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

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

$$[{\mathcal{D}} \mu (\sigma)] \equiv [{\mathcal{D}} x (\sigma)] [{... ...D}} P (\sigma)] [{\mathcal{D}} \pi (\sigma)] [{\mathcal{D}} N (\sigma)] \quad ,$$ (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

$$S [X , P , \xi , \pi , N] = \frac{1}{2} \int _{X} P _{\mu \nu} dx ^{\mu} \wedge dx ^{\nu} + \frac{1}{2} \int _{\Xi} \pi _{ab} d \xi ^{a} \wedge d \xi ^{b} +$$

$$\qquad - \frac{1}{2} \int _{\Sigma} d ^{2} \sigma N ^{ab} (\sigma) \left[ \pi _{ab} - \epsilon _{ab} H (P) \right] \quad .$$ (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,

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

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

$$K [C , C _{0} ; A] = \int _{0} ^{\infty} d {\mathcal{E}} e ^{i {\mathcal{E}} A} \int _... ...al{D}} x (\sigma)] [{\mathcal{D}} P (\sigma)] [{\mathcal{D}} N (\sigma)] \times$$

$$\quad \times \exp \left\{ \frac{i}{2} \int _{X} P_{\mu\nu} dx ^{\mu} \wedge dx ^{\nu} + \right .$$

$$\qquad \left . - \frac{i}{2} \epsilon _{ab} \int _{\Sigma} d ^{5} \sigma N ^{ab} (\sigma) \left[ {\mathcal{E}} - H (P) \right] \right\}$$ (11)

$$\equiv 2 i m ^{2} \int _{0} ^{\infty} d{\mathcal{E}} e ^{i {\mathcal{E}} A} G [C , C _{0} ; {\mathcal{E}}]$$ (12)

$$G [C , C _{0} ; {\mathcal{E}}] = \int _{C _{0}} ^{C} [{\mathcal{D}} x] [{\mathcal{D}} N] \exp \lef... ...dot{x} ^{\mu \nu} \dot{x} _{\mu \nu} + N(\sigma) {\mathcal{E}} \right] \right\}$$

$$= \int _{C _{0}} ^{C} [{\mathcal{D}} x(\sigma)] [{\mathcal{D}} N(\sigma)] \exp \left\{ - i S _{Schild} [x , N] \right\} \quad .$$ (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

$$G[C , C _{0} ; {\mathcal{E}}] \simeq \int _{C _{9}} ^{C} ... ...{- \dot{x} ^{\mu \nu} \dot{x} _{\mu \nu}} \right\} \quad .$$ (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

$$K[T , C _{9} ; A] = \left( \frac{m ^{2}}{2 i \pi A} \right) ^{3/2} \exp \left[ \frac{... ... \sigma ^{\mu \nu} (C) - \sigma ^{\mu\nu} (C _{0}) \right) ^{2} \right] \quad ,$$

$$\sigma ^{\mu \nu} (C) \equiv \oint _{C} x ^{\mu} dx ^{\nu} \quad ,$$ (15)

we obtain through Eqs. (12), (13) a non-perturbative definition of the Nambu-Goto propagator (14):

$$G[C , C _{0} ; m ^{2}] = \frac{1}{2 i m ^{2}} \int _{0} ... ...nfty} dA e ^{- i m ^{2} A} K [C , C _{0} ; A] \quad ,$$ (16)

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