2.3 Gaussian Loop-wavepacket

The quantum state represented by Eq.(12) is completely de-localized in loop space, which means that all the string shapes are equally probable, or that the string has no definite shape at all. Even though the wave functional (22) is a solution of equation (7), it does not have an immediate physical interpretation: at most, it can be used to describe a flux in loop space rather than to describe a single physical object. Physically acceptable one-string states are obtained by a suitable superposition of ``elementary'' plane wave solutions. The quantum state closest to a classical string will be described by a Gaussian wave packet,

 

$$\Psi [C _{0} ; 0] = \left[ \frac{1}{2 \pi ( \Delta \sigma ) ^{2}}... ...eft( \frac{i}{2} \oint _{C _{0}} x ^{\mu} d x ^{\nu} P _{\mu \nu} \right) \cdot$$

$$\qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdot \exp \left... ...ma) ^{2}} \left( \oint _{C _{0}} x^{\mu} dx ^{\nu} \right) ^{2} \right] \quad ,$$ (23)

where $\Delta \sigma$ represents the width, or position uncertainty in loop space, corresponding to an uncertainty in the physical shape of the loop. By inserting Eq. (23) into Eq. (8), and integrating out $\sigma ^{\mu \nu} (C _{0})$, we find

 

$$\Psi [C ; A] = \left[ \frac{1}{2 \pi (\Delta \sigma) ^{2}} \right... .../4} \frac{1}{\left( 1 + i A / m ^{2} (\Delta \sigma) ^{2} \right) ^{3/2}} \cdot$$

$$\qquad \qquad \cdot \exp \left\{ \frac{1}{\left(1 + i A / m ^{2} ... ...ta \sigma) ^{2}} \sigma ^{\mu \nu} (C) \sigma _{\mu \nu} (C) \right . + \right.$$

$$\quad \qquad \qquad \qquad \left. \left . + \frac{i}{2} \oint _{C... ...\frac{i A}{4 m ^{2}} P _{\mu \nu} (C) P ^{\mu \nu} (C) \right] \right\} \quad .$$ (24)

The wave functional represented by equation (24) spreads throughout loop space in conformity with the laws of quantum mechanics. In particular, the center of the wave packet moves according to the stationary phase principle, i.e.

$$\sigma ^{\mu \nu} (C) - \frac{A}{m ^{2}} P ^{\mu \nu} (C) = 0 \quad ,$$ (25)

and the width broadens as A increases

$$\Delta \sigma (A) = \Delta \sigma \left( 1 + A ^{2} / 4 m ^{4} (\Delta \sigma) ^{4} \right) ^{1/2} \quad .$$ (26)

Thus, as discussed previously, A represents a measure of the ``time-like'' distance between the initial and final string loop. Then, $m ^{2} (\Delta \sigma) ^{2}$ represent the wavepacket mean life. As long as $A \ll m ^{2} (\Delta \sigma) ^{2}$, the wavepacket maintains its original width $\Delta \sigma$. However, as A increases with respect to $m ^{2} (\Delta \sigma) ^{2}$, the wavepacket becomes broader and the initial string shape decays in the background space. Notice that, for sharp initial wave packets, i.e., for $(\Delta \sigma) \ll 2 \pi \alpha '$, the shape shifting process is more ``rapid'' than for large wave packets. Hence, strings with a well defined initial shape will sink faster into the sea of quantum fluctuations than broadly defined string loops.