4. The string kernel wave equation

The purpose of this section is to show how to derive the functional wave equation for K [ x (s) , x ₀ (s) ; A] from the corresponding path integral. This equation describes how the string responds to a variation of the final boundary $\xi ^{a} = \xi ^{a} (s)$, just as the ordinary Schrödinger equation describes a particle reaction to a shift of the time interval end-point. As we have seen in section 2, the string ``natural'' evolution parameter is the area A of the string manifold, so that functional or area derivatives generate ``translations'' in loop space, or string deformations in Minkowski space. Thus, we expect the functional wave equation to be of order one in $\partial/\partial A$, and of order two in $\delta / \delta x ^{\mu} (s)$, or $\delta / \delta \sigma ^{\mu\nu} (s)$⁵.
The standard procedure to arrive at the kernel wave equation goes through a recurrence relation satisfied by the discretized version of the Jacobi path integral [14], [15]. However, such a construction is well defined only when the action is a polynomial in the dynamical variables. For a non-linear action such as the Nambu-Goto area functional, a lattice definition of the path integral is much less obvious. Moreover, the continuum functional wave equation is recovered through the highly non-trivial limit of vanishing lattice step [14], [15]. In any case, the whole procedure seems disconnected from the classical approach to string dynamics, whereas we would like to see a logical continuity between quantum and classical dynamics. Against this background, it seems useful to offer an alternative path integral derivation of the string functional wave equation which is deeply rooted in the Hamiltonian formulation of string dynamics discussed in section 2, and is basically derived from the same Jacobi variational principle which we have consistently adopted so far.
The kernel variation under infinitesimal deformations of the field variables is

$$\delta K [ x (s) , x _{0} (s) ; A] = \frac{i}{\hbar} \int ... ...(\sigma)] \, \delta S \exp \left( {i S / \hbar} \right) \ .$$ (25)

As usual, only boundary variations will contribute to equation (25) if we restrict the fields to vary within the family of classical solutions corresponding to a given initial string configuration. Then,

$$\delta S _{\mathrm{cl.}}[ C ; A ] = \oint _{C} p _{\mu \nu} \, \delta x ^{\mu} (s) \, d x ^{\nu} - E \, dA \ .$$ (26)

From equations (25) and (26), we obtain

  

$$\frac{\partial}{\partial A} K [ x (s) , x _{0} (s) ; A ] - \frac{i E}{\hbar} K [ x (s) , x _{0} (s) ; A] \ ,$$ = (27)

$$\frac{\delta}{\delta x ^{\mu} (s)} K [ x (s) , x _{0} (s) ; A ] \frac{i}{\hbar} \int _{x _{0} (s)} ^{x(s)} \int _{\xi _{0} (s)} ^... ...sigma)] p _{\mu \nu} x^{\, \prime \, \nu} \exp \left( {i S / \hbar} \right) \ .$$ = (28)

Then, by comparison of (27), (28) and (10), one obtains immediately the kernel wave equation

 

$$- \frac{\hbar ^{2}}{2 m ^{2}} \left( \int _{0} ^{1} ds \sqrt{x ^{... ...{2}} {\delta x ^{\mu} (s) \ \delta x _{\mu} (s)} K [ x (s) , x _{0} (s) ; A ] =$$

$$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = i \hbar \frac{\partial}{\partial A} K [ x (s) , x _{0} (s) ; A] .$$ (29)

Thus, K [ x (s) , x ₀ (s) ; A] can be determined either by solving the functional wave equation (29), or by evaluating the path integral (17).
Once equation (29) is given, it is straightforward to show that G[ C , C ₀ ; m ²] satisfies the following equation

$$\left[ - \hbar ^{2} \left( \int _{0} ^{1} \! \! \! ds \s... ...ight] G[ C , C _{0} ; m ^{2} ] = - \delta [ C - C _{0}] .$$ (30)

Therefore, G[ C , C ₀ ; m ²] can be identified with the Green function for the string.