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Next: 5.2 Integrating the path Up: 5. Computing the kernel Previous: 5. Computing the kernel

   
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:

 \begin{displaymath}K _{0} [ x (s) , x _{0} (s) ; A]
=
\mathcal{N}
A ^{\alpha}...
...\left(
i I [ x (s) , x _{0} (s) ; A ]
/
\hbar
\right)
\ ,
\end{displaymath} (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,
  
$\displaystyle \frac{2 \alpha m ^{2}}{A}$ = $\displaystyle -
\left(
\int _{0} ^{1} ds
\sqrt{x ^{\, \prime \, 2}}
\right) ^{-...
...ime \, 2}}}
\frac{\delta ^{2} I}{\delta x ^{\mu} (s) \ \delta x _{\mu} (s)} \ ,$ (32)
$\displaystyle 2 m ^{2}
\frac{\partial I}{\partial A}$ = $\displaystyle -
\left(
\int _{0} ^{1} ds
\sqrt{x ^{\, \prime \, 2}}
\right) ^{-...
...}
\frac{\delta I}{\delta x ^{\mu} (s)}
\frac{\delta I}{\delta x _{\mu} (s)}
\ .$ (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

 \begin{displaymath}\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}
\ .
\end{displaymath} (34)

Then, from the above definition we obtain
  
$\displaystyle \frac{\delta \sigma ^{\mu \nu} [C]}{ \delta x ^{\alpha} (s)}$ = $\displaystyle \delta _{\alpha} {} ^{\mu} x ^{\, \prime \, \nu} (s)
-
\delta _{\alpha} {} ^{\nu} x ^{\, \prime \, \mu} (s)
\ ,$ (35)
$\displaystyle \frac{\delta ^{2} \sigma ^{\mu \nu} [C]}
{\delta x ^{\alpha} (s) \delta x ^{\beta} (u)}$ = $\displaystyle \left(
\delta _{\alpha} {}^{\mu} \delta _{\beta} {}^{\nu}
-
\delt...
...ha} {}^{\nu} \delta _{\beta} {}^{\mu}
\right)
\frac{d}{ds}
\delta ( s - u )
\ .$ (36)

Next, we introduce the trial solution
 
S cl.[ x (s) , x 0 (s) ; A] = $\displaystyle \frac{\beta}{4 A}
\left(
\sigma ^{\mu \nu} [C]
-
\sigma ^{\mu \nu...
...]
\right)
\left(
\sigma _{\mu \nu} [C]
-
\sigma _{\mu \nu} [C _{0}]
\right)
\ ,$  
  $\textstyle \equiv$ $\displaystyle \frac{\beta}{4 A}
\Sigma ^{\mu \nu} [ C - C _{0}]
\Sigma _{\mu \nu} [ C - C _{0}]$ (37)

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

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

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

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

Equations (33) and (32) now give

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

Finally, if we define the loop space Dirac delta function

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

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

The above equation, in turn, leads us to the following representation of the Nambu-Goto closed string propagator
 
    $\displaystyle \int _{x _{0} (s)} ^{x(s)} [D x ^{\mu} (\sigma)]
\exp
\left\{
-
\...
...} \sigma
\sqrt{-\frac{1}{2}
{\dot{x}}^{\mu \nu}
{\dot{x}}_{\mu \nu}}
\right\}
=$  
    $\displaystyle \qquad =
\int _{0} ^{\infty} \! \! \! \! \! \! d A \,
e ^{- i m ^...
... A}
\Sigma ^{\mu \nu} [ C - C _{0} ]
\Sigma _{\mu \nu} [ C - C _{0} ]
\right)
.$ (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.
next up previous
Next: 5.2 Integrating the path Up: 5. Computing the kernel Previous: 5. Computing the kernel

Stefano Ansoldi
Department of Theoretical Physics
University of Trieste
TRIESTE - ITALY