2.1 The loop propagator and the loop Schrodinger equation

A useful starting point of our discussion is the exact form of the string quantum kernel discussed in Ref. [3], namely

$$K [ C , C _{0} ; A ] = \left( \frac{m ^{2}}{2 i \pi A} \r... ... _{\mu \nu} (C) - \sigma _{\mu \nu} (C _{0}) \right] \right\}$$ (1)

where m ² is defined in terms of the string tension, $m ^{2} = 1 / 2 \pi \alpha '$, $\sigma _{\mu\nu}(C)$ is the area element of the loop $C : \, x ^{\mu} = x ^{\mu} (s)$ [6], i.e.

$$\sigma ^{\mu \nu} (C) \equiv \oint _{C} x ^{\mu} d x^{\nu}$$ (2)

and A is the proper area of the string manifold, invariant under reparametrization of the world-sheet coordinates $\{ \xi ^{0} , \xi ^{1} \}$

$$A = \frac{1}{2} \epsilon _{mn} \int _{{\mathcal{S}}} d \xi ^{m} \wedge d \xi ^{n} \quad .$$ (3)

The geometric doublet ( $\sigma _{\mu \nu} , A$) represents the set of dynamical variables in our formulation of string dynamics. This choice makes it possible to develop a Hamilton-Jacobi theory of string loops which represents a natural extension of the familiar formulation of classical and quantum mechanics of point-particles [2], [3]. With hindsight, the analogy becomes transparent when one compares Eq. (1) with the amplitude for a relativistic particle of mass m to propagate from $x _{0} ^{\mu}$ to $x ^{\mu}$ in a proper-time lapse T:

$$K ( x , x _{0} ; T) = \left( \frac{m}{2 \pi T} \right) ^{2}... ...\frac{i m}{2 T} \vert x - x _{0} \vert ^{2} \right] \quad .$$ (4)

This comparison suggests the following correspondence between dynamical variables for particles and strings: a particle position in spacetime is labelled by four real numbers $x ^{\mu}$ which represent the projection of the particle position vector along the coordinate axes. In the string case, the conventional choice is to consider the position vector for each constituent point , and then follow their individual dynamical evolution in terms of the coordinate time x ⁰, or the proper time $\tau$. As a matter of fact, the canonical string quantization is usually implemented in the proper time gauge $x^0=\tau$. However, this choice explicitly breaks the reparametrization invariance of the theory, whereas in the Hamilton-Jacobi formulation of string dynamics, we have insisted that reparametrization invariance be manifest at every stage. The form (1) of the string propagator reflects that requirement. We shall call $\sigma (C)$ the string configuration tensor which plays the same role as the position vector in the point-particle case. Indeed, the six components of $\sigma (C)$ represent the projection of the loop area onto the coordinate planes in spacetime. Likewise, the reparametrization invariant evolution parameter for the string turns out to be neither the coordinate time, nor the proper time of the constituent points, but the proper area A of the whole string manifold. As a matter of fact, the string world-sheet is the spacetime image, through the embedding $x ^{\mu} = x ^{\mu} (\sigma ^{0} , \sigma ^{1})$, of a two dimensional manifold of coordinates $(\sigma ^{0} , \sigma ^{1})$. Thus, just as the proper time $\tau$ is a measure of the timelike distance between the final and initial position of a point particle, the proper area A is a measure of the timelike, or parametric distance between C and C ₀, i.e., the final and the initial configuration of the string. This idea was originally proposed by Eguchi [5].
It may seem less clear which interior area must be assigned to any given loop, as one can imagine infinite different surfaces having the loop as a unique boundary. However, once we accept the idea to look at the area of a surface S _C as a sort of time label for its boundary $\partial S _{C}$, then the arbitrariness in the assignment of S _C corresponds to the usual freedom to reset the initial instant of time for our clock. The important point is that, once the initial area has been chosen, the string clock measures area lapses. In summary, then, $\left[ \sigma ^{\mu \nu} (C) - \sigma ^{\mu \nu} (C _{0}) \right] ^{2}$ represents the ``spatial distance squared'' between C and C <sub>0</sub>, and A represents the classical time lapse for the string to change its shape from C <sub>0</sub> to C.
The quantum formulation of string-dynamics based on these non-canonical variables was undertaken in [3] with the evaluation of the string kernel and the derivation of the Schrödinger loop equation. Presently, we are interested in the quantum fluctuations of a loop. By this, we mean a shape-changing transition, and we would like to assign a probability amplitude to any such process. In order to do this, we make use of ``areal, or loop, derivatives'', as developed, for instance, by Migdal [6]. It may be useful to review briefly how loop derivatives work, since they are often confused with ordinary functional derivatives in view of the formal relation

$$\frac{\delta}{\delta x ^{\mu} (s)} = x ^{\prime \nu} (s) \frac{\delta}{\delta \sigma ^{\mu\nu} (s)} \quad .$$ (5)

However, there is a basic difference between these two types of operation. To begin with, an infinitesimal shape variation corresponds to ``cutting'' the loop C at a particular point, say y, and then joining the two end-points to an infinitesimal loop $\delta C _{y}$. Accordingly,

$$\delta \sigma ^{\mu \nu} (C ; y) = \oint _{\delta C _{y}} x ^{\mu} dx ^{\mu} \approx dy ^{\mu} \wedge dy ^{\nu}$$ (6)

where $dy ^{\mu} \wedge dy ^{\nu}$ is the elementary oriented area subtended by $\delta C _{y}$. A suggestive description of this procedure, due to Migdal [6], is that of adding a ``petal'' to the original loop. Then, we can speak of an ``intrinsic distance'' between the deformed and initial strings, as the infinitesimal, oriented area variation, $\delta \sigma ^{\mu \nu} (C ; y) \equiv dy ^{\mu} \wedge dy ^{\nu}$. ``Intrinsic'', here, means that the (spacelike) distance is invariant under reparametrization and/or embedding transformations. Evidently, there is no counterpart of this operation in the case of point-particles, because of the lack of spatial extension. Note that, while $\delta x ^{\mu} (s)$ represents a smooth deformation of the loop $x ^{\mu} = x ^{\mu} (s)$, the addition of a petal introduces a singular ``cusp'' at the contact point. Moreover, cusps produce infinities in the ordinary variational derivatives but not in the area derivatives [6].
Non-differentiability is the hallmark of fractal objects. Thus, anticipating one of our results, quantum loop fluctuations, interpreted as singular shape-changing transitions resulting from ``petal addition'', are responsible for the fractalization of the string. Evidently, in order to give substance to this idea, we must formulate the shape uncertainty principle for loops, and the centerpiece of this whole discussion becomes the loop wave functional $\Psi (C ; A)$, whose precise meaning we now wish to discuss.
Suppose the shape of the initial string is approximated by the loop configuration $C _{0} : \, x ^{\mu} = x ^{\mu} _{0} (s)$. The corresponding ``wave packet'' $\Psi(C _{0} ; 0)$ will be concentrated around C ₀. As the areal time increases, the initial string evolves, sweeping a world-sheet of parametric proper area A. Once $C _{0} : \, x ^{\mu} = x ^{\mu} _{0} (s)$ and A are assigned, the final string $x ^{\mu} = x ^{\mu} (s)$ can attain any of the different shapes compatible with the given initial condition and with the extension of the world-sheet. Each geometric configuration corresponds to a different ``point'' in loop space [3]. Then, $\Psi (C ; A)$ will represent the probability amplitude to find a string of shape $C : \, x ^{\mu} = x ^{\mu} (s)$ as the final boundary of the world-surface, of proper area A, originating from C ₀. From this vantage point, the quantum string evolution is a random shape-shifting process which corresponds, mathematically, to the spreading of the initial wave packet $\Psi(C _{0} ; 0)$ throughout loop space. The wavefunctional $\Psi (C ; A)$ can be obtained either by solving the loop Schrödinger equation

$$- \frac{1}{4 m ^{2}} \left( \oint _{C} dl (s) \right) ^{-1}... ... (s)} = i \frac{\partial \Psi [C ; A]}{\partial A} \quad ,$$ (7)

where, $dl (s) \equiv ds \sqrt{x ^{\prime} (s) ^{2}}$ is the invariant element of string length, or by means of the amplitude (1), summing over all the initial string configurations. This amounts to integrate over all the allowed loop configurations $\sigma (C _{0})$:

 

$$\Psi [C ; A] \sum _{C _{0}} K [C , C _{0} ; A] \Psi [C _{0} ; 0]$$ =

$$\left( \frac{m ^{2}}{2 i \pi A} \right) ^{3/2} \int [{\mathcal{D}... ...\left( \sigma (C) - \sigma (C _{0}) \right) ^{2} \right] \Psi [C _{0} ; 0] \, .$$ = (8)

Equation (7), we recall, is the quantum transcription, through the Correspondence Principle,

  

$$\rightarrow - i \frac{\partial}{\partial A}$$ H (9)

$$P _{\mu \nu} (s) \rightarrow i \frac{\delta}{\delta \sigma ^{\mu \nu} (s)}$$ (10)

of the classical relation between the area-hamiltonian H and the loop momentum density $P _{\mu \nu} (s)$ [3]:

$$H (C) = \frac{1}{4 m ^{2}} \left( \oint _{C} dl (s) \right) ^{-1} \oint _{C} dl (s) P _{\mu \nu} (s) P ^{\mu \nu} (s)$$ (11)

Once again, we note the analogy between Eq.(11) and the familiar energy momentum relation for a point particle, H = p ² / 2 m. We shall comment on the ``non-relativistic'' form of Eq. (11) in the concluding section of this paper. Presently, we limit ourselves to note that the difference between the point-particle case and the string case, stems from the spatial extension of the loop,and is reflected in Eq. (11) by the averaging integral of the momentum squared along the loop itself. Equation (11) represents the total loop energy instead of the energy of a single constituent string bit. Just as the particle linear momentum gives the direction along which a particle moves and the rate of position change, so the loop momentum describes the deformation in the loopshape and the rate of shape change. The corresponding hamiltonian describes the energy variation as the loop area varies, irrespective of the actual point along the loop where the deformation takes place.
Accordingly, the hamiltonian (9) represents the generating operator of the loop area variations, and the momentum density (10) represents the generator of the deformations in the loop shape at the point $x ^{\mu} (s)$.
From the above discussion, we are led to conclude that:

1.

  deformations may occur randomly at any point on the loop;

2.

  the antisymmetry in the indices $\mu$, $\nu$ guarantees that $P _{\mu \nu} (s)$ generates orthogonal deformations only, i.e. $P _{\mu \nu} (s) x ^{\prime\ mu} (s) x^{\prime \nu} (s) \equiv 0$;

3.

  shape changes cost energy because of the string tension and the fact that, adding a small loop, or ``petal'', increases the total length of the string;

4.

  the energy balance condition is provided by equation (11) at the classical level and by equation (7) at the quantum level. In both cases the global energy variation per unit proper area is obtained by a loop average of the double deformation at single point.

1), 2), 3), 4) represent the distinctive features of the string quantum shape shifting phenomenon.