )$. The non-linearity of the Nambu-Goto action can be ``softened'' by assigning an auxiliary metric field $\gamma _{ab} ( \tau , \sigma )$ over the string manifold, and then writing the action in the Howe-Tucker form [1]. After this reshuffling of variables, the original string model is converted into a local field theory and is quantized through canonical, or path integral methods [2]. Quantum fluctuations around a classical solution eventually give rise to a spectrum of elementary particles, and the string itself acquires the status of fundamental building block of everything in the universe.

On the other hand, one may regard a string as an elementary physical system by itself, and focus on the geometric and topological properties of the string manifold. Vortices in a super-conducting medium [3] and cosmic strings [4] are two noteworthy examples of this geometrical approach. Quantum fluctuations are now interpreted as transitions between different string configurations. In particular, the quantum propagation kernel acquires the meaning of probability amplitude for the string shape to evolve from an initial configuration, represented by the non self-intersecting spatial loop $C _{0} : 0 \le s \le 1 \rightarrow x ^{\mu} _{0} = x ^{\mu} (s) , \, x ^{\mu} _{0} (0) = x ^{\mu} _{0} (1)$, to a final, non self-intersecting configuration $C: x ^{\mu} = x ^{\mu} (s)$.
Thus, in this functional approach, spatial deformations of the string shape are mapped into ``translations'' in the space of all possible loop configurations, and our major concern is to develop a ``Hamiltonian'' theory for the quantum mechanics of strings in loop space. As a matter of fact, the main purposes of this paper can be stated as follows:

1.

to give a path integral definition for the string kernel;

2.

to prove that the path integral definition is equivalent to a functional wave equation defined over a ``space of loops'', and to show that the same kernel can be derived from both formulations;

3.

to determine the exact form of the free string propagation kernel.

Accordingly, the paper is organized as follows.
In section 2, we obtain the Jacobi functional equation for a closed string. This equation is the essential link between the classical and quantum theory because it provides the WKB approximation to the whole functional ``Schrödinger'' equation.
In section 3, we define the Feynman path integral for the quantum propagation kernel. We find that the reparametrization invariant sum over world-sheets, weighed by the exponential of the Nambu-Goto action, is a Jacobi path integral, i.e. a sum over string histories at ``fixed energy'' $E = 1/4 \pi \alpha '$.
In section 4, we present a path integral derivation of the string kernel wave equation. No discretization procedure is involved. Instead, we use Jacobi's variational principle to derive the functional Schrödinger equation for the string in a manifestly reparametrization invariant form.
In section 5, we compute the quantum kernel for a free string in two different ways. In subsection 5.1, we solve the kernel ``Schrödinger'' equation by exploiting the role of the Jacobi equation as the classical limit of the full quantum equation. In subsection 5.2, the string quantum kernel is obtained from the path integral using a ``trick'' which bypasses the use of semi-classical approximation or phase space discretization.
Sect.6 is devoted to a brief summary of the results and to the discussion of their possible generalization.