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Next: 4 Plücker coordinates and Up: 3 Eguchi's Areal Quantization Previous: 3.1 The original formulation


3.2 Quantum Mechanics in Loop Space

The Eguchi quantization program is essentially a sort of quantum mechanics formulated in a space of string loops, i.e., a space in which each point represents a possible geometrical configuration of a closed string. To establish a connection with the Nambu-Goto, or Polyakov path-integral, it is advantageous to start from the quantum kernel

    $\displaystyle K [C , C _{2} ; A]
=
\int _{C _{0}} ^{C}
\int _{\gamma _{0}} ^{\gamma} [{\mathcal{D}} \mu (\sigma)]
\exp \left( i S [x , \xi , p , \pi , N] \right)$ (8)
    $\displaystyle [{\mathcal{D}} \mu (\sigma)]
\equiv
[{\mathcal{D}} x (\sigma)]
[{...
...D}} P (\sigma)]
[{\mathcal{D}} \pi (\sigma)]
[{\mathcal{D}} N (\sigma)]
\quad ,$ (9)

where the histories connecting the initial string $C _{0}$ to the final one $C$ are weighted by the exponential of the reparametrized Schild action
$\displaystyle S [X , P , \xi , \pi , N]$ $\textstyle =$ $\displaystyle \frac{1}{2}
\int _{X}
  P _{\mu \nu}  
dx ^{\mu} \wedge dx ^{\nu}
+
\frac{1}{2}
\int _{\Xi}
  \pi _{ab}  
d \xi ^{a} \wedge d \xi ^{b}
+$  
    $\displaystyle \qquad -
\frac{1}{2}
\int _{\Sigma} d ^{2} \sigma
  N ^{ab} (\sigma)
\left[ \pi _{ab} - \epsilon _{ab} H (P) \right]
\quad .$ (10)

The ``dictionary'' used in the above equation is as follows:
  1. $P _{\mu \nu}$ is the momentum canonically conjugated to the world-sheet area element;
  2. $H (P) = P ^{\mu \nu} P_{\mu \nu} / 4 m ^{4} $ is the Schild Hamiltonian density, and
  3. $m ^{2} = 1 / 4 \pi \alpha '$ is the string tension.
Furthermore, the original world-sheet coordinates $\xi ^{a}$ have been promoted to the role of dynamical variables, i.e., they now represent fields $\xi ^{a} (\sigma)$ defined over the string manifold,

\begin{displaymath}
\xi ^{a} \longrightarrow \xi ^{a} (\sigma ^{m})
\quad , \qquad
m = 0 , 1
\end{displaymath}

and $\pi _{ab}$, the momentum conjugate to $\xi ^{a}$, has been introduced into the Hamiltonian form of the action. The relevant dynamical quantities in loop space are listed in Table 3.2.


Table: Loop Space functionals and boundary fields
   
$
H [C]
=
(4 m ^{2} l _{C}) ^{-1}
\oint _{C} d \mu (s)
P _{\mu \nu} (s)
P ^{\mu \nu} (s)
$ (Schild) Loop Hamiltonian
   
$
H (s)
=
(4 m{^2} l _{C}) ^ {-1}
T _{\mu \nu} (s)
P ^{\mu \nu} (s)
$ (Schild) String Hamiltonian
   
$
d \mu (s)
\equiv
\sqrt{x ^{\prime   2}(s)}
x ^{\prime   \mu}
\: , \:
x ^{\prime \mu}
\equiv
d x ^{\mu} / ds
$ loop invariant measure
   
$
l _{C}
\equiv
\oint _{C} d \mu(s)
$ loop proper length
   
$
P _{\mu \nu} (s)
=
m ^{2}
\epsilon ^{mn} \partial _{m} x _{\mu} \partial _{n} x _{\nu}
$ area momentum density
   
$
P _{\mu \nu} [C]
\equiv
l _{C} ^{-1}
\oint _{C} d \mu(s)
P_{\mu \nu} (s)
$ loop momentum
   

The enlargement of the canonical phase space endows the Schild action with the full reparametrization invariance under the transformation $\sigma ^{m} \longrightarrow \zeta ^{m} (\sigma)$, while preserving the polynomial structure in the dynamical variables, which is a necessary condition to solve the path integral. The regained reparametrization invariance forces the new (extended) hamiltonian to be weakly vanishing, i.e., $H (P) - \epsilon ^{ab} \pi _{ab} / 2 \approx 0$. The quantum implementation of this condition is carried out by means of the Lagrange multiplier $N ^{ab} (\sigma)$.

By integrating out $\pi ^{ab}$ and $\xi ^{a}$ one obtains

$\displaystyle K [C , C _{0} ; A]$ $\textstyle =$ $\displaystyle \int _{0} ^{\infty} d {\mathcal{E}}
e ^{i {\mathcal{E}} A}
\int _...
...al{D}} x (\sigma)]
[{\mathcal{D}} P (\sigma)]
[{\mathcal{D}} N (\sigma)]
\times$  
    $\displaystyle \quad \times
\exp
\left\{
\frac{i}{2}
\int _{X}  
P_{\mu\nu}  
dx ^{\mu} \wedge dx ^{\nu}
+
\right .$  
    $\displaystyle \qquad
\left .
-
\frac{i}{2}
\epsilon _{ab}
\int _{\Sigma} d ^{5} \sigma  
N ^{ab} (\sigma)
\left[ {\mathcal{E}} - H (P) \right]
\right\}$ (11)
  $\textstyle \equiv$ $\displaystyle 2 i m ^{2}
\int _{0} ^{\infty} d{\mathcal{E}}
e ^{i {\mathcal{E}} A}
G [C , C _{0} ; {\mathcal{E}}]$ (12)
$\displaystyle G [C , C _{0} ; {\mathcal{E}}]$ $\textstyle =$ $\displaystyle \int _{C _{0}} ^{C} [{\mathcal{D}} x] [{\mathcal{D}} N]
\exp
\lef...
...dot{x} ^{\mu \nu} \dot{x} _{\mu \nu}
+
N(\sigma) {\mathcal{E}}
\right]
\right\}$  
  $\textstyle =$ $\displaystyle \int _{C _{0}} ^{C} [{\mathcal{D}} x(\sigma)] [{\mathcal{D}} N(\sigma)]
\exp
\left\{
- i
S _{Schild} [x , N]
\right\}
\quad .$ (13)

The above expressions show the explicit relation between the fixed area string propagator $K [C , C _{0} ; A]$, and the fixed ``energy'' string propagator $G [C , C _{2} ; {\mathcal{E}}]$ without recourse to any ad hoc averaging prescription in order to eliminate the $A$ parameter dependence. Moreover, the saddle point value of the string propagator (13) is evaluated to be
\begin{displaymath}
G[C , C _{0} ; {\mathcal{E}}]
\simeq
\int _{C _{9}} ^{C}
...
...{- \dot{x} ^{\mu \nu} \dot{x} _{\mu \nu}}
\right\}
\quad .
\end{displaymath} (14)

Since ${\mathcal{E}}$ has dimension of inverse length square, in natural units, we can set the string tension equal to $m ^{2}$, and then (14) reproduces exactly the Nambu-Goto path integral. This result allows us to establish the following facts:
  1. Eguchi's approach corresponds to quantizing a string by keeping fixed the area of the string histories in the path integral, and then taking the average over the string tension values;
  2. the Nambu-Goto approach, on the other hand, corresponds to quantizing a string by keeping fixed the string tension and then taking the average over the world-sheet areas;
  3. the two quantization schemes are equivalent in the saddle point approximation.
Finally, since the Schild propagator $K [C , C _{0} ; A]$ can be computed exactly
$\displaystyle K[T , C _{9} ; A]$ $\textstyle =$ $\displaystyle \left(
\frac{m ^{2}}{2 i \pi A}
\right) ^{3/2}
\exp
\left[
\frac{...
...
\sigma ^{\mu \nu} (C) - \sigma ^{\mu\nu} (C _{0})
\right) ^{2}
\right]
\quad ,$  
$\displaystyle \sigma ^{\mu \nu} (C)$ $\textstyle \equiv$ $\displaystyle \oint _{C} x ^{\mu} dx ^{\nu}
\quad ,$ (15)

we obtain through Eqs. (12), (13) a non-perturbative definition of the Nambu-Goto propagator (14):
\begin{displaymath}
G[C , C _{0} ; m ^{2}]
=
\frac{1}{2 i m ^{2}}
\int _{0} ...
...nfty} dA  
e ^{- i m ^{2} A}  
K [C , C _{0} ; A]
\quad ,
\end{displaymath} (16)

where $K [C , C _{0} ; A]$ is given by Eq. (15).


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Next: 4 Plücker coordinates and Up: 3 Eguchi's Areal Quantization Previous: 3.1 The original formulation

Stefano Ansoldi