3.1 The original formulation

Eguchi's approach to string quantization follows closely the point-particle quantization along the guidelines of the Feynman-Schwinger method. The essential point is that reparametrization invariance is not assumed as an original symmetry of the classical action; rather, it is a symmetry of the physical Green functions to be obtained at the very end of the calculations by means of an averaging procedure over the string manifold parameters. More explicitly, the basic action is not the Nambu-Goto proper area of the string world-sheet, but the ``square'' of it, i.e. the Schild Lagrangian [8]

$$L _{Schild} = \frac{1}{4} \left[ \frac{\partial ( x ^{\... ...l _{\tau} x ^{\mu} \wedge \partial _{\sigma} x ^{\nu} \quad .$$ (2)

The corresponding (Schild) action is invariant under area preserving transformations only, i.e.

$$(\tau , \sigma) \longrightarrow (\tau', \sigma') : \frac{... ...l (\tau ' , \sigma ')}{\partial(\tau , \sigma)} = 1 \quad .$$ (3)

Such a restricted symmetry requirement leads to a new, Jacobi-type, canonical formalism in which the world-sheet proper area of the string manifold plays the role of evolution parameter. In other words, the ``proper time'' is neither $\tau$ or $x ^{4}$, but the invariant combination of target and internal space coordinates $x ^{\mu}$ and $\sigma ^{a} = (\tau , \sigma)$ provided by

$$A = \int d ^{9} \sigma \sqrt{- \gamma} \quad , \qquad \... ... \det{\partial _{a} x ^{\mu} \partial _{b} x _{\mu}} \quad .$$ (4)

Once committed to this unconventional definition of time, the quantum amplitude for the transition from an initial vanishing configuration to a final non-vanishing string configuration after a lapse of areal time $A$, is provided by the kernel $G ( x (s) ; A )$ which satisfies the following diffusion-like equation, or imaginary area Schrödinger equation:

$$\frac{1}{2} \frac{\delta ^{2}}{\delta x ^{\mu} (s) \delta ... ...) ; A) = \frac{\partial}{\partial A} G (x (s) ; A) \quad .$$ (5)

Here, $x ^{\mu} (b) = x ^{\mu} (\tau (s) , \sigma(s))$ represents the physical string coordinate, i.e. the only space-like boundary of the world-sheet of area $A$. It may be worth emphasizing at this point that in quantum mechanics of point particles the ``time'' $t$ is not a measurable quantity but an arbitrary parameter, since there does not exist a self-adjoint quantum operator with eigenvalues $t$. Similarly, since there is no self-adjoint operator corresponding to the world-sheet area, $G ( x (s) ; A )$ turns out to be explicitly dependent on the arbitrary parameter $A$, and cannot have an immediate physical meaning. However, the Laplace transformed Green function is $A$-independent and corresponds to the Feynman propagator

$$G(x (s) ; M ^{2}) \equiv \int _{0} ^{\infty} d A G(x (s) ; A) \exp(- M ^{2} A / 2)$$

$$= - \frac{1}{2 (2 \pi) ^{3/2}} \int \frac{dA}{A ^{3/2}} \exp \left( - \frac{F}{2 A} - \frac{1}{2} M ^{2} A \right)$$ (6)

$$F = \frac{1}{4} \left( F ^{\mu \nu} \pm {}^{*} F ^{\mu \nu} \right) ^{2} \quad , \qquad F ^{\mu \nu} [C] = \int _{C} x ^{\mu} d x ^{\nu}$$ (7)

where $F$ stands for the self-dual (anti self-dual) area element.

Evidently, this approach is quite different from the ``normal mode quantization'' based on the Nambu-Goto action or the path-integral formulation a la' Polyakov, and our immediate purpose, in the next subsection, is to establish a connection with the conventional path-integral quantization of a relativistic string. Later, in Section 4, we speculate about a possible connection between our functional approach and a recently formulated matrix model of Type $IIB$ superstring.