5.1 Integrating the functional wave equation

It is possible to compute K [ x (s) , x ₀ (s) ; A] exactly in the ``free'' case because the Lagrangian corresponding to the Hamiltonian (2) is quadratic with respect to the generalized velocities $\dot{x} ^{\mu \nu}$. Previous experience with this class of Lagrangians suggests the following ansatz for the string quantum kernel:

$$K _{0} [ x (s) , x _{0} (s) ; A] = \mathcal{N} A ^{\alpha}... ...\left( i I [ x (s) , x _{0} (s) ; A ] / \hbar \right) \ ,$$ (31)

where $\mathcal{N}$ is a normalization constant, and $\alpha$ a real number. Substituting this ansatz into eq.(29) gives two independent equations for the amplitude and the phase respectively,

  

$$\frac{2 \alpha m ^{2}}{A} - \left( \int _{0} ^{1} ds \sqrt{x ^{\, \prime \, 2}} \right) ^{-... ...ime \, 2}}} \frac{\delta ^{2} I}{\delta x ^{\mu} (s) \ \delta x _{\mu} (s)} \ ,$$ = (32)

$$2 m ^{2} \frac{\partial I}{\partial A} - \left( \int _{0} ^{1} ds \sqrt{x ^{\, \prime \, 2}} \right) ^{-... ...} \frac{\delta I}{\delta x ^{\mu} (s)} \frac{\delta I}{\delta x _{\mu} (s)} \ .$$ = (33)

Comparing equations (33) and (11), we see that $I = S _{\mathrm{cl.}}[ x (s) , \, x _{0} (s) ; A]$ and (33) is just the classical Jacobi equation. Therefore, the main problem is to determine the form of S _cl. in the string case. We do so by analogy with the relativistic point-particle case, where S _cl. is a functional of the world-line length element. Accordingly, we first introduce the oriented surface element as a functional of the surface boundary C

$$\sigma ^{\mu \nu} [ C ] \equiv \oint _{C} x ^{\mu} d x ^{\n... ...int _{0} ^{1} d u \, x ^{\mu} (u) \frac{d x ^{\nu}}{d u} \ .$$ (34)

Then, from the above definition we obtain

  

$$\frac{\delta \sigma ^{\mu \nu} [C]}{ \delta x ^{\alpha} (s)} \delta _{\alpha} {} ^{\mu} x ^{\, \prime \, \nu} (s) - \delta _{\alpha} {} ^{\nu} x ^{\, \prime \, \mu} (s) \ ,$$ = (35)

$$\frac{\delta ^{2} \sigma ^{\mu \nu} [C]} {\delta x ^{\alpha} (s) \delta x ^{\beta} (u)} \left( \delta _{\alpha} {}^{\mu} \delta _{\beta} {}^{\nu} - \delt... ...ha} {}^{\nu} \delta _{\beta} {}^{\mu} \right) \frac{d}{ds} \delta ( s - u ) \ .$$ = (36)

Next, we introduce the trial solution

 

$$\frac{\beta}{4 A} \left( \sigma ^{\mu \nu} [C] - \sigma ^{\mu \nu... ...] \right) \left( \sigma _{\mu \nu} [C] - \sigma _{\mu \nu} [C _{0}] \right) \ ,$$ S _cl.[ x (s) , x ₀ (s) ; A] =

$$\equiv \frac{\beta}{4 A} \Sigma ^{\mu \nu} [ C - C _{0}] \Sigma _{\mu \nu} [ C - C _{0}]$$ (37)

where $\beta$ is a second parameter to be fixed by the equations (32), (33). By taking into account (35), (36), we find

$$\frac{\delta S _{\mathrm{cl.}}}{\delta x ^{\mu} (s)} = \fra... ... \Sigma _{\mu \nu} [ C - C _{0} ] x ^{\prime \, \nu} (s) \ .$$ (38)

Note that the dependence on the parameter s is only through the factor $x^{\prime\,\nu}(s)$. Then,

$$\frac{\delta ^{2} S _{\mathrm{cl.}}}{\delta x _{\mu} (s) \delta x ^{\mu}(s)} = \frac{3}{A} x ^{\, \prime \, 2} (s) \ .$$ (39)

Equations (33) and (32) now give

$$\beta = - \frac{2 \alpha}{3} m ^{2} \ , \qquad \alpha = -3/2 \ .$$ (40)

Finally, if we define the loop space Dirac delta function

$$\delta [ C - C _{0} ] \equiv \lim _{\epsilon \rightarrow 0}... ...\nu} [ C - C _{0} ] \Sigma _{\mu \nu} [ C - C _{0} ] \right)$$ (41)

then, the kernel normalization constant is fixed by the boundary condition (16), and we finally obtain the promised expression of the quantum kernel as an exact evaluation of the path integral

$$K [ x (s) , x _{0} (s) ; A ] = \left( \frac{m ^{2}}{2 i \p... ...[ C - C _{0} ] \Sigma _{\mu \nu} [ C - C _{0} ] \right) \ .$$ (42)

The above equation, in turn, leads us to the following representation of the Nambu-Goto closed string propagator

 

$$\int _{x _{0} (s)} ^{x(s)} [D x ^{\mu} (\sigma)] \exp \left\{ - \... ...} \sigma \sqrt{-\frac{1}{2} {\dot{x}}^{\mu \nu} {\dot{x}}_{\mu \nu}} \right\} =$$

$$\qquad = \int _{0} ^{\infty} \! \! \! \! \! \! d A \, e ^{- i m ^... ... A} \Sigma ^{\mu \nu} [ C - C _{0} ] \Sigma _{\mu \nu} [ C - C _{0} ] \right) .$$ (43)

Note that, since no approximation was used to obtain equation (43), the above representation can also be interpreted as a new definition of the Nambu-Goto path integral. This definition is based on the classical Jacobi formulation of string dynamics rather than on the customary discretization procedure.